Ion mobility

The ion mobility (nowadays mostly a symbol because it denotes the chemical potential ) is defined in physical chemistry as the migration speed of ions (see also: mobility ) in a solvent at a certain temperature, standardized to the electrical field strength . It is usually given in the unit centimeter per second (cm / s). ${\displaystyle \mu }$ ${\displaystyle v}$ ${\displaystyle \mu }$ ${\displaystyle E}$ ${\displaystyle u}$

definition

Ion mobility is now generally referred to as the speed of migration that ions exhibit in an electric field. ${\displaystyle u}$

${\displaystyle v={\frac {u}{E}}}$

with: the electric field strength in V / cm (voltage per electrode spacing in cm). ${\displaystyle E={\frac {U}{l}}}$ ${\displaystyle U}$ ${\displaystyle l}$

Physicists sometimes still use the symbols of classical physics:

${\displaystyle \mu ={\frac {v}{E}}}$

at the physical speed of the ion. ${\displaystyle v}$

The ion mobility thus indicates the standardized migration speed of ions of a certain type in water at a reference temperature to be named. Strictly speaking, the concentration should also be stated. The values ​​in the table always relate to (limit ion mobility in analogy to limit equivalent conductivity). The specific conductivity of electrolytes is related to it. While the migration speed increases with increasing field strength , the ion mobility always remains a constant (hence the normalization to ). ${\displaystyle E}$ ${\displaystyle c=0}$ ${\displaystyle u}$ ${\displaystyle E}$ ${\displaystyle v}$ ${\displaystyle E}$

The product of ion mobility and Faraday constant is the limit equivalent conductivity (limit conductivity). The product of ion mobility and electric field strength is the migration speed of the ion at ideal dilution. ${\displaystyle F}$ ${\displaystyle E}$

Note: Today, the ionic migration speed is usually named with a symbol and the ion mobility with a symbol . The unit of the migration speed is [cm / s] and that of the ion mobility is S · cm ² / (A · s). Ion mobility and migration speed have the same numerical values ( ) for the electric field strength (or ), depending on the units of ion mobility used. The units are unaffected and different. ${\displaystyle u}$ ${\displaystyle v}$ ${\displaystyle E=1\,\mathrm {V/cm} }$ ${\displaystyle E=1\,\mathrm {V/m} }$ ${\displaystyle \{u\}=\{v\}}$

The symbol today is usually the chemical potential in physical chemistry , so it should be used for ion mobility. ${\displaystyle \mu }$ ${\displaystyle v}$

Numerical values

${\displaystyle u}$here the migration speed of the ion is calculated for a field strength of 1 V / cm. It moves faster or slower at any other field strength. The ion mobility is an isothermal constant, independent of the field strength and not a speed. The temperature coefficient is the relative increase in ion mobility or migration speed or equivalent conductivity ( limit conductivity ) per Kelvin temperature increase, related (normalized) to a reference value (here the value for 18 ° C is used:) . For its definition see temperature dependence of the transfer number . The α values ​​apply equally to the temperature dependence of , and . ${\displaystyle v}$ ${\displaystyle \alpha }$ ${\displaystyle \alpha _{i,18\,^{\circ }\mathrm {C} }}$ ${\displaystyle v}$ ${\displaystyle u}$ ${\displaystyle \lambda }$

The cations are sorted in order of increasing valency in their main or sub-group. The anions are initially sorted according to their valence, then according to the atomic number of the central atom. The organic anions are sorted at the end of the table according to their valency, number of carbon atoms and number of hydrogen.

table

Temperature in ° C	ion	Symbol (formula)	${\displaystyle z}$	${\displaystyle v}$in S m 2 / (A s) (= m 2 / (V s))	${\displaystyle v}$in S cm 2 / (A s) (= cm 2 / (V s))	${\displaystyle u}$in cm / s (only applies to: E  = 1 V / cm)	Limiting conductivity (equivalent conductivity for c  = 0) in S · cm 2 / mol ${\displaystyle (1/z)\lambda _{i,25\ ^{\circ }\mathrm {C} ,\ \mathrm {unendlich} }}$	Temperature coefficient (based on the values ​​of 18 ° C) in K −1 ${\displaystyle \alpha _{i}^{18}}$	Temperature coefficient (extrapolated, based on the values ​​of 25 ° C) in K −1 ${\displaystyle \alpha _{i}^{25}}$
25th	Hydronium	H + or H 3 O +	+1	3.627e-7th	3.627e-3	3.627e-3	349.8	0.0159	0.0143
25th	lithium	Li +	+1	4th.04e-8th	4th.04e-4th	4th.04e-4th	38.69	0.0210	0.0173
25th	ammonium	NH 4 +	+1	7th.62e-8th	7th.62e-4th	7th.62e-4th	73.4	0.0187	0.0169
25th	Trimethylammonium cation	NH (CH 3 ) 3 +	+1	4th.89e-8th	4th.89e-4th	4th.89e-4th	47.2
25th	Tetramethylammonium cation	N (CH 3 ) 4 +	+1	4th.30the-8th	4th.30the-4th	4th.30the-4th	41.5
25th	Tetraethylammonium cation	N (C 2 H 5 ) 4 +	+1	3.45e-8th	3.45e-4th	3.45e-4th	33.3	0.0264	0.0223
25th	Tetrapropylammonium cation	N (C 3 H 7 ) 4 +	+1	2.38e-8th	2.38e-4th	2.38e-4th	23
25th	Hydroxylammonium (I) cation	[NH 3 (OH)] +	+1	?	?	?	?
25th	Hydrazinium (I) cation	N 2 H 5 +	+1	?	?	?	?
25th	sodium	Na +	+1	5.193e-8th	5.193e-4th	5.193e-4th	50.11	0.0198	0.0174
25th	potassium	K +	+1	7th.618e-8th	7th.618e-4th	7th.618e-4th	73.52	0.0187	0.0165
25th	Rubidium	Rb +	+1	7th.98e-8th	7th.98e-4th	7th.98e-4th	77	0.02306	0.0180
25th	Cesium	Cs +	+1	7th.98e-8th	7th.98e-4th	7th.98e-4th	77	0.0218	0.0170
25th	Vanadyl (I) cation (colorless vanadium (V) oxide cation) e.g. B. (VO 2 ) 2 SO 4	V V O 2 +	+1	?	?	?	?
25th	Copper (I)	Cu 1+	+1	?	?	?
25th	Silver (I)	Ag +	+1	6th.42e-8th	6th.42e-4th	6th.42e-4th	61.92	0.0209	0.0200
25th	Thallium (I)	TI +	+1	7th.74e-8th	7th.74e-4th	7th.74e-4th	74.7	0.02183	0.0189
25th	beryllium	Be 2+	+2	4th.66e-8th	4th.66e-4th	4th.66e-4th	45
25th	Dihydroxylammonium (II) cation e.g. B. as [NH 3 (OH) 2 ] 2+ SO4 2−	[NH 3 (OH) 2 ] 2+	+2	?	?	?	?
25th	Hydrazinium (II) cation	N 2 H 6 2+	+2	?	?	?	?
25th	magnesium	Mg 2+	+2	5.49e-8th	5.49e-4th	5.49e-4th	53.06	0.0217	0.0189
25th	Calcium	Ca 2+	+2	6th.17the-8th	6th.17the-4th	6th.17the-4th	59.5	0.0206	0.0180
25th	strontium	Sr 2+	+2	6th.163e-8th	6th.163e-4th	6th.163e-4th	59.46	0.02501	0.0216
25th	barium	Ba 2+	+2	6th.59e-8th	6th.59e-4th	6th.59e-4th	63.64	0.0223	0.0193
25th	Titanyl (II) cation (titanium IV oxide cation)	Ti IV O 2+	+2	?	?	?	?
25th	Vanadium (II) (purple)	V 2+	+2	?	?	?	?
25th	Vanadyl (II) cation (blue vanadium (IV) oxide cation) e.g. B. V IV O 2+ SO 4 2−	V IV O 2+	+2	?	?	?	?
25th	Manganese (II)	Mn 2+	+2	5.54e-8th	5.54e-4th	5.54e-4th	50	0.0308	0.0254
25th	Iron (II)	Fe 2+	+2	7th.05e-8th	7th.05e-4th	7th.05e-4th	53.5	0.0289	0.0240
25th	Cobalt (II)	Co 2+	+2	5.08e-8th	5.08e-4th	5.08e-4th	49	0.0127	0.0286
25th	Nickel (II)	Ni 2+	+2	5.6the-8th	5.6the-4th	5.6the-4th	49	0.0286	0.0238
25th	Copper (II)	Cu 2+	+2	5.87e-8th	5.87e-4th	5.87e-4th	54	0.0329	0.0268
25th	Zinc (II)	Zn 2+	+2	5.47e-8th	5.47e-4th	5.47e-4th	53	0.0176	0.0157
25th	Cadmium (II)	Cd 2+	+2	5.6the-8th	5.6the-4th	5.6the-4th	54	0.02819	0.0212
25th	Mercury (I)	[Hg I ] 2 2+ = [Hg I -Hg I ] 2+	real + 2 / + 1 statistically per atom!	7th.11e-8th	7th.11e-4th	7th.11e-4th	68.6
25th	Mercury (II)	Hg 2+	+2	6th.59e-8th	6th.59e-4th	6th.59e-4th	63.6
25th	Lead (II)	Pb 2+	+2	7th.25the-8th	7th.25the-4th	7th.25the-4th	65	0.0211	0.0184
25th	radium	Ra 2+	+2	6th.92e-8th	6th.92e-4th	6th.92e-4th	66.8	0.0257	0.0220
25th	Uranyl cation	[UO 2 ] 2+	+2	1.74e-8 (unsure)	1.74e-4 (unsure)	1.74e-4 (unsure)	16.8 (uncertain)
25th	aluminum	Al 3+	+3	6th.53e-8th	6th.53e-4th	6th.53e-4th	63	0.0821	0.0522
25th	Titanium (III)	Ti 3+	+3	?	?	?	?
25th	Vanadium (III) (green)	V 3+	+3	?	?	?	?
25th	Chrome (III)	Cr 3+	+3	6th.94e-8th	6th.94e-4th	6th.94e-4th	67
25th	Hexammine cobalt (III) cation	[Co III (NH 3 ) 6 ] 3+	+3	1.06e-7th	1.06e-3	1.06e-3	102
25th	Iron (III)	Fe 3+	+3	7th.05e-8th	7th.05e-4th	7th.05e-4th	68	0.0164	0.0147
25th	Lanthanum (III)	La 3+	+3	7th.22nde-8th	7th.22nde-4th	7th.22nde-4th	69.5
25th	Cerium (III)	Ce 3+	+3	7th.23e-8th	7th.23e-4th	7th.23e-4th	69.8
25th	Cerium (IV)	Ce 4+	+4	?	?	?	?
25th	Tetrafluoroborate (tetrafluoroboric acid ion)	[BF 4 ] -	−1	?	?	?	?
25th	Tricyano carban anion	(CN) 3 C -	−1	4th.81e-8th	4th.81e-4th	4th.81e-4th	46.4	0.0293	0.0243
25th	Dicyano- azanide (−1) -anion (also: dicyano-amide-anion)	(CN) 2 N -	−1	5.63e-8th	5.63e-4th	5.63e-4th	54.3	0.0240	0.0205
25th	Metaborate macromolecular meta-boric acid ion (dioxoborate)	[BO 2 ] n -	−1	?	?	?	?
25th	Bicarbonate	HCO 3 -	−1	4th.61e-8th	4th.61e-4th	4th.61e-4th	44.5
25th	Cyanide (cyanide)	CN -	−1	8th.08e-8th	8th.08e-4th	8th.08e-4th	82
25th	Cyanate (cyanate)	OCN -	−1	6th.70e-8th	6th.70e-4th	6th.70e-4th	64.6	0.02555	0.0217
25th	Thiocyanate (rhodanide, thiocyanate)	SCN -	−1	6th.84e-8th	6th.84e-4th	6th.84e-4th	66	0.0226	0.0195
25th	Carbaminat also carbamate (ion of amidocarbonic acid )	(NH 2 ) COO -	−1	?	?	?	?
25th	Azide	N 3 -	−1	7th.20the-8th	7th.20the-4th	7th.20the-4th	69.5
25th	nitrite	NO 2 -	−1	7th.5e-8th	7th.5e-4th	7th.5e-4th	72	0.0315	0.0258
25th	nitrate	NO 3 -	−1	7th.41e-8th	7th.41e-4th	7th.41e-4th	71.44	0.0219	0.0178
25th	Fulminate (fulmic acid ion, pop acid ion)	CNO -	−1	?	?	?	?
25th	hydroxide	OH -	−1	2.052e-7th	2.052e-3	2.052e-3	197.6	0.0206	0.0180
25th	Deuteroxide ( deuterated hydroxide ion)	OD -	−1	1.23e-7th	1.23e-3	1.23e-3	119
25th	Hydrogen peroxide	HO 2 -	−1	?	?	?
25th	fluoride	F -	−1	5.742e-8th	5.742e-4th	5.742e-4th	55	0.0255	0.0217
25th	Hydrogen difluoride	HF 2 -	−1	7th.8the-8th	7th.8the-4th	7th.8the-4th	75
25th	Trihydrogen orthosilicate	H 3 SiO 4 -	−1	?	?	?	?
25th	Dihydrogen phosphate	H 2 PO 4 -	−1	3.73e-8th	3.73e-4th	3.73e-4th	36	0.0408	0.0317
25th	Hexafluorophosphate (hexafluorophosphoric acid ion)	[P V F 6 ] -	−1	?	?	?	?
25th	Hexafluoroarsenate (hexafluoroarsenic V acid ion)	[As V F 6 ] -	−1	?	?	?	?
25th	Hexafluoroantimonate (hexafluoroantimonic acid ion)	[Sb V F 6 ] -	−1	?	?	?	?
25th	Hydrogen sulfide	HS -	−1	6th.7the-8th	6th.7the-4th	6th.7the-4th	65	0.02005	0.0176
25th	Hydrogen sulfite	HSO 3 -	−1	6th.01e-8th	6th.01e-4th	6th.01e-4th	58
25th	Hydrogen sulfate	HSO 4 -	−1	5.18the-8th	5.18the-4th	5.18the-4th	50
25th	chloride	Cl -	−1	7th.918e-8th	7th.918e-4th	7th.918e-4th	76.34	0.0225	0.0189
25th	Hypochlorite	ClO -	−1	?	?	?	?
25th	Chlorite	ClO 2 -	−1	5.39e-8th	5.39e-4th	5.39e-4th	52
25th	Chlorate	ClO 3 -	−1	6th.77e-8th	6th.77e-4th	6th.77e-4th	65.3	0.0243	0.0208
25th	Perchlorate	ClO 4 -	−1	7th.05e-8th	7th.05e-4th	7th.05e-4th	68.0	0.0215	0.0187
25th	Trioxovanadate-V (metavanadate-V)	V V O 3 -	−1	?	?	?	?
25th	Permanganate (violet-dark red manganate-VII)	Mn VII O 4 -	−1	6th.32e-8th	6th.32e-4th	6th.32e-4th	61	0.0216	0.0187
25th	Dicyanocuprate-I (also Dicyanocuprate-I)	[Cu I (CN) 2 ] -	−1	?	?	?	?
25th	Dihydrogen arsenate (dihydrogen arsenate-V)	H 2 As V O 4 -	−1	3.52e-8th	3.52e-4th	3.52e-4th	34
25th	Dicyanoargentate-I	[Ag I (CN) 2 ] -	−1	?	?	?	?
25th	Dichloroargentate-I	[Ag I Cl 2 ] -	−1	?	?	?	?
25th	Antimonite (Antimonate III) ion of antimony acid	Sb III O 2 -	−1	?	?	?	?
25th	Trioxoantimonate-V (meta-antimonate-V) ion of the metaantimony (V) acid HSbO 3	Sb V O 3 -	−1	?	?	?	?
25th	hydrous tetroxoantimonate-V (orthoantimonate-V) ion of ortho-antimonic acid H 3 SbO 4 The hydrogen contained is chemically bound water, not a hydrogen salt !	H 2 Sb V O 4 - = H 2 O * Sb V O 3 -	−1	3.2e-8th	3.2e-4th	3.2e-4th	31
25th	Hexahydroxoantimonate (Hexahydroxoantimonate-V) ion of antimonic acid	Sb V (OH) 6 -	−1	?	?	?	?
25th	bromide	Br -	−1	8th.09e-8th	8th.09e-4th	8th.09e-4th	78.3	0.0237	0.0203
25th	Tribromide	Br 3 -	−1	4th.46e-8th	4th.46e-4th	4th.46e-4th	43
25th	Hypobromite ion of hypobromous acid	BrO -	−1	?	?	?	?
25th	Bromite ion of the bromous acid	BrO 2 -	−1	?	?	?	?
25th	Bromate	BrO 3 -	−1	5.8the-8th	5.8the-4th	5.8the-4th	55.7	0.02041	0.0179
25th	Iodide	I -	−1	7th.96e-8th	7th.96e-4th	7th.96e-4th	76.8	0.0234	0.0200
25th	Triiodide	J 3 -	−1	?	?	?	?
25th	Iodate	IO 3 -	−1	4th.30the-8th	4th.30the-4th	4th.30the-4th	41.5	0.0275	0.0231
25th	Tetroxoperiodate (metaperiodate)	JO 4 -	−1	5.636e-8th	5.636e-4th	5.636e-4th	54.38	0.0157	0.0141
25th	water-containing pentoxoperiodate z. B. LiH 2 JO 5 (formerly: "Mesoperiodate", "Mesoperiodic acid H 3 JO 5 ")	H 2 JO 5 - = H 2 O · [JO 4 ] -	−1	?	?	?	?
25th	hydrous heptoxoperiodate z. B. LiH 6 JO 7 (formerly: "Paraperiodate", "Paraperiodic acid H 7 JO 7 ")	H 6 JO 7 - = H 2 O · [H 4 JO 6 ] -	−1	?	?	?	?
25th	Perrhenate (colorless rhenate VII)	Re VII O 4 -	−1	5.667e-8th	5.667e-4th	5.667e-4th	54.68	0.02513	0.0214
25th	Hydrogentellurate VI	H 5 Te VI O 6 - = Te VI O (OH) 5 -	−1	?	?	?	?
25th	Dicyanoaurate-I	[Au I (CN) 2 ] -	−1	?	?	?	?
25th	Tetrachloroaurate III ion of tetrachloroauric (III) acid	[Au III Cl 4 ] -	−1	?	?	?	?
25th	Tetraborate tetraboric acid ion	B 4 O 5 (OH) 4 2− (earlier incorrectly: B 4 O 7 2− )	−2	?	?	?	?
25th	Carbonate (carbonate)	CO 3 2−	−2	7th.15the-8th	7th.15the-4th	7th.15the-4th	74	0.0214	0.0186
25th	peroxide	O 2 2−	−2	?	?	?
25th	Metasilicate	SiO 3 2−	−2	?	?	?	?
25th	Dihydrogen orthosilicate	H 2 SiO 4 2−	−2	?	?	?	?
25th	Hexafluorosilicate (hexafluorosilicic acid ion)	[SiF 6 ] 2−	−2	?	?	?	?
25th	Hydrogen phosphate	HPO 4 2−	−2	5.9e-8th	5.9e-4th	5.9e-4th	57
25th	sulfide	S 2−	−2	5.6the-8th	5.6the-4th	5.6the-4th	54
25th	Sulfoxylate (ion of sulfoxylic acid)	S II O 2 2−	−2	?	?	?	?
25th	Sulfite ("sulfate IV")	S IV O 3 2−	−2	7th.46e-8th	7th.46e-4th	7th.46e-4th	72
25th	Sulfate ("sulfate VI")	S VI O 4 2−	−2	8th.29e-8th	8th.29e-4th	5.742e-4th	80	0.0240	0.0205
25th	Peroxomonosulfate (Peroxosulfat-VI)	S VI O 5 2−	−2	?	?	?	?
25th	disulfide	S 2 2−	−2	?	?	?	?
25th	Thiosulfite (ion of thiosulfurous acid)	S 2 O 2 2−	−2	?	?	?	?
25th	Thiosulfate (disulphur-II-acid ion)	S II 2 O 3 2−	−2	8th.8the-8th	8th.8the-4th	8th.8the-4th	84.9
25th	Dithionite (di-sulfur-III-acid-ion)	S III 2 O 4 2−	−2	6th.89e-8th	6th.89e-4th	6th.89e-4th	66.5
25th	Disulfite (ion of disulfurous acid, disulfuric acid) disulfate IV	S IV 2 O 5 2−	−2	?	?	?	?
25th	Dithionate (di-sulfur-V-acid-ion)	S V 2 O 6 2−	−2	?	?	?	?
25th	Disulfate (di-sulfur-VI-acid-one)	S VI 2 O 7 2−	−2	?	?	?	?
25th	Peroxodisulfate (peroxo-di-sulfuric acid-VI-ion)	S VI 2 O 8 2−	−2	?	?	?
25th	Trithionate (ion of trithionic acid, thio-disulfuric acid)	S 3 O 6 2−	−2	?	?	?	?
25th	Tetrathionate (ion of tetrathionic acid)	S 4 O 6 2−	−2	?	?	?	?
25th	Tetracyanozincate II	[Zn II (CN) 4 ] 2−	−2	?	?	?	?
25th	Chromate (mono-chromate-VI)	Cr VI O 4 2−	−2	8th.7the-8th	8th.7the-4th	8th.7the-4th	83	0.0238	0.0204
18!	Dichromate (di-chromate VI)	Cr VI 2 O 7 2−	−2	4th.66e-8th	4th.66e-4th	4th.66e-4th	at 18 ° C: 45
25th	Manganate (green manganate-VI)	Mn VI O 4 2−	−2	?	?	?	?
25th	Ferrate VI	[Fe VI O 4 ] 2−	−2	?	?	?	?
25th	Selenite (ion of selenious acid)	SeO 3 2−	−2	?	?	?	?
25th	Selenate (selenic acid ion)	SeO 4 2−	−2	7th.85e-8th	7th.85e-4th	7th.85e-4th	75.7	0.0235	0.0202
25th	Selenide	Se 2−	−2	?	?	?	?
25th	Molybdate (molybdate VI)	Mo VI O 4 2−	−2	7th.67e-8th	7th.67e-4th	7th.67e-4th	74.5
25th	Tetracyanocadmat-II	[Cd II (CN) 4 ] 2-	−2	?	?	?	?
25th	Stannate-IV (metastannate-IV) ion of tin-IV-acid	[Sn IV O 3 ] 2−	−2	?	?	?	?
25th	Telluride	Te 2−	−2	?	?	?	?
25th	water-containing hexoxoperiodate (water-containing orthoperiodate) no hydrogen salt , the hydrogen contained here is chemically bound water, e.g. B. in salt Na 2 H 3 IO 6	H 3 IO 6 2−	−2	?	?	?	?
25th	Wolframate (Wolframate VI)	W VI O 4 2−	−2	7th.19the-8th	7th.19the-4th	7th.19the-4th	69.4	0.02518	0.0214
25th	Hexachloroplatinate-IV ion of hexachloroplatinic (IV) acid	[Pt IV Cl 6 ] 2−	−2	?	?	?	?
25th	Tetraiodomercurate II	[Hg II I 4 ] 2−	−2	?	?	?	?
25th	Diuranate (Di-Uranate VII) e.g. B. as ammonium diuranate	[U VII 2 O 7 ] 2−	−2	?	?	?	?
25th	Orthoborate orthoboric acid ion	BO 3 3−	−3	?	?	?	?
25th	Hexafluoroaluminate	[AlF 6 ] 3−	−3	?	?	?	?
25th	Hydrogen orthosilicate	HSiO 4 3−	−3	?	?	?	?
25th	Phosphite	PO 3 3−	−3	?	?	?	?
25th	phosphate	PO 4 3−	−3	7th.25the-8th	7th.25the-4th	7th.25the-4th	70.0
25th	Tetroxovanadate ( Orthovanadate -V)	V V O 4 3−	−3	?	?	?	?
25th	Hexacyanoferrate (III) anion (anion of the red blood liquor salt)	[Fe III (CN) 6 ] −3	−3	1.047e-7th	1.047e-3	1.047e-3	101
25th	Arsenite (arsenate-III)	As III O 3 3−	−3	?	?	?	?
25th	Arsenate (Arsenat-V)	As V O 4 3−	−3	?	?	?	?
25th	Orthosilicate	SiO 4 4−	−4	?	?	?	?
25th	Divanadate (ion of divanadic acid)	V 2 O 7 4−	−4	?	?	?	?
25th	Hexacyanoferrate (II) anion (anion of the yellow blood liquor salt)	[Fe II (CN) 6 ] 4−	−4	1.150e-7th	1.150e-3	1.150e-3	111
25th	anhydrous hexoxoperiodate (anhydrous orthoperiodate) e.g. B. in salt Ba 5 (IO 6 ) 2	IO 6 5−	−5	?	?	?	?
25th	Formate (methanate) formic acid ion	CHO 2 - = HCOO -	−1	5.64e-8th	5.64e-4th	5.64e-4th	54.6	0.0196	0.0173
25th	Acetate (acetate, ethanate, ethanate) acetic acid ion	C 2 H 3 O 2 - = CH 3 COO -	−1	4th.24e-8th	4th.24e-4th	4th.24e-4th	40.9	0.0241	0.0206
25th	Monochloro acetate chloroacetic acid ion	C 2 H 2 O 2 Cl - = CH 2 ClCOO -	−1	4th.13e-8th	4th.13e-4th	4th.13e-4th	39.8
25th	Dichloroacetate dichloroacetic acid ion	C 2 HO 2 Cl 2 - = CHCl 2 COO -	−1	3.94e-8th	3.94e-4th	3.94e-4th	38
25th	Trichloroacetate Trichloroacetic acid ion	C 2 O 2 Cl 3 - = CCl 3 COO -	−1	3.63e-8th	3.63e-4th	3.63e-4th	35
25th	Cyano acetate cyanoacetic acid ion	C 3 H 2 O 2 N - = CH 2 (CN) COO -	−1	4th.33e-8th	4th.33e-4th	4th.33e-4th	41.8
25th	n-propionate (n-propanate) n- propanoic acid ion	C 3 H 5 O 2 - = C 2 H 5 COO -	−1	3.71e-8th	3.71e-4th	3.71e-4th	35.8
25th	Lactate ( lactic acid ion )	C 3 H 5 O 3 - = C 2 H 4 (OH) COO -	−1	?	?	?	?
25th	n-butyrate (n-butanate) n- butanoic acid ion	C 4 H 7 O 2 - = C 3 H 7 COO -	−1	3.38e-8th	3.38e-4th	3.38e-4th	32.6
25th	Picrate ( picric acid ion )	C 6 H 2 N 3 O 7 - = C 6 H 2 (NO 2 ) 3 O -	−1	3.15the-8th	3.15the-4th	3.15the-4th	31	0.0333	0.0270
25th	o-chloro-benzoate (o- chlorobenzoic acid ion )	C 7 H 4 ClO 2 - = C 6 H 4 ClCOO -	−1	3.16e-8th	3.16e-4th	3.16e-4th	30.5
25th	Benzoate ( benzoic acid ion )	C 7 H 5 O 2 - = C 6 H 5 COO -	−1	3.36e-8th	3.36e-4th	3.36e-4th	32.3
25th	o-nitro-benzoate (o- nitrobenzoic acid ion )	C 7 H 5 NO 4 - = C 6 H 4 (NO 2 ) COO -	−1	3.29e-8th	3.29e-4th	3.29e-4th	31.7
25th	3,5-dinitro-benzoate (3,5- dinitrobenzoic acid ion )	C 7 H 4 NO 6 - = C 6 H 4 (NO 2 ) 2 COO -	−1	2.97e-8th	2.97e-4th	2.97e-4th	28.7
25th	Salicylate ( salicylic acid ion )	C 7 H 5 O 3 - = C6H4 (OH) COO -	−1	3.6the-8th	3.6the-4th	3.6the-4th	35
25th	Ethylbenzene - p - sulfonate	C 8 H 8 O 3 S - = C 6 H 4 (C 2 H 5 ) SO 3 -	−1	3.04e-8th	3.04e-4th	3.04e-4th	29.3
25th	n- butylbenzene - p - sulfonate	C 10 H 13 SO 3 - = C 6 H 4 (C 4 H 9 ) SO 3 -	−1	2.65e-8th	2.65e-4th	2.65e-4th	25.6
25th	n- octyl - benzene - p - sulphonate	C 14 H 21 SO 3 - = C 6 H 4 (C 8 H 17 ) SO 3 -	−1	2.39e-8th	2.39e-4th	2.39e-4th	23.1
25th	Tetra-phenyl borate	C 24 H 16 B - = [B (C 6 H 4 ) 4 ] -	−1	?	?	?	?
25th	Oxalate (ethandate) Oxalic acid ion	C 2 O 4 2− = (COO - ) 2	−2	7th.69e-8th	7th.69e-4th	7th.69e-4th	72.5	0.0231	0.0199
25th	Malonate ( malonic acid ion )	C 3 H 2 O 4 2− = CH 2 (COO - ) 2	−2	6th.3e-8th	6th.3e-4th	6th.3e-4th	61
25th	Maleate ( maleic acid ion )	C 4 H 2 O 4 2− = (CH) 2 (COO - ) 2	−2	?	?	?	?
25th	Succinate ( succinic acid ion )	C 4 H 4 O 4 2− = (CH 2 ) 2 (COO - ) 2	−2	6th.2e-8th	6th.2e-4th	6th.2e-4th	60
25th	Malate ( malic acid ion )	C 4 H 4 O 5 2− = C 2 H 3 (OH) (COO - ) 2	−2	?	?	?	?
25th	Tartrate ( tartaric acid ion )	C 4 H 4 O 6 2− = (CHOH) 2 (COO - ) 2	−2	6th.6the-8th	6th.6the-4th	6th.6the-4th	64	0.0234	0.0200
25th	o-phthalate (1,2- phthalic acid ion )	C 8 H 4 O 4 2− = C 6 H 4 (COO - ) 2	−2	5.39e-8th	5.39e-4th	5.39e-4th	52
25th	Citrate (citrate) citric acid ion	C 6 H 5 O 7 3− = C 3 H 4 (OH) (COO - ) 3	−3	7th.41e-8th	7th.41e-4th	7th.41e-4th	71.5	0.02404	0.0206

The table is based on values ​​of the limit conductivities for 25 ° C from the book Tables for Chemistry (Hübschmann, 1991) and some values ​​from electrochemistry (Milazzo, 1952). The values ​​were calculated from this. The temperature coefficients mentioned by Huebschmann obviously refer to different (not mentioned temperatures), therefore the α-values ​​(for 18 ° C) were calculated from the conductivity values ​​of 18 ° C and 25 ° C and normalized to the value of 18 ° C. The values ​​of the temperature coefficient for 25 ° C were determined by extrapolation and are each somewhat smaller because they are normalized to 25 ° C. They are likely to be slightly less accurate than the alpha values ​​for 18 ° C. Further conductivity values ​​come from the books "Chemietechnik" (Bierwerth) and "Physical chemistry for technicians and engineers" (Näser / Lempe / Regen). The Faraday constant was set at 96485.3 for the conversions. Rommel calls the migration speed of the hydrogen ion 3.27 · 10 ⁻³  cm / s at E  = 1 V / cm, which corresponds to an ion mobility of v  = 3.27 · 10 ⁻³  cm² / Vs.

As you can see, complex ions have greater ion mobility and equivalent conductivities than “monoatomic” ions, so they migrate faster with the same field strength . ${\displaystyle E}$

The mobility of dissolved ions depends on their size, charge, the hydration shell and other interactions with the solvent. The ion mobility of common inorganic cations and anions is of the order of 5 · 10 ⁻⁸  m ² / (s · V).

Two exceptions are striking: the hydroxide ions OH ^- and the hydronium ions H ⁺ have a mobility that is four or seven times higher. This is due to the formation of hydrogen bonds and ion migration through the Grotthuss mechanism .

Notes on the various units of ion mobility and the reasons for their definition ${\displaystyle v}$

The relationship applies:

${\displaystyle 1\,\mathrm {\frac {m^{2}}{Vs}} =1\,\mathrm {\frac {Sm^{2}}{As}} =10000\,\mathrm {\frac {cm^{2}}{Vs}} =10000\,\mathrm {\frac {Scm^{2}}{As}} }$

At a field strength of 1 V / m or 1 V / cm, the ion mobility is identical in value to the speed of movement of the ion in the corresponding unit (m / s or cm / s). Because it applies: ${\displaystyle v}$ ${\displaystyle E}$

${\displaystyle v={\frac {u}{E}}={\frac {u\cdot l}{U}}}$

with the voltage and the electrode gap . ${\displaystyle U}$ ${\displaystyle l}$

The ion mobility is the migration speed of the ion normalized to the field strength . It is therefore an isothermal constant for each type of ion in a solvent, while the migration speed of the ion is proportional to the electric field strength . Ions with different ion mobility can therefore never migrate at the same speed in the same electric field. ${\displaystyle E}$ ${\displaystyle u(i)}$ ${\displaystyle i}$ ${\displaystyle E}$ ${\displaystyle v}$

Physicists apparently like to use the “big” unit, that is, units without unit prefixes , such as m ² / (V · s) which corresponds to S · m ² / (A · s). Technicians, on the other hand, like to calculate with practically relevant units or unit attachments such as S · cm ² / (A · s) (corresponds to cm ² / (V · s)), since the electrode areas and electrode spacings are often given in cm² and cm. In any case, the reference temperature must be specified, otherwise the values ​​are worthless (as is the case with the limit conductivities ). ${\displaystyle T}$

Notes on outdated symbols of the linked quantities

Today the following applies in the SI system:

• time ${\displaystyle t}$
• speed ${\displaystyle v}$
• Amount of substance ${\displaystyle n}$
• Equivalent conductivity , limiting conductivity of individual ions or (often with +/- index) ${\displaystyle \lambda }$ ${\displaystyle \Lambda }$
• molar conductivity of individual ions or (often with +/- index) ${\displaystyle \lambda _{m}}$ ${\displaystyle \Lambda _{m}}$

But technicians still use it as a symbol of speed. In electrochemistry, the symbol is the speed of migration of the ion. Physicists and technicians use symbols as an alternative (physical speed of the SI system). In electrochemistry, however, the migration speed normalized to the field strength is the so-called ion mobility . The ion mobility is an isothermal constant for each ion. The ion migration speed increases with the electric field strength (voltage per electrode spacing ), while the ion mobility is / remains constant. Today you should use n + and n- for the transfer number if possible, since the time is in the SI system. is also the valency (alternatively:) of an ion and in the SI system the amount of substance (mole). The small ( ) or large lambda ( ) today is the equivalent conductivity for the respective concentration (with index “c”) or for ideal dilution (limit conductivity, provided with index “infinite”). The large lambda is sometimes used as the sum of the equivalent conductivity of all ions. so the following also applies today: ${\displaystyle w}$ ${\displaystyle u}$ ${\displaystyle v}$ ${\displaystyle v}$ ${\displaystyle v}$ ${\displaystyle u}$ ${\displaystyle E}$ ${\displaystyle U}$ ${\displaystyle l}$ ${\displaystyle v}$ ${\displaystyle t}$ ${\displaystyle n}$ ${\displaystyle z}$ ${\displaystyle n}$ ${\displaystyle \lambda }$ ${\displaystyle \Lambda }$ ${\displaystyle \Lambda }$

• Transfer numbers n +, n-
• Charge number, valence n, e.g.
• Often the total equivalent conductivity of all ions is named with the large lambda (without index +/-) ${\displaystyle \Lambda }$

earlier:

• time ${\displaystyle t}$
• Speed (physics) or (technology) ${\displaystyle v}$ ${\displaystyle w}$
• Transfer number or ${\displaystyle n}$ ${\displaystyle t}$
• Ion mobility (also called "ion conductivity"): w, w +, w- or u (cation) and v (anion) ("u" and "v" at Milazzo in "Elektrochemie") as well as u for cation and anion
• Ion migration speed (also referred to as "migration ability"): w +, w- (in Milazzo in "Elektrochemie" p. 42 and in Keune in "chimica" p. 139) and v (as physical speed; Keune "chimica")
• Equivalent conductivity , limiting conductivity of individual ions or (often with +/- indice) ${\displaystyle \lambda }$ ${\displaystyle \Lambda }$
• Often the total equivalent conductivity of all ions is named with the large lambda (without index +/-) ${\displaystyle \Lambda }$
• molar conductivity μ

Because of these many changes and ambiguities, such an overview is urgently needed. In Milazzo's book, too, translations from Italian into German were incorrectly translated. Ion mobility and migration speed are often confused. Always note the units. At a field strength of 1 V / cm (or 1 V / m), the migration speeds of the ions with the ion mobilities in S · cm ² / (A · s) (or S · m ² / (A · s)) are numerically the same .

Relationship between ion mobility, migration speed and equivalent conductivity

The relationships between ion mobility in [Scm ² / As], migration speed in [cm / s] in the electric field in [V / cm] and the isothermal equivalent conductivity in [Scm ² / mol] of an ion (cation or anion), which is constant for every ion concentration are: ${\displaystyle v}$ ${\displaystyle u}$ ${\displaystyle E}$ ${\displaystyle \lambda }$

ionic migration speed ${\displaystyle u_{\pm }=v_{\pm }\cdot E={\frac {E\cdot \lambda _{\pm }}{F}}}$

Ion mobility ${\displaystyle v_{\pm }={\frac {u_{\pm }}{E}}={\frac {\lambda _{\pm }}{F}}}$

Electric field strength , Faraday constant . For E = 1 [V / cm] and are numerical values ​​of the same size (but still have different units). The limit equivalent conductivity (lambda-infinite) only applies to c = 0 mol / liter, approximately also below c = 0.01 mol / l. ${\displaystyle E}$ ${\displaystyle F}$ ${\displaystyle u}$ ${\displaystyle v}$

For practitioners, the ion mobility was often given in the unit [cm / s], ie the unit of the migration speed. This can only mean that the field strength was E 1 [V / cm]. In these cases, check whether another field strength has been mentioned. ${\displaystyle v}$ ${\displaystyle u}$

Concentration dependence of all three quantities

The linked parameters ion mobility, migration speed of the ion and equivalent conductivity (of the salt or ion) are concentration-dependent. With decreasing molar concentration or ionic strength, these quantities drift towards their maximum values ​​(limit values) for ideal dilution (c = 0). So the maximum equivalent conductivity (limit conductivity), the maximum migration speed and the maximum ion mobility.

Tabulated values ​​of the ion mobility apply to ideal dilution and are therefore proportional to the limit conductivity and maximum (normalized) migration speed at E = 1V / cm.

Really diluted solutions therefore have smaller (measurable) values ​​than the theoretical values ​​for ideal dilution calculated from limit conductivity values ​​or ion mobility values. All statements also apply to the molar conductivity .

Relationship to the molar conductivity

If the valence of an ion is multiplied by its equivalent conductivity / limit equivalent conductivity, one obtains the molar conductivity / molar limit conductivity of this ion i. The sum of the molar values ​​for cation and anion then form the molar conductivity / molar limit conductivity of this salt. ${\displaystyle \lambda _{m,i}}$ ${\displaystyle \Lambda _{m,{\text{Salz}}}}$

The molar conductivity (of an ion i) is the product of the ionic valency (number of charges), Faraday constant and ion mobility : ${\displaystyle z_{i}}$ ${\displaystyle F}$ ${\displaystyle v_{i}}$

${\displaystyle \lambda _{m,i}^{(c,T)}=z_{i}\cdot F\cdot v_{i}^{(c,T)}=n_{i,{\text{Salz}}}^{(c,T)}\cdot \Lambda _{m,{\text{Salz}}}^{(c,T)}}$

${\displaystyle n_{i}}$is here the transfer number of the ion i in the salt, i.e. the proportion that this ion makes up in the current transport in this solution. The sum of all transfer numbers of all ions participating in the current transport is always one!

The following applies to the salt:

${\displaystyle \Lambda _{m,{\text{Salz}}}^{(c,T)}=z\cdot F\cdot \left(v_{\text{Kation}}^{(c,T)}+v_{\text{Anion}}^{(c,T)}\right)=\lambda _{m,{\text{Kation}}}^{(c,T)}+\lambda _{m,{\text{Anion}}}^{(c,T)}=\Lambda _{m,{\text{Salz}}}^{(c,T)}\cdot \left(n_{\text{Kation, Salz}}^{(c,T)}+n_{\text{Anion, Salz}}^{(c,T)}\right)}$

${\displaystyle z}$is the charge exchange number in the latter equation. All specific values ​​are dependent on concentration and temperature.

Notes on the limit conductivities and conductivities at higher molar concentrations

The limit conductivity values ​​for equivalent conductivity or molar conductivity of the ion can be calculated directly from the tabulated values ​​of the ion mobility .

Limit conductivities are the values ​​that can be effectively determined in ideally diluted solutions (c = 0 mol / liter). Practically valid at concentrations below 0.001 mol / liter. At higher concentrations or ionic strengths , values ​​of the equivalent conductivity or molar conductivity are determined which are always lower than the limit conductivity values ​​that can be calculated from the ion mobility. In practice, equivalent conductivities or molar conductivities for higher molar concentrations can only be calculated approximately, according to the Debye-Hückel-Onsager law for the conductivity of ions . Accordingly, it generally applies to all equivalent conductivities:

${\displaystyle \Lambda _{\text{äqui}}(c)=\Lambda _{\text{äqui}}^{0}-(A_{1}\cdot \Lambda _{\text{äqui}}^{0}+A_{2})\cdot {\sqrt {I}}}$

It contains and constants that only depend on temperature, the dielectric constant of the solvent and the valencies of the ions. is the ionic strength (mean concentration weighted quadratically according to the valences). Division of this equation by the limit equivalent conductivity leads to the equation of the conductivity coefficient (see subchapter conductivity coefficient). ${\displaystyle A_{1}}$ ${\displaystyle A_{2}}$ ${\displaystyle \varepsilon }$ ${\displaystyle z}$ ${\displaystyle I}$

If you want to get exact values, you only have to try to determine the specific electrolytic conductivity by measuring the real concentrated solution and then calculate the total equivalent conductivity or molar total conductivity.

If a measured value of the equivalent conductivity or molar conductivity of the salt (i.e. the total conductivity) has been determined, the conductivity coefficient can be determined, which represents the quotient of conductivity at real concentration and the limit conductivity.

In order to determine that of the individual ions from the total conductivities, the transfer numbers (at the respective concentration!) Must be determined by measurement.

The individual conductivity (molar or equivalent) of the ion i is the product of the measured conversion number n of the cation / anion i of a salt and the total conductivity (molar or equivalent) of the salt at the molar concentration and temperature present. If the salt dissociates into three ions, three conversion numbers must be determined (or two and one can be calculated).

The conductivity coefficient f _λ

The conductivity coefficient f _λ (of a concentration) is defined as the quotient of the equivalent conductivity of an ion type at the molar concentration and the limit equivalent conductivity of this ion type in an ideally diluted solution (c = 0 mol / liter): ${\displaystyle c}$

${\displaystyle f_{\lambda }^{\text{(c)}}={\frac {\lambda _{\pm }^{\text{(c)}}}{\lambda _{\pm }^{\text{(c=0)}}}}}$

Empirically and theoretically according to Debye-Hückel-Onsager, the following relationship to the effective concentration / ionic strength I was found:

• ${\displaystyle f_{\lambda }=1-{K\cdot {\sqrt {I}}}}$

K is a constant and I is the ionic strength , an "average effective concentration" weighted quadratically according to the ionic valencies.

For 1-1-valent electrolytes, this equation could already be confirmed from Kohlrausch's square-root law of equivalent conductivity (by dividing by the limit equivalent conductivity).

In very weakly dissociated electrolytes, the conductivity coefficient largely corresponds to the degree of dissociation , since in solutions of these electrolytes the concentration of the dissociated portion of the electrolyte present as ions is very low, i.e. there is a low ion concentration, as in ideally diluted solutions. The mean activity coefficient of these ions in an almost ideally diluted solution should therefore be almost 1.

In strong electrolytes, one can assume that the conductivity coefficient is the product of the mean activity coefficient and the degree of dissociation .

See also:

• Conversion number # Dependence on the defined conductivity coefficient fλ and
• Transfer number # Example for application reference .

Importance of the conductivity coefficient

according to its definition equation (above):

• If you multiply the limit equivalent conductivity (limit conductivity) of an electrolyte by the actual conductivity coefficient (at a given temperature and concentration) of this "real concentrated" solution, you get the actual equivalent conductivity of this solution (at a given temperature and concentration).

This is important for higher concentrations, since at higher molar concentrations the equivalent conductivities actually to be expected can no longer be calculated with known models. The conductivity coefficient itself can only be calculated from real measured values ​​of real solutions. It is therefore not a model value, but a real value. For its calculation from the measured specific conductivity and the equivalent conductivity calculated from it, the limit equivalent conductivity (limit conductivity) is also required.

• If the conductivity coefficient is multiplied by 100%, the numerical value obtained indicates how many percent of the maximum ions that are present - assuming 100% dissociation in the real concentrated solution - participate in the conduction mechanism.

For ideal dilution, the conductivity coefficient therefore tends towards 1 (corresponds to 100% participation of all ions in the conduction mechanism).

Numerical values ​​of conductivity coefficients of various electrolytes at low and medium concentrations

The values ​​for 19 and 25 ° C in the table below were calculated from the equivalent conductivity values ​​given by Huebschmann for various molar concentrations by normalizing (dividing) the respective limit equivalent conductivity. The values ​​for 18 ° C were calculated from the equivalent conductivity values ​​in the small conductivity primer (Rommel, 1980). The equivalent concentrations used by Rommel were converted into molar concentrations for the divalent electrolytes sulfuric acid and copper sulfate by dividing by 2. Some missing limit conductivity values ​​were supplemented / calculated from other sources.

Conductivity coefficients of various electrolytes at low and medium concentrations

Ionic Value ratio	temperature in ° C	electrolyte	Tendency to dissociate	c  = 1 in mol / l	c  = 0.5 in mol / l	c  = 0.25 in mol / l	c  = 0.2 in mol / l	c  = 0.1 in mol / l	c  = 0.05 in mol / l	c  = 0.025 in mol / l	c  = 0.02 in mol / l	c  = 0.01 in mol / l	c  = 0.005 in mol / l	c  = 0.0025 in mol / l	c  = 0.002 in mol / l	c  = 0.001 in mol / l	c  = 0.0005 in mol / l	c  = 0.00025 in mol / l	c  = 0.0002 in mol / l	c  = 0.0001 in mol / l	c  = 0.00005 in mol / l	c  = 0 in mol / l	Limit equivalent conductivity of the electrolyte in S · cm 2 / mol ${\displaystyle \Lambda ^{0,(T)}}$
1: 1	18th	HCl	very strong	0.790	0.858	-	0.898	0.921	0.940	-	0.961-	0.969	0.979	-	0.984	0.987	0.990	-	-	-	-	1	381
1: 1	18th	KCl	strong	0.7556	0.7871	-	0.8301	0.8609	0.8900	-	0.9224	0.9408	0.9562	-	0.9708	0.9785	0.9846	-	0.9900	0.9923	-	1	130.1
1: 1	18th	NaCl	strong	0.682	0.742	-	0.805	0.844	0.878	-	0.914	0.9358	0.9523	-	0.9688	0.9771	0.9834	-	0.9890	0.9917	-	1	109.0
1: 1	18th	LiCl	strong	0.641	0.715	-	0.788	0.833	0.871	-	0.909	0.931	0.949	-	0.967	0.976	0.983	-	0.989	0.992	-	1	98.9
1: 1	18th	KNO 3	strong	0.636	0.705	-	0.780	0.8285	0.8688	-	0.9107	0.9344	0.9526	-	0.9692	0.9771	0.9834	-	0.9897	0.9921	-	1	126.5
1: 2	18th	H 2 SO 4	medium strength	-	0.516	0.535	-	0.558	0.587	0.660	-	0.746	0.803	0.860	-	0.915	0.941	-	-	-	-	1	383.5
2: 2	18th	CuSO 4	weak	-	0.224	-	-	0.328	0.381	0.445	-	0.543	0.623	0.704	-	0.799	0.857	0.90000	-	0.9383	0.9557	1	115
1: 1	19th	ENT 3	very strong	0.796	-	-	-	0.928	-	-	-	0.976	-	-	-	0.995	-	-	-	-	-	1	377
1: 1	19th	HCl	very strong	0.7894	-	-	-	0.9205	-	-	-	0.9704	-	-	-	0.9887	-	-	-	-	-	1	381.3
1: 2	19th	Na 2 CO 3	very strong	0.686	-	-	-	0.913	-	-	-	0.933	-	-	-	-	-	-	-	-	-	1	103.1
1: 1	19th	KOH	strong	0.774	-	-	-	0.896	-	-	-	0.959	-	-	-	0.984	-	-	-	-	-	1	237.7
1: 1	19th	AI	strong	0.7902	-	-	-	0.8696	-	-	-	0.9413	-	-	-	0.9779	-	-	-	-	-	1	131.1
1: 1	19th	KBr	strong	-	-	-	-	0.8632	-	-	-	0.9403	-	-	-	0.9781	-	-	-	-	-	1	132.3
1: 1	19th	KCl	strong	0.756	-	-	-	0.8615	-	-	-	0.9415	-	-	-	0.9792	-	-	-	-	-	1	130.0
1: 1	19th	NaCl	strong	0.6821	-	-	-	0.8440	-	-	-	0.9353	-	-	-	0.9771	-	-	-	-	-	1	109.0
1: 1	19th	KNO 3	strong	0.636	-	-	-	0.8285	-	-	-	0.9344	-	-	-	0.9779	-	-	-	-	-	1	126.5
1: 1	19th	AgNO 3	strong	0.439	-	-	-	0.814	-	-	-	0.9309	-	-	-	0.9771	-	-	-	-	-	1	115.8
1: 1	19th	H 3 C-COONa	medium strength	0.531	-	-	-	0.787	-	-	-	0.905	-	-	-	0.969	-	-	-	-	-	1	77.6
1: 2	19th	Na 2 SO 4	medium strength	-	-	-	-	0.704	-	-	-	0.888	-	-	-	0.9587	-	-	-	-	-	1	111.3
1: 2	19th	H 2 SO 4	medium strength	-	-	-	-	0.6064	-	-	-	0.803	-	-	-	0.938	-	-	-	-	-	1	384.7
2: 2	19th	MgSO 4	weak	0.255	-	-	-	0.438	-	-	-	0.671	-	-	-	0.879	-	-	-	-	-	1	113.5
2: 2	19th	ZnSO 4	weak	0.234	-	-	-	0.399	-	-	-	0.6410	-	-	-	0.8670	-	-	-	-	-	1	113.5
2: 2	19th	CuSO 4	weak	0.226	-	-	-	0.3846	-	-	-	0.6289	-	-	-	0.8640	-	-	-	-	-	1	114.0
(1: 1 as hypothetical NH 4 OH)	19th	NH 3 solution	very weak	0.0037	-	-	-	0.014	-	-	-	0.0397	-	-	-	0.116	-	-	-	-	-	1	242
1: 1	19th	H 3 C-COOH	very weak	0.00372	-	-	-	0.0132	-	-	-	0.0409	-	-	-	0.117	-	-	-	-	-	1	349.5
1: 1	25th	HCl	very strong	-	-	-	-	0.9177	0.9359	-	0.9549	0.9662	0.9751	-	-	0.9883	-	-	-	0.9913	-	1	426.4
1: 1	25th	KCl	strong	0.7452	-	-	-	0.8603	0.8897	-	0.9229	0.9424	0.9576	-	-	0.9803	-	-	-	0.9861	-	1	149.9
1: 1	25th	AgNO 3	strong	-	-	-	-	0.8181	0.8639	-	0.9101	0.9352	0.9535	-	-	0.9783	-	-	-	0.9847	-	1	133.4
1: 2	25th	Na 2 SO 4	medium strength	-	-	-	-	0.7608	0.7513	-	0.8208	0.8643	0.9005	-	-	0.9543	-	-	-	0.9665	-	1	130.1
2: 1	25th	MgCl 2	medium strength	-	-	-	-	0.7504	0.7966	-	0.8504	0.8852	0.9143	-	-	0.9591	-	-	-	0.9707	-	1	129.4
1: 1	25th	HCOOH	very weak	-	-	-	-	0.0410	0.0576	-	0.0598	0.124	0.171	-	-	0.3412	-	-	-	0.4322	-	1	404.4
1: 1	25th	H 3 C-COOH	very weak	-	-	-	-	0.0135	0.0186	-	0.0289	0.0409	0.0573	-	-	0.149	-	-	-	0.170	-	1	390.9

Since potassium chloride solution is used as a calibration standard for conductivity measurements, its values ​​are highlighted in bold. All electrolytes were  sorted in descending order according to the conductivity coefficient at the concentration c = 0.1 mol / l. The naming of the tendency to dissociate was arbitrary for this concentration:

• very strong: > 0.9 ${\displaystyle f_{\lambda }}$
• strong: 0.8 <<0.9 ${\displaystyle f_{\lambda }}$
• medium strength: 0.5 <<0.8 ${\displaystyle f_{\lambda }}$
• weak: 0.1 <<0.5 ${\displaystyle f_{\lambda }}$
• very weak: <0.1 ${\displaystyle f_{\lambda }}$

As you can see, 1: 1 electrolytes in particular tend to almost completely dissociate and the ions participate in the electrolytic conductivity. The more multivalent ions there are in the electrolyte, the lower the conductivity coefficients are compared to 1: 1 electrolytes at the same molar concentration. Therefore the ionic strength I was defined and inserted into the improved Kohlrausch law (improved square root law) instead of the molar concentration :

${\displaystyle \Lambda _{i}^{(c,I,T)}=\Lambda _{i}^{0,(T)}{-A\cdot {\sqrt {I}}}}$

If you divide this law by the total limit conductivity of the electrolyte (to the temperature T ) you get the formula already mentioned above for the calculation of conductivity coefficients of an electrolyte up to ionic strengths of a maximum of I  = 0.01 mol / l valid: ${\displaystyle \Lambda _{i}^{0}}$

${\displaystyle {f_{\lambda }}_{i}^{(c,I,T)}=1-{K\cdot {\sqrt {I}}}}$

K and A are constants of the models that must be determined. Here i is not a single ion, but the entire electrolyte.

For higher ionic strengths / concentrations, conductivity coefficients can only be determined from measured values ​​of the specific conductivity (in S / cm) of the solutions, as is done in the table above. Even with the very weak electrolytes acetic acid and ammonia solution, it is very clear that at 19 ° C and a molar concentration of only 1 mmol / l, only a little more than 11% of the hypothetically maximum ions present in the conduction process.

Numerical values ​​of specific conductivity, equivalent conductivity and conductivity coefficients of various electrolytes at high concentrations

At high molar concentrations, the conductivity that is actually to be expected can no longer be calculated from the ion mobility for ideal dilution. Then tabular values ​​of the specific conductivity or the conductivity coefficient are required.

An old lexicon of electrical engineering gives under the heading "conductivity" a table with values ​​of the specific conductivity in S / cm for some highly concentrated aqueous electrolytes at 18 ° C:

Specific conductivity in S / cm for some highly concentrated aqueous electrolytes at 18 ° C

w, concentration in Mass percentage	NaCl	ZnSO 4	CuSO 4	AgNO 3	KOH	HCL	ENT 3	H 2 SO 4
5	0.067	0.019	0.019	0.026	0.172	0.395	0.258	0.209
10	0.121	0.032	0.032	0.048	0.315	0.630	0.461	0.392
15th	0.164	0.042	0.042	0.068	0.425	0.745	0.613	0.543
20th	0.196	0.047	-	0.087	0.499	0.762	0.711	0.653
25th	0.214	0.048	-	0.106	0.540	0.723	0.770	0.717
30th	-	0.044	-	0.124	0.542	0.662	0.785	0.740
40	-	-	-	0.157	0.450	0.515	0.733	0.680
50	-	-	-	0.186	-	-	0.631	0.541
60	-	-	-	0.210	-	-	0.513	0.373
70	-	-	-	-	-	-	0.396	0.216
80	-	-	-	-	-	-	0.267	0.111

The values ​​in bold are absolute conductivity maxima of these electrolytes. On the basis of existing conductivity values ​​of the specific conductivity, equivalent conductivity values ​​and conductivity coefficients can be calculated from these values ​​if the molar concentrations of the solutions have been determined from books of tables and the limit equivalent conductivity values ​​are known. This is to be done in the following table for hydrochloric acid , nitric acid , sulfuric acid , potassium hydroxide and sodium chloride solution .

In the table below, the values ​​marked in bold are the original (measured) specific conductivity values ​​for the mass percentages given in the above data source. Since the substance concentrations from various tables could only be found for 20 ° C (in some cases interpolation), the conductivity values ​​were extrapolated to 20 ° C with the determined temperature coefficient k (for ideal dilution). The calculated conductivity values ​​and the conductivity coefficient can therefore contain relative errors of up to several percent. Conductivity values ​​/ equivalent conductivity values ​​/ limit conductivity values ​​derived / calculated from other data sources have been put in brackets.

solution	w, concentration in Mass percentage	${\displaystyle \kappa }$ in S / cm (at 18 ° C)	k in 1 / K (at 18 ° C)	${\displaystyle \kappa }$ in S / cm (at 20 ° C, extrapolated using k)	c in mol / l (at 20 ° C)	${\displaystyle c^{*}}$in mol / cm 3 (at 20 ° C)	z	${\displaystyle \Lambda _{c}}$in S cm 2 / mol (at 20 ° C)	${\displaystyle f_{\lambda }}$ (at 20 ° C)	${\displaystyle \kappa _{hyp}}$ in S / cm (at 20 ° C)
hydrochloric acid	0	0	≈0.017	0	0	0	1	(≈387.8)	1	0
	1	(0.0813 <x <0.0949)	≈0.017	(0.0842 <x <0.0982)	0.275	0.000275	1	(306 <x <357)	(0.789 <x <0.921)	0.107
	5	0.395	≈0.017	≈0.408	1.404	0.001404	1	≈291	≈0.750	0.544
	10	0.630	≈0.017	≈0.651	2.871	0.002871	1	≈227	≈0.585	1.11
	15th	0.745	≈0.017	≈0.770	4,414	0.004414	1	≈174	≈0.449	1.71
	20th	0.762	≈0.017	≈0.788	6.022	0.006022	1	≈131	≈0.338	2.33
	25th	0.723	≈0.017	≈0.748	7.706	0.007706	1	≈97.1	≈0.250	2.99
	30th	0.662	≈0.017	≈0.685	9.416	0.009416	1	≈72.7	≈0.187	3.65
	40	0.515	≈0.017	≈0.533	13.14	0.01314	1	≈40.6	≈0.105	5.10
nitric acid	0	0	≈0.0169	0	0	0	1	(≈383)	1	0
	1	(0.0503 <x <0.0538) but must be greater than 0.0516 (= 0.258 / 5)!	≈0.0169	(0.0521 <x <0.0557) must be greater than 0.0534 (= 0.267 / 5)!	0.159	0.000159	1	(328 <x <350)	(0.856 <x <0.914)	0.0609
	5	0.258	≈0.0169	≈0.267	0.814	0.000814	1	≈328	≈0.856	0.312
	10	0.461	≈0.0169	≈0.477	1.673	0.001673	1	≈285	≈0.744	0.641
	15th	0.613	≈0.0169	≈0.634	2,580	0.002580	1	≈246	≈0.642	0.988
	20th	0.711	≈0.0169	≈0.735	3,539	0.003539	1	≈208	≈0.543	1.36
	25th	0.770	≈0.0169	≈0.796	4,550	0.004550	1	≈175	≈0.457	1.74
	30th	0.785	≈0.0169	≈0.812	5.618	0.005618	1	≈145	≈0.379	2.15
	31	( 0.865 at 25 ° C) Maximum value!	≈0.0169	≈0.792	5.838	0.005838	1	(≈136)	(≈0.355)	2.23
	40	0.733	≈0.0169	≈0.758	7.911	0.007911	1	≈95.8	≈0.250	3.03
	50	0.631	≈0.0169	≈0.652	10.39	0.01039	1	≈62.8	≈0.164	3.98
	60	0.513	≈0.0169	≈0.530	13.01	0.01301	1	≈40.7	≈0.106	4.98
	70	0.396	≈0.0169	≈0.409	15.70	0.01570	1	≈26.1	≈0.0681	6.01
	80	0.267	≈0.0169	≈0.276	18.44	0.01844	1	≈15.0	≈0.0392	7.06
	90	?	≈0.0169	?	?	?	1	?	?	?
	100	( 0.015 at 25 ° C)	≈0.0169	(≈0.0137)	24.01	0.02401	1	(≈0.571)	(≈0.00149)	9.20
sulfuric acid	0	0	≈0.01534	0	0	0	2	(≈390.6)	1	0
	1	(≈0.0488)	≈0.01534	(≈0.0503)	0.103	0.000103	2	(≈244.0)	(≈0.6247)	0.0805
	5	0.209	≈0.01534	≈0.2154	0.526	0.000526	2	≈204.8	≈0.524	0.4109
	10	0.392	≈0.01534	≈0.4040	1.087	0.001087	2	≈185.8	≈0.476	0.8492
	15th	0.543	≈0.01534	≈0.5597	1.685	0.001685	2	≈166.1	≈0.425	1,316
	20th	0.653	≈0.01534	≈0.6730	2,324	0.002324	2	≈144.8	≈0.371	1,816
	25th	0.717	≈0.01534	≈0.7390	3.004	0.003004	2	≈123.0	≈0.315	2,347
	30th	0.740 maximum!	≈0.01534	≈0.7627	3.728	0.003728	2	≈102.3	≈0.262	2.912
	40	0.680	≈0.01534	≈0.7009	5.313	0.005313	2	≈65.96	≈0.169	4.151
	50	0.541	≈0.01534	≈0.5576	7.113	0.007113	2	≈39.20	≈0.100	5.557
	60	0.373	≈0.01534	≈0.3844	9.166	0.009166	2	≈20.97	≈0.0537	7.160
	70	0.216	≈0.01534	≈0.2226	11.49	0.01149	2	≈9.687	≈0.0248	8.976
	80	0.111	≈0.01534	≈0.1144	14.07	0.01407	2	≈ 4.065	≈0.0104	10.99
	90	?	≈0.01534	?	16.65	0.01665	2	?	?	13.01
	100	( 0.0826 at 25 ° C) 1)	≈0.01534	(≈0.0763)	18.68	0.01868	2	(≈2.04)	(≈0.00522)	14.59
Potassium hydroxide	0	0	≈0.0201	0	0	0	1	(≈242.5)	1	0
	1	(0.0333 <x <0.0382) but must be greater than 0.0344 (= 0.172 / 5)	≈0.0201	(0.0347 <x <0.0398) but must be greater than 0.0358 (= 0.179 / 5)	0.1796	0.0001796	1	(193 <x <222)	(0.79 <x <0.915)	0.04333
	5	0.172	≈0.0201	≈0.179	0.9274	0.0009274	1	≈193	≈0.796	0.2249
	10	0.315	≈0.0201	≈0.328	1.929	0.001929	1	≈170	≈0.701	0.4678
	15th	0.425	≈0.0201	≈0.442	3.013	0.003013	1	≈147	≈0.606	0.7307
	20th	0.499	≈0.0201	≈0.519	4.190	0.004190	1	≈124	≈0.511	1.016
	25th	0.540	≈0.0201	≈0.562	5.480	0.005480	1	≈103	≈0.425	1.329
	30th	0.542	≈0.0201	≈0.564	6.885	0.006885	1	≈81.9	≈0.338	1.670
	40	0.450	≈0.0201	≈0.468	10.06	0.01006	1	≈46.5	≈0.192	2,440
Sodium chloride solution	0	0	≈0.0214	0	0	0	1	(≈111.3)	1	0
	1	(0.0130 <x <0.0155) but must be greater than 0.0134 (= 0.067 / 5)!	≈0.0214	(0.0136 <x <0.0162) but must be greater than 0.0140 (= 0.070 / 5)!	0.172	0.000172	1	(79.0 <x <94.0)	(0.710 <x <0.845)	0.01914
	5	0.067	≈0.0214	≈0.070	0.886	0.000886	1	≈79.0	≈0.710	0.09861
	10	0.121	≈0.0214	≈0.126	1,832	0.001832	1	≈68.8	≈0.618	0.2039
	15th	0.164	≈0.0214	≈0.171	2.845	0.002845	1	≈60.1	≈0.540	0.3166
	20th	0.196	≈0.0214	≈0.204	3,926	0.003926	1	≈52.0	≈0.467	0.4369
	25th	0.214	≈0.0214	≈0.223	5.085	0.005085	1	≈43.9	≈0.394	0.5659
	26.498 (saturated at 20 °)	-	-	(0.2260)	≈5.447 (extrapolated)	≈0.005447 (extrapolated)	1	(≈41.49) (extrapolated)	(≈0.3728) (extrapolated)	0.6063
Silver nitrate solution	0	0	≈0.0214	0	0	0	1	(≈115.9)	1	0
	1	(≈0.0058)	≈0.0214	(≈0.0060)	0.05930	0.00005930	1	(≈101.7)	(≈0.877)	0.006873
	5	0.026	≈0.0214	≈0.027	0.3066	0.0003066	1	≈88.1	≈0.760	0.03553
	10	0.048	≈0.0214	≈0.050	0.6407	0.0006407	1	≈78.0	≈0.673	0.07426
	15th	0.068	≈0.0214	≈0.071	1.0056	0.0010056	1	≈70.6	≈0.609	0.1165
Copper sulfate solution	0	0	≈0.0276	0	0	0	2	(≈120.8)	1	0
	1	(x> 0.00539)	≈0.0276	(x> 0.0057)	0.063248	0.000063248	2	(x> 45.06)	(> 0.3730)	0.01528
	5	0.019	≈0.0276	≈0.020	0.32924	0.00032924	2	≈30.4	≈0.252	0.07954
	10	0.032	≈0.0276	≈0.034	0.69357	0.00069357	2	≈24.5	≈0.203	0.1676
	15th	0.042	≈0.0276	≈0.044	1.097 (extrapolated)	0.001097 (extrapolated)	2	≈20.0 (extrapolated)	≈0.166 (extrapolated)	0.2650
Zinc sulfate solution	0	0	≈0.0214	0	0	0	2	(≈115.9)	1	0
	1	(x> 0.0056)	≈0.0214	(x> 0.0059)	0.06253	0.00006253	2	(x> 47.2)	(> 0.407)	0.01449
	5	0.019	≈0.0214	≈0.020	0.3255	0.0003255	2	≈30.7	≈0.265	0.07545
	10	0.032	≈0.0214	≈0.033	0.6857	0.0006857	2	≈24.1	≈0.208	0.1589
	15th	0.042	≈0.0214	≈0.044	1.0848 (extrapolated)	0.0010848 (extrapolated)	2	≈20.3 (extrapolated)	≈0.175 (extrapolated)	0.2515

1) Another source (Bierwerth) gives the value 0.010 S / cm for the specific conductivity for 100% sulfuric acid at 25 ° C and 0.015 S / cm for pure nitric acid. This presumably applies to extremely anhydrous sulfuric acid (removal of the last ppm of water).

Since the specific conductivity is the product of the current ( concentration-dependent ) equivalent conductivity , the charge exchange number (valence) z and the molar concentration per cubic centimeter , the currently effective equivalent concentration can be calculated by dividing the measured specific conductivity through the molar concentration per cubic centimeter and the charge exchange number is shared. ${\displaystyle \kappa }$ ${\displaystyle \Lambda _{c}}$ ${\displaystyle c^{*}}$ ${\displaystyle c^{*}}$ ${\displaystyle z}$

for a constant temperature the following relationship applies:

${\displaystyle \kappa ^{(c)}=z\cdot c^{*}\cdot \Lambda _{c}^{(c)}=z\cdot c^{*}\cdot \Lambda _{0}\cdot f_{\lambda }^{(c)}}$

The products and are the equivalent concentration per liter and the equivalent concentration per cubic centimeter . ${\displaystyle z\cdot c=c_{\text{äqui}}}$ ${\displaystyle z\cdot c^{*}=c_{\text{äqui}}^{*}}$ ${\displaystyle c_{\text{äqui}}}$ ${\displaystyle c_{\text{äqui}}^{*}}$

${\displaystyle \kappa _{\text{hyp}}}$is the hypothetical conductivity value that would result if all particles / molecules were assumed to be 100% dissociated as ions and all would take part in the conductivity process as if they were ideally diluted. It is the product of limit equivalent conductivity (limit conductivity), charge exchange number and molar concentration per cubic centimeter . ${\displaystyle z}$ ${\displaystyle c^{*}}$

In the "extrapolation" from 18 ° C to 20 ° C of the concentrations and equivalent conductivities, it was not taken into account that the volume expands and the molar concentration drops slightly as a result. The actual equivalent concentrations would therefore have to be minimally higher than those calculated here.

used data sources:

• Chemistry table book for determining the substance concentrations from mass concentrations for 20 ° C.
• Tables for chemistry , for determining the temperature coefficients (with the calculated transfer figures for ideal dilution) and determining the limit conductivity values ​​for 20 ° C.
• Material data and characteristic values ​​of process engineering, for the determination of densities and substance concentrations for potassium hydroxide at 20 ° C.
• Chemical engineering table book , conductivity of anhydrous sulfuric acid and nitric acid and saturated sodium chloride solution at 25 ° C or 20 ° C.

Determination of equivalent conductivity, limiting conductivity and ion mobility of an ion type by measurements

If the values ​​for equivalent conductivity / limiting conductivity, ion mobility of an ion type are not known, they can be determined by measuring the specific conductivity at different concentrations and the desired temperature, if the transfer numbers of the cation and anion of the salt used have been determined by a Hittorf experiment (at same concentration and temperature!). The equivalent conductivity of an ion is the product of the conversion number of the ion (in the salt / for this salt) and the equivalent conductivity of the whole salt at this concentration and temperature. In order to determine the limit conductivity and the (limit) ion mobility, the measurements mentioned must be made at different concentrations in each case for concentrations graduated from a decade, for example 0.0001 mol / l, 0.001 mol / l, 0.01 mol / l, see:

• Application reference of the transfer number
• Concentration dependence of the number of transfers
• Kohlrausch's square root law of equivalent conductivity

The equivalent conductivity values ​​/ limit conductivity values ​​and the (limit) ion mobility values ​​can only be calculated from the measured values ​​of the specific conductivity and transfer numbers. They are values ​​of mathematical models that cannot be measured directly! This also applies to the molar (limit) conductivity.

The temperature dependence of the electrolytic conductivity

While metals show increasing resistance , i.e. decreasing conductivity , with increasing temperature, the conductivity of ionic aqueous electrolyte solutions (acids, bases, salt solutions) increases with increasing temperature. This also applies to molten salts. This is explained by the decrease in the viscosity of the solvent (or the melt) and the increase in the associated ion mobility or equivalent conductivity or . There is an approximately linear increase in water of: ${\displaystyle v_{i}}$ ${\displaystyle \lambda _{i}}$ ${\displaystyle \lambda _{\mathrm {Salz} }}$

• Ion mobility , e.g. B. in S · cm² / (A · s) ${\displaystyle v_{i}}$
• molar conductivity , in S · cm² / mol ${\displaystyle \lambda _{m}}$
• Equivalent conductivity , in S · cm² / mol ${\displaystyle \lambda _{ev}}$
• Migration speed u of an ion (with the same field strength E), in cm / s
• specific conductivity , in S / cm ${\displaystyle \kappa }$

in the temperature range between 18 ° C and about 90 ° C in ideally diluted solutions. All five named - interlinked - variables can therefore be described with a (largely) constant coefficient with regard to their temperature dependence. For the total conductivity of the solution as specific conductivity or equivalent conductivity or molar conductivity, the following applies to the constant temperature coefficient of the entire solution with reference temperature : ${\displaystyle k_{\mathrm {Salz} }^{T0}}$ ${\displaystyle T_{0}}$

${\displaystyle k_{\mathrm {Salz} }^{T0}={\frac {\lambda _{\mathrm {ev,Salz} }^{(T2)}-\lambda _{\mathrm {ev,Salz} }^{(T0)}}{(T_{2}-T_{0})\cdot \lambda _{\mathrm {ev,Salz} }^{(T0)}}}={\frac {\kappa _{\mathrm {Salz} }^{(T2)}-\kappa _{\mathrm {Salz} }^{(T0)}}{(T_{2}-T_{0})\cdot \kappa _{\mathrm {Salz} }^{(T0)}}}={\frac {\lambda _{\mathrm {m,Salz} }^{(T2)}-\lambda _{\mathrm {m,Salz} }^{(T0)}}{(T_{2}-T_{0})\cdot \lambda _{\mathrm {m,Salz} }^{(T0)}}}}$

To calculate the k values, either specific conductivities in [S / cm] of the entire solution (of the salt) or molar conductivity or equivalent conductivities of the entire solution (of the salt) can be used! ${\displaystyle \kappa }$ ${\displaystyle \Lambda _{m}}$ ${\displaystyle \Lambda _{ev}}$

For each individual ion i the following applies analogously to the constant temperature coefficient of the reference temperature : ${\displaystyle \alpha _{i}^{T0}}$ ${\displaystyle T_{0}}$

${\displaystyle \alpha _{i}^{T0}={\frac {\lambda _{i}^{(T2)}-\lambda _{i}^{(T0)}}{(T_{2}-T_{0})\cdot \lambda _{i}^{(T0)}}}}$

The following then applies to the new conductivity of an ion i at a changed temperature T :

${\displaystyle \lambda _{i}^{(c,T)}=\lambda _{i}^{(c,T0)}\cdot (1+\alpha _{i}^{T0}\cdot (T-T_{0}))}$

${\displaystyle T_{0}}$is the reference temperature with the reference conductivity or in addition. ${\displaystyle \lambda ^{T0}}$ ${\displaystyle \kappa ^{T0}}$

For all of the salt (the solution):

${\displaystyle \lambda _{\mathrm {Salz} }^{(c,T)}=\lambda _{\mathrm {Kation} }^{(c,T0)}\cdot (1+\alpha _{\mathrm {Kation} }^{T0}\cdot (T-T_{0}))+\lambda _{\mathrm {Anion} }^{(c,T0)}\cdot (1+\alpha _{\mathrm {Anion} }^{T0}\cdot (T-T_{0}))=\lambda _{\mathrm {Salz} }^{(c,T0)}\cdot (1+k^{T0}\cdot (T-T_{0}))}$

or for the specific conductivity in S / cm:

${\displaystyle \kappa _{\mathrm {Salz} }^{(c,T)}=\kappa _{\mathrm {Salz} }^{(c,T0)}\cdot (1+k^{T0}\cdot (T-T_{0}))}$

The equivalent conductivities only ever apply to one concentration / equivalent concentration ! The alpha values ​​for an ion are by definition constants, i.e. independent of temperature. But they are -really- somewhat temperature-dependent and not completely independent of the concentration. For each temperature coefficient or k , the reference temperature must be given as an index, since it was calculated - in this model - - by normalization (division) to a reference conductivity (at the reference temperature), e.g. B. and for reference temperature 18 ° C. The constant k - of the entire solution - is composed proportionally with the conversion numbers of the ions in this salt from the alpha values ​​of cation and anion: ${\displaystyle \alpha _{i}}$ ${\displaystyle T_{0}}$ ${\displaystyle \alpha _{i}^{18}}$ ${\displaystyle k^{18}}$

${\displaystyle K^{(c,T0)}=n_{\mathrm {Kation} }^{(c,T0)}\cdot \alpha _{\mathrm {Kation} }^{T0}+n_{\mathrm {Anion} }^{(c,T0)}\cdot \alpha _{\mathrm {Anion} }^{T0}}$

Usually, the alpha values ​​are (approximately) regarded as independent of the concentration, in this case too the k-value of the solution is concentration-dependent, since the transfer numbers are concentration-dependent. The higher the reference temperature, the smaller the α values ​​of the ion or the k values ​​of the solution! They are always positive because conductivity increases with increasing temperature. The values ​​for different reference temperatures can be converted into one another (if the values ​​are assumed to be constants).

All of the formulas mentioned apply of course only to ideally diluted solutions and approximately ideally diluted solutions of strong electrolytes , which are therefore as completely dissociated as possible . Full validity would therefore at dissociation of 100% and activity coefficients of present. Usually up to a molar concentration of a maximum of 0.01 mol / l for monovalent electrolytes or up to 0.01 mol / l of the calculated ionic strength for polyvalent ions in the electrolyte. Strictly speaking, every ion in an ideally diluted solution also has slightly different alpha values ​​at different temperatures (weak exponential function). As a result, the k value of the solution is not entirely constant either. ${\displaystyle f_{a}=1}$

If you multiply the assumed constant k values ​​or alpha values ​​by 100%, you get the percentage increase in conductivity / ion mobility / migration speed per Kelvin temperature increase.

Further theory, tables and diagrams on alpha values ​​and k values ​​can be found in "The Little Conductivity Primer". An article on the consideration of the non-linear temperature profile in conductivity measurements using a polynomial model can be found in "Conductivity Primer, An Introduction to Conductometry".

Conversion of temperature coefficients to other reference temperatures

one defines two different reference temperatures and : ${\displaystyle T_{0}}$ ${\displaystyle T_{00}}$

${\displaystyle T_{00}=\delta +T_{0}}$

So one can derive the equation for converting the alpha values ​​from the defining equations of the temperature coefficients or by analogy with the k values:

${\displaystyle k^{T00}={\frac {{\tfrac {\lambda ^{T2}}{\lambda ^{T0}+\lambda ^{T0}\cdot k^{T0}\cdot \delta }}-1}{T_{2}-(T_{0}+\delta )}}={\frac {\lambda ^{T2}-\lambda ^{T00}}{\lambda ^{T00}\cdot (T_{2}-T_{00})}}={\frac {{\tfrac {\lambda ^{T2}}{\lambda ^{T00}}}-1}{T_{2}-T_{00}}}}$

${\displaystyle \delta }$is any positive or negative temperature difference here. is any temperature, but usually higher than both reference temperatures. One of them is conductivity ${\displaystyle T_{2}}$ ${\displaystyle T_{2}}$ ${\displaystyle \lambda ^{T2}}$

The temperature coefficients alpha of an ion, or k of a solution, each have smaller (positive) numerical values ​​if their reference temperature is higher.

Example: The limit equivalent conductivities for potassium ions are: 65 (18 ° C) and 73.5 (25 ° C) S · cm² / mol. The temperature difference is 7 K. The temperature coefficient for a reference temperature of 18 ° C is:

${\displaystyle \alpha _{K+}^{18}={\tfrac {(73,5-65)\,\mathrm {S\cdot cm^{2}/mol} }{7\,\mathrm {K} \cdot 65\,\mathrm {S\cdot cm^{2}/mol} }}=0,0187\,\mathrm {K^{-1}} }$.

For a further 7 K increase in temperature, i.e. 25 ° C + 7 K = 32 ° C, a conductivity of 73.5 S · cm² / mol + (73.5 S · cm² / mol - 65 S · cm² / mol) = 82 S · cm² / mol to be expected for constant alpha values, i.e. constant increase in absolute conductivity. The alpha value for 25 ° C reference temperature is therefore:

${\displaystyle \alpha _{K+}^{25}={\tfrac {(82-73,5)\,\mathrm {S\cdot cm^{2}/mol} }{7\,\mathrm {K} \cdot 73,5\,\mathrm {S\cdot cm^{2}/mol} }}=0,0165\,\mathrm {K^{-1}} }$.

This was the most logical way.

Sample with the formula:

${\displaystyle \alpha _{\text{Kaliumion}}^{25^{\circ }\mathrm {C} }={\frac {{\tfrac {82}{65+65\cdot \alpha ^{18\,^{\circ }\mathrm {C} }\cdot 7\,\mathrm {K} }}-1}{32\,^{\circ }\mathrm {C} -(18\,^{\circ }\mathrm {C} +7\,\mathrm {K} )}}=0{,}0165\,\mathrm {K^{-1}} }$.

The k value of a solution of weakly dissociated electrolytes

If an electrolyte is only slightly or very slightly dissociated, there is usually a strong dependence of the degree of dissociation on the temperature. The dissociation, i.e. the formation of ions, increases with the temperature in these cases (sometimes strongly). The conductivity coefficient therefore increases in the direction of 1.

With such electrolytes, the relative increase in conductivity / ion mobility / migration speed (with the same field strength E) (related to the temperature change) can be greater than the largest alpha value of cation or anion. In these cases, the k value of the solution can assume a significantly larger value than the alpha values ​​of the ions.

Näser names some k values ​​for typical types of electrolyte (without temperature information):

• strong acids k ≈ 0.016 K ⁻¹ (strong electrolyte)
• strong bases k ≈ 0.019 K ⁻¹ (strong electrolyte)
• Salt solutions k ≈ 0.022 K ⁻¹ (strong electrolyte)
• Water k = 0.058 K ⁻¹ (very weak electrolyte)

In very weak electrolytes, it is known that even minor impurities have a strong effect on the increase in conductivity. For natural inland, drinking or surface water, the value according to the "little conductivity guide" for water should be between 0.0191 (at 0 ° C) and 0.0217 (at 35 ° C). At 25 ° C: (The reference temperature was always 25 ° C!). However, these values ​​contradict the value given by Näser for water (without temperature information). ${\displaystyle k^{25}}$ ${\displaystyle k^{25}=0,0210}$

In weak and, above all, very weak electrolytes, the increase in conductivity can be made up proportionally from the increase in the degree of dissociation and the increase in conductivity due to the temperature coefficients of the ions, taking into account the transfer numbers of the ions.

Typical weak electrolytes are weak organic and inorganic acids such as carbonic acid , hydrocyanic acid , hypochlorous acid , acetic acid , longer-chain alkanoic acids , as well as ammonia solution and the salts of weak acids with weak bases.

The concentration dependence of the temperature coefficient

There is obviously a dependence of the temperature coefficient of the solution, i.e. the k-value, on the concentration of the electrolyte. For example, Rommel gives the following values ​​for several electrolytes at different concentrations (extract):

Concentration dependence of the temperature coefficient

Mass fraction in%	NaOH	H 2 SO 4	NH 4 NO 3	NaCl
5	0.0201	0.0121	0.0203	0.0217
10	0.0217	0.0128	0.0194	0.0214
15th	0.0249	0.0136	-	0.0212
20th	0.0299	0.0145	0.0179	0.0216
25th	0.0375	0.0154	-	0.0227
30th	0.0450	0.0162	0.0168	-
35	0.0550	0.0170	-	-
40	0.0648	0.0178	0.0160	-

The values ​​given in the original literature in% / K were converted by dividing by 100 into k values ​​in the unit K ⁻¹ . These are temperature coefficients for the reference temperature of 18 ° C (probably also at 18 ° C), i.e. values. ${\displaystyle k^{18}}$

As can be seen, the ammonium nitrate solution even has k values ​​that decrease with increasing concentration. If the k values ​​of the solution are concentration dependent, the alpha values ​​of the ions must also be concentration dependent.

Further information on the temperature coefficients

Since the increase in electrolytic conductivity - with increasing temperature - is actually a (weak) exponential function, all ionic electrolytes also have a somewhat larger temperature coefficient at a higher temperature (which is usually assumed to be approximately constant). The temperature coefficient at a fixed reference temperature increases exponentially with increasing measurement temperature (increasing exponential curve).

When changing the (mathematical) reference temperature of the temperature coefficient, however, the increase in the reference temperature always leads to a lower numerical value for the temperature coefficient, regardless of the current measurement temperature.

Ion mobility and electrolytic conductivity in superheated water

The conductivity of electrolyte solutions always increases as the temperature rises. For very high temperatures, it is difficult to find values ​​for equivalent conductivities or ion mobilities in specialist literature. The standard work by Milazzo ( Elektrochemie. 1952, Tab. 11, p. 48), on the other hand, provides some informative values ​​for several electrolytes:

electrolyte	Equivalent concentration in mol / l ${\displaystyle c_{ev}}$	18 ° C	50 ° C	75 ° C	100 ° C	128 ° C	156 ° C	218 ° C	281 ° C	306 ° C
KCl	0.08	113.5	-	-	341.5	-	498	638	723	720
AgNO 3	0.08	96.5	-	-	294	-	432	552	614	604
Ba (NO 3 ) 2	0.08	81.6	-	-	275.5	-	372	449	430	-
MgSO 4	0.08	52	-	-	136	-	133	-	75.2	-
H 2 SO 4	0.002	353.9	501.3	560.8	571.0	551	536	563 *	-	637

The equivalent conductivity values ​​for the various temperatures have the unit S · cm ² / mol. The equivalent concentration is the product of the charge exchange number (valence) and the molar concentration. For the divalent electrolytes sulfuric acid, magnesium sulfate and barium nitrate, the molar concentration is half of the specified equivalent concentration . The values ​​marked in bold are local or absolute maxima of the equivalent conductivity. The value marked with * is a second local maximum of sulfuric acid, which presumably results from its two dissociation stages (hydrogen sulfate ion!). The equivalent conductivities of potassium chloride and sulfuric acid increase from 18 ° C to 306 ° C by approx. 534% (KCl) and approx. 80% (sulfuric acid). The conversion figures at these temperatures are not known. However, it can be assumed that these cannot change fundamentally (compared to 18 ° C) and therefore the ion mobilities must have increased approximately at the same rate with increasing temperature as the equivalent conductivities.

At temperatures above 100 ° C, the aqueous electrolyte solution is of course under increased pressure ( vapor pressure of the water!).

Ion mobility and electrolytic conductivity of ions in non-aqueous solvents

Some salts, acids, bases (including organic ones) also dissolve in polar non-aqueous solvents, such as absolute ethanol , pyridine , benzene , dimethyl sulfate , dimethylformamide and many others , especially when the solvent has a high dielectric constant , i.e. is itself very polar. In these cases, ions are usually also present in the solution, so that it has electrical conductivity.

Walden's rule

1887 found Paul Walden The Walden rule out, after which the product of the limit equivalent conductance (limit conductivity) of an ion or electrolyte with the dynamic viscosity of the respective solvent is largely constant:

${\displaystyle \lambda _{Ion}^{(T)}\cdot \eta _{{\text{Lösemittel}}_{1}}^{(T)}=\lambda _{\text{Ion}}^{(T)}\cdot \eta _{{\text{Lösemittel}}_{2}}^{(T)}={\text{const.}}}$

For an ion. Or analogously for a (e.g. binary) electrolyte:

${\displaystyle \Lambda _{Salz}^{(T)}\cdot \eta _{{\text{Lösemittel}}_{1}}^{(T)}=\Lambda _{\text{Salz}}^{(T)}\cdot \eta _{{\text{Lösemittel}}_{2}}^{(T)}={\text{const.}}}$

Milazzo states that the above general Walden's rule turns into the following equation in some cases:

${\displaystyle \lambda _{\text{Ion}}^{(T)}\cdot (\eta _{{\text{Lösemittel}}_{1}}^{(T)})^{a}=\lambda _{\text{Ion}}^{(T)}\cdot (\eta _{{\text{Lösemittel}}_{2}}^{(T)})^{a}={\text{const.}}}$

It is valid for a constant exponent: . ${\displaystyle a}$ ${\displaystyle a<1}$

In general, Walden's rule means that with increasing temperature the dynamic viscosity of the solvent decreases (as with every liquid!) And therefore the associated equivalent conductivity / ion mobility in this solvent must increase. The increase in the limit conductivity with increasing temperature - in a solvent - is therefore already part of the Waldensian rule. The Waldensche rule should also apply to molten salt. However, it should only apply to different solvents if the ions are crystalline in size, i.e. only slightly hydrated in aqueous solution.

It was later found out that this rule also applies approximately to different temperatures in different solvents (generalized Walden's rule):

${\displaystyle \lambda _{Ion}^{(T1)}\cdot \eta _{{\text{Lösemittel}}_{1}}^{(T1)}=\lambda _{\text{Ion}}^{(T2)}\cdot \eta _{{\text{Lösemittel}}_{2}}^{(T2)}={\text{const.}}}$

${\displaystyle T_{2}}$and here are therefore different temperatures of the two solvents. ${\displaystyle T_{1}}$

If the dynamic viscosities of water and a suitable second solvent (for a salt or an ion) are known, the expected equivalent conductivity (at this temperature) in the new solvent (or the molten salt) can be calculated. ${\displaystyle \eta }$

Since the equivalent conductivity is the product of the ion mobility and the Faraday constant F, Walden's rule can be reformulated accordingly: ${\displaystyle \lambda _{i}}$ ${\displaystyle v_{i}}$

${\displaystyle (v_{i}^{(T)}\cdot F)\cdot \eta _{\text{Lösemittel}}^{(T)}={\text{const.}}}$

The Waldensche rule is dealt with extensively with numerical examples and tables at Milazzo.

Example numerical values ​​for Walden's rule

According to Milazzo, Walden found the rule when comparing the products of limit equivalent conductivity or dynamic viscosity of about 30 different solvents in which the salt tetraethylammonium iodide is soluble. In the following, “Walden products” for tetraethylammonium iodide / its cation at different temperatures in water and at constant temperature in different solvents are named in three tables. ${\displaystyle \lambda ^{0}}$ ${\displaystyle \Lambda ^{0}}$ ${\displaystyle \eta }$

Tetraethylammonium cation dissolved in water

	0 ° C	18 ° C	25 ° C	50 ° C	70 ° C	100 ° C
Limit equivalent conductivity in S · cm 2 / mol ${\displaystyle \lambda _{+}^{0}}$	16.2	28.1	33.3	53.4	-	103
Walden product ( ) in S · cm 2 · P / mol ${\displaystyle \lambda _{+}^{0}*\eta }$	0.290	0.296	0.298	0.294	-	0.293
Walden product ( ) in S · cm² · Pa · s / mol ${\displaystyle \lambda _{+}^{0}*\eta }$	0.0290	0.0296	0.0298	0.0294	-	0.0293

Tetraethylammonium iodide dissolved in benzonitrile

	0 ° C	18 ° C	25 ° C	50 ° C	70 ° C	100 ° C
Walden product ( ) in S · cm 2 · P / mol ${\displaystyle \Lambda ^{0}*\eta }$	0.65	-	0.66	0.66	0.63	-
Walden product ( ) in S · cm² · Pa · s / mol ${\displaystyle \Lambda ^{0}*\eta }$	0.065	-	0.066	0.066	0.063	-

Tetraethylammonium iodide dissolved in different solvents ("at constant temperature", all values ​​apparently at the same temperature, which was not mentioned; exceptions: water and phenol)

Temperature in ° C	Solvent	dynamic viscosity of the solvent in mPas ${\displaystyle \eta }$	Limit equivalent conductivity in S · cm 2 / mol ${\displaystyle \Lambda ^{0}}$	Walden product ( ) in S · cm 2 · P / mol ${\displaystyle \Lambda ^{0}*\eta }$	Walden product ( ) in S cm 2 Pa s / mol ${\displaystyle \Lambda ^{0}*\eta }$
oA / 18-25	water	oA / 1.002	oA / 94.2	0.981	0.0981 / 0.0944
oA / 18-25	Methanol	oA / 0.584	oA / 124.0	0.630	0.0630 / 0.0724
oA	Ethanol	oA	oA	0.586	0.0586
50	phenol	oA	oA	0.631	0.0631
oA / 18-25	acetone	oA / 0.325	oA / 225.0	0.662	0.0662 / 0.0731
oA	Methyl ethyl ketone (MEK)	oA	oA	0.620	0.0620
oA	Acetonitrile	oA	oA	0.643	0.0643
oA	Ethyl - cyanoacetic - ester	oA	oA	0.646	0.0646
oA	Benzonitrile	oA	oA	0.659	0.0659
oA	ortho - toluene - nitrile	oA	oA	0.650	0.0650
oA	Ethylene chloride	oA	oA	0.604	0.0604
oA	Nitromethane	oA	oA	0.685	0.0685
oA	Nitrobenzene	oA	oA	0.673	0.0673
oA	Pyridine	oA	oA	0.760	0.0760
Remarks: oA = not specified Values ​​separated by a slash come from a second source. The sources were the books "Elektrochemie" (Milazzo, Austria 1952, Springer-Verlag, pp. 49-50) and "Physical Chemistry for Technicians and Engineers" (Näser / Lempe / Regen, GDR, 1960, pp. 343-344).

As can be seen from the numerical value of the equivalent conductivity of tetraethylammonium iodide in the solvent acetone, the total conductivity (225.0) has increased by about 139% compared to aqueous solution (94.2; other sources give 110.1 for 25 ° C). If the conversion numbers in acetone were known, the limiting equivalent conductivity of the (tetraethylammonium cation and iodide anion in acetone and, ultimately, their ion mobility in acetone) could be calculated. In water, their conversion numbers at 25 ° C are 0.302 (tetraethylammonium cation) and 0.698 (iodide anion). The percentage increase in the total conductivity is divided like the transfer numbers in relation to the number 1. Under the (daring) assumption that the transfer numbers are identical in acetone as in water, limit conductivities of 33.3 S · cm² / mol x (1 + 139% / 100%) x 0.302 = 47.3 S x cm² / mol for the tetraethylammonium cation and 76.8 S x cm² / mol x (1 + 139% / 100%) x 0.698 = 151 .3 S · cm² / mol for the iodide anion in acetone. The ion mobility of the cation / anion under this assumption - in acetone - would be +139% · 0.302 = +42% (tetraethylammonium ion) and +139% · 0.698 = +97% (iodide ion) greater than the ion mobility in water.

However, a change in the conversion numbers of cation and anion is likely when the solvent is changed. Only the measurement of the transfer rates - in non-aqueous solvents - by a Hittorf experiment can bring clarity here.

Numerical values ​​of the specific conductivity and equivalent conductivity of molten salts

As in highly concentrated aqueous solutions, the conductivity of molten salts cannot be calculated from the ion mobility in water at ideal dilution. As an approximation, this is only possible using Wald's rule. Therefore tabular values ​​for nominal temperatures are necessary. A number of values ​​can be found in the book "Elektrochemie" (Milazzo, 1951, pp. 53-54, Table 15). Some values ​​are taken from the book "Tabular Book of Chemical Technology" (Bierwerth, 2005, p. 91). Some melting temperatures were taken from the "Brockhaus ABC Chemie" lexicon. Some melting temperatures are higher than the stated measuring temperature. This could be due to the fact that the melt was pressurized during the experiment , since low-melting salts often have high vapor pressures and therefore tend to evaporate or evaporate (see: Melting pressure ) as in the case of aluminum chloride . In these cases, the trial melting temperature may actually decrease compared to the atmospheric melting temperature.

Numerical values ​​of the specific conductivity and equivalent conductivity of molten salts

Measurement temperature in ° C ${\displaystyle T_{0}}$	Molten salt	specific conductivity in S / cm ${\displaystyle T_{0}}$	Temperature coefficient k in K −1	Melting temperature in ° C ${\displaystyle T_{S}}$
≈190 (at 2.5 atm )	AlCl 3	0.00000056	-	192.6 at air pressure
> 194	MoCl 5	0.0000018	-	194
194	NbCl 5	0.0000002	-	194
221	TaCl 5	0.0000003	-	221
230	AgNO 3	0.74	-	209
236	TeCl 4	0.12	0.092	224
250	WCl 5	0.67	0.034	248
263	SnCl 2	0.89	0.064	246
266	BiCl 3	0.44	0.032	230
> 275	WCl 6	0.0000019	-	275
294	HgCl 2	0.00052	0.0096	276
389	CdI 2	0.19	0.11	388
430	Cu 2 Cl 2 (bimolecular CuCl )	3.27	0.00749	422
450	CuCl (presumably bimolecular)	3.3	-	432
450	TICl	1.17	0.0299	430
451	BeCl 2	0.0032	81	440
460	ZnCl 2	0.051	0.29	313
508	PbCl 2	1.48	0.0311	501
529	Hg 2 Cl 2	1.0	0.018	525
550	AgCl	4.65	-	457.5
550	AgBr	3.00	-	434
570	UCl 4	0.34	0.082	589
571	CdBr 2	1.06	0.0189	567
576	CdCl 2	1.93	0.0104	568
594	InCl 3	0.42	0.21	586
600	AgCl	4.44	0.00414	457.5
600	AgBr	3.39	0.00501	434
600	AgI	2.17	0.00281	552
605	Cu 2 I 2 (bimolecular CuI )	1.82	0.00978	605
660	CsCl	1.14	0.0175	646
710	AI	1.35	0.0170	680
714	YCl 3	0.40	0.050	680
729	MgCl 2	1.05	0.0162	708
733	RbCl	1.49	0.0141	715
750	KBr	1.65	-	730
760	KBr	1.66	0.0120	730
780	LiCl	7.59	0.00132	613
795	CaCl 2	1.99	0.0176	772
800	KCl	2.19	0.00959	776
814	ThCl 4	0.67	0.027	765
850	NaCl	3.66	0.00601	801
860	Theatrical Version	4.14	0.0109	856
868	LaCl 3	1.14	0.0289	860
900	NaCl	3.77	-	801
900	KCl	2.40	-	776
900	SrCl 2	1.98	0.0146	873
905	LiF	20.3	0.0493	870
959	ScCl 3	0.56	0.050	939
> 962	BaCl 2	1.71	0.0175	962
1000	BaCl 2	2.05	-	962
1000	NaF	3.15	0.0263	992

For temperatures deviating from the measurement temperature, the conductivity to be expected can be calculated using the temperature coefficient k of the melt: ${\displaystyle T_{0}}$

${\displaystyle \kappa _{Salz}^{(T)}=\kappa _{Salz}^{(T_{0})}*(1+k^{(T_{0})}*(T-T_{0}))}$

The temperature coefficient usually applies over many hundreds of Kelvin . The exact values ​​from which it was calculated was not specified. The value mentioned here was calculated from the constant "b" mentioned by Milazzo for the measuring temperature T. For this purpose, "b" was divided by 100 and also divided (normalized) by "a" (specific conductivity at measuring temperature). The peculiarly high value for k for beryllium chloride probably results from a strong increase in the degree of dissociation even with a slight increase in temperature (weakly dissociated electrolyte!). Overall, most of the temperature coefficients k are in the known order of magnitude of ions in aqueous solution (≈0.02).

Some salts are even in the melt (mostly only up to just above the melting point) almost not at all dissociated in ions (molecular melt) and therefore have very low conductivity. For example aluminum chloride. At higher temperatures, the conductivity can then increase sharply in such cases or even rise suddenly.

The equivalent conductivity can be calculated if the density of the molten salt is known, or the molar volume (in cubic centimeters per mol) or the molar concentration (per cubic centimeter). Milazzo gives values ​​of the molar volume at the melting temperature for several salts. In addition, he gives the molar volume for many salts at the respective measuring temperature of the specific conductivity.

Equivalent conductivity of some salts at melting temperature (Milazzo, p. 54, Tab. 16)

Molten salt	Melting temperature in ° C ${\displaystyle T_{S}}$	Equivalent conductivity in S · cm 2 / mol at ${\displaystyle \Lambda }$ ${\displaystyle T_{S}}$	Molar mass M in g / mol	Molar volume of the melt in cm 3 / mol ${\displaystyle v_{M}}$	Density of the melt in g / cm 3 ${\displaystyle \rho _{Schmelze}}$	Density of the salt at 20 ° C in g / cm 3 ${\displaystyle \rho }$
LiCl	613	166	42.39	28.0 (613 ° C) / 28.3 (780 ° C)	1.51 / 1.50	2.068
NaCl	801	133.5	58.44	51.7 (801 ° C) / 37.7 (850 ° C)	1.13 / 1.55	2.17
KCl	776	103.5	74.55	61.2 (776 ° C) / 48.8 (800 ° C)	1.22 / 1.53	1.984
RbCl	715	78.2	120.92	70.5 (715 ° C) / 53.7 (733 ° C)	1.72 / 2.25	2.76
CsCl	646	66.7	186.36	77.5 (646 ° C) / 59.9 (660 ° C)	2.40 / 3.11	3.97
BeCl 2	440	0.086	79.92	53.8 / 52.7 (451 ° C)	1.49 / 1.52	1.901 (25 ° C)
MgCl 2	708	28.2	95.21	81.4 (708 ° C) / 56.6 (729 ° C)	1.17 / 1.68	2.32
CaCl 2	772	51.9	110.98	88.0 (772 ° C) / 60 (795 ° C)	1.26 / 1.85	2.15
SrCl 2	873	55.7	158.53	62.6 (873 ° C) / 58.7 (900 ° C)	2.53 / 2.70	3.05
BaCl 2	962	64.6	208.23	<75.6 (> 962 ° C) / 66.3 (approx. 1000 ° C)	2.35 / 2.68	3,913
AlCl 3	approx. 190 (at 2.5 atm )	0.000015	133.34	80 (approx. 190 ° C) / 101 (approx. 190 ° C)	1.67 / 1.32	2.44
ScCl 3	939	15th	151.31	80 (959 ° C) / 91 (959 ° C)	1.89 / 1.66	2.39
YCl 3	680	9.5	195.26	71 (714 ° C) / 77.5 (714 ° C)	2.75 / 2.52	2.81
LaCl 3	860	29.0	245.26	99.3 (860 ° C) / 77.8 (868 ° C)	2.47 / 3.15	3.842 (25 ° C)
ThCl 4	765	16	373.85	95 (814 ° C) / no details	3.93 / not specified	4.59

Notes on the table: The "double values " separated by a slash are firstly the value of the molar volume of the molten salt calculated from the equivalent conductivity given by Milazzo (for melting temperature) . Second, the value given directly by Milazzo (presumably valid for the measuring temperature) of the specific conductivity - see first table. For the calculation, the specific conductivity was converted from the measuring temperature to the melting temperature using the temperature coefficient k . With some salts this was not possible, here the specific conductivity at the measuring temperature was used directly. The double values ​​of the densities of the salt melts were calculated from both values ​​of the molar volumes. ${\displaystyle \Lambda }$ ${\displaystyle \kappa }$

Note: The stated and / or calculated values ​​(equivalent conductivity / molar volume / density of the melt) from the original sources definitely contain inaccuracies or considerable errors. The molar volume of a liquid / melt must always increase with increasing temperature. The density of the melt must decrease. Therefore, the calculated values ​​and those given in the original source are only correct in terms of magnitude and are to be understood as guide values. It is certainly difficult to precisely determine the density of a molten salt at high temperatures. In this respect, the densities of the melts mentioned here, the mentioned molar volumes of the melts and ultimately the mentioned equivalent conductivities of the salt melts are to be understood as guide values.

Calculation of the equivalent conductivity of molten salts

The sizes linked here are:

• the directly measurable specific conductivity in S / cm ${\displaystyle \kappa }$
• the calculable molar conductivity of the melt in S · cm ² / mol ${\displaystyle \Lambda _{m}}$
• the calculable equivalent conductivity of the melt in S · cm ² / mol ${\displaystyle \Lambda _{\text{äqui}}}$
• the molar volume of the melt in cm ³ / mol ${\displaystyle v_{m}^{*}}$
• the molar concentration (*: per cubic centimeter) of the melt in mol / cm ³ ${\displaystyle c^{*}}$
• Equivalent concentration (per cubic centimeter) of the melt in mol / cm ³ or equivalents / cm ³ ${\displaystyle c_{\text{äqui}}^{*}}$
• the equivalent volume of the melt in cm ³ / equivalent ${\displaystyle v_{\text{äqui}}^{*}}$
• the molar mass M of the salt in g / mol
• the density of the melt in g / cm ³ ${\displaystyle \rho _{\text{Schmelze}}^{*}}$
• Charge exchange number (effective valence of the ions, for barium chloride is for example 2) ${\displaystyle z}$

the following relationships now apply:

${\displaystyle {\frac {1}{c^{*}}}=v_{m}^{*}}$

${\displaystyle z\cdot c^{*}=c_{\text{äqui}}^{*}}$

${\displaystyle {\frac {1}{z\cdot c^{*}}}=v_{\text{äqui}}^{*}={\frac {v_{m}^{*}}{z}}={\frac {M}{z\cdot \rho _{\text{Schmelze}}}}}$

${\displaystyle \Lambda _{\text{m}}={\frac {\kappa \cdot M}{\rho _{\text{Schmelze}}}}}$

${\displaystyle \Lambda _{\text{äqui}}=\kappa \cdot v_{\text{äqui}}^{*}={\frac {\kappa \cdot v_{m}^{*}}{z}}={\frac {\kappa }{z\cdot c^{*}}}={\frac {\kappa \cdot M}{z\cdot \rho _{\text{Schmelze}}}}}$

and ultimately:

${\displaystyle \rho _{\text{Schmelze}}={\frac {\kappa \cdot M}{z\cdot \Lambda _{\text{äqui}}}}}$.

The values ​​of the molar conductivity and the equivalent conductivity are concentration-dependent and temperature-dependent!

If the transfer numbers of the ions in the melt could be determined by a Hittorf experiment , the equivalent conductivities of the cations and anions in the melt at the measuring temperature could be determined, and ultimately even the ion mobilities. In high-temperature melts, however, this will not be possible in practice.

Comparison of the specific conductivity and equivalent conductivity of solution and molten salt

The comparison is made here using the example of sodium chloride . It clearly shows the importance of equivalent conductivity and its relationship to specific conductivity. Although the temperature of the molten salts is much higher than that of the aqueous solutions, their equivalent conductivity is only slightly higher than that of the ideally diluted aqueous solution ( c = 0). This is due to the much higher dynamic viscosity of the molten salt compared to that of water at 19 ° C or 25 ° C. The ions have a greater mechanical resistance (frictional force) to overcome when diffusing through the melt under the influence of the electric field E (field strength).

Electrolyte NaCl	Temperature T in ° C	specific conductivity in S / cm ${\displaystyle \kappa }$	molar concentration c in mol / l	molar concentration in mol / cm 3 ${\displaystyle c^{*}}$	Equivalent conductivity in S · cm 2 / mol at ${\displaystyle \Lambda }$ ${\displaystyle T}$	Molar volume of the melt in cm 3 / mol ${\displaystyle v_{m}}$	Density of the melt in g / cm 3 ${\displaystyle \rho _{Schmelze}}$
Molten salt	900	3.77	?	?	> 142 expected	?	?
Molten salt	850	3.66	≈ 26.5 (uncertain)	≈ 0.0265 (uncertain)	≈ 138.1 (uncertain)	≈ 37.7 (uncertain)	≈ 1.55 (uncertain)
Molten salt	801	≈ 2.58 (quite safe)	≈ 19.3 (uncertain)	≈ 0.0193 (uncertain)	≈ 133.5 (uncertain)	≈ 51.7 (uncertain)	≈ 1.13 (uncertain)
aqueous solution ( ideally diluted )	25th	${\displaystyle 0^{*}}$	0	0	126.4 ( limit conductivity in water at 25 ° C)	-	-
aqueous solution	25th	0.0001237	0.001	0.000001	123.7	-	-
aqueous solution	25th	0.001185	0.01	0.00001	118.5	-	-
aqueous solution	25th	0.010067	0.1	0.0001	106.7	-	-
aqueous solution	25th	0.0858	1.0	0.001	85.8	-	-
aqueous solution ( ideally diluted )	19th	${\displaystyle 0^{*}}$	0	0	109.0 ( limit conductivity in water at 19 ° C)	-	-
aqueous solution	19th	0.0001065	0.001	0.000001	106.5	-	-
aqueous solution	19th	0.0010195	0.01	0.00001	101.95	-	-
aqueous solution	19th	0.00920	0.1	0.0001	92.0	-	-
aqueous solution	19th	0.07435	1.0	0.001	74.35	-	-
${\displaystyle *:}$The intrinsic conductivity of ultrapure water at room temperature is in the order of magnitude of> 50 nS / cm.

The same applies here to melt and solution as to any ionic electrolyte:

${\displaystyle \kappa ^{(c,T)}=z\cdot c^{*}\cdot \Lambda _{\text{äqui, c}}^{(T)}}$

The charge exchange number z is 1 for sodium chloride .

Note: The stated and / or calculated values ​​for melts (equivalent conductivity / molar volume / density) from the original sources definitely contain inaccuracies or considerable errors. The molar volume of a liquid / melt must always increase with increasing temperature. The density of the melt must decrease. Therefore, the calculated values ​​and those given in the original source are only correct in terms of magnitude and are to be understood as guide values. It is certainly difficult to precisely determine the density of a molten salt at high temperatures. In this respect, the densities of the melts mentioned here, the mentioned molar volumes of the melts and ultimately the mentioned equivalent conductivities of the salt melts are to be understood as guide values.

used data sources:

• Electrochemistry. 1951.
• Chemistry tables. 1991.
• Chemical engineering table book. 2005.

Notes on the migration speed of an ion

The migration rate and is dependent on the electric field strength E . Therefore, the last one must always be given. The associated value of the migration speed for the unit field strength E = 1 V / cm (or E = 1 V / m) is often given. In this special case, the numerical values ​​of migration speed u and ion mobility v are the same, but have different units. If the field strength for a migration speed is not stated, this can mean that the field strength was the unit field strength. To be on the safe side, compare the number mentioned with the numerical value of the ion mobility. They should be the same.

In general, the indication of the ion mobility is v preferable because they are not on the field strength E is dependent. The ion mobility is constant for a temperature (at a molar concentration).

In the models for the calculation, it is assumed that an ion migrates to the electrode on a direct (shortest) linear path. But this is not the case. An ion actually migrates steadily in the direction of the electrode, but treads a zigzag course (disordered movements). The path actually covered is therefore longer than the theoretical (shortest) path. The migration speeds calculated or measured by observation are therefore apparent migration speeds. The real migration speed must be greater, but cannot be determined.

The partial current J _{i introduced} by an ion i in the electric field E

Each type of ion i - which can be discharged at the electrode potentials at the electrodes (see decomposition potential ) - brings a current (partial current) J _i to the total current of the electrolyte in the electric field E. The current (partial current) J _i is composed of the product of Faraday constant F , field strength E (or electrical voltage U per electrode spacing l ), electrode surface A (in square centimeters), the molar ionic concentration (amount of substance of the ion i in relation to cubic centimeters), the charge number (charge exchange number of the ion) z (or n ) and the ion mobility v of the ion together:

${\displaystyle J_{i}=F\cdot E\cdot A\cdot c_{i}\cdot z_{i}\cdot v_{i}}$ With ${\displaystyle E={\frac {U}{l}}}$

The total current I _tot ( electrolysis current ) is made up of the partial currents J _{i of} all the ions i actually discharged. Which ions are discharged depends on the electrode potentials and the current density .

Nernst-Einstein relationship

The mobility v (or ) of ions in electric fields is described by the Einstein-Smoluchowski relationship in connection with diffusion coefficients from Fick's 1st law . This representation is sometimes referred to as the "Nernst-Einstein relationship": ${\displaystyle \mu }$

${\displaystyle \mu =v={\frac {q}{k\cdot T}}\cdot D}$

With

• Charge of the particles ${\displaystyle q}$
• the Boltzmann constant ${\displaystyle k}$
• the absolute temperature ${\displaystyle T}$
• Diffusion coefficient of the particles in the medium. ${\displaystyle D}$

Influence of ion mobility on the transfer number of an ion

→ Main article : Transfer number

The transfer number depends on the ion concentration (alternatively: on the molality ) and the ion mobility. If the concentration of the ion species is  high, a large proportion of the electrical current can be transported by these ions. Regarding ion mobility: Fast ions are able to transport a larger part of the current than slow ions. Hydronium ions and hydroxide ions can transport much more current than other ions because they use a special charge exchange mechanism ("extra conductivity"). In real terms, they migrate much more slowly than theoretically calculated, and largely only pass their charges on to neighboring ions of the same solvent water. The maximum value of this extra conductivity is reached at around 150 ° C in water. ${\displaystyle i}$

See also: Relationship between the transfer number and the diffusion coefficient of the ion

use

The different ion mobility is used in various electrophoresis methods to separate ionic substances in an electric field and e.g. B. separately feed a measurement.

The ion mobility in the gas phase plays an important role in analytical instruments such as the ion mobility spectrometers . Here the different drift speed of ions in an external electric field is used to achieve a separation of different analytes after their ionization .

literature

• John Eggert, L. Hock, G.-M. Schwab: Textbook of physical chemistry. 9th edition. S. Hirzel Verlag, Stuttgart 1968.
• Peter W. Atkins, Julio de Paula: Physical chemistry. 4th edition. Wiley-VCH, Weinheim 2006.
• Hübschmann, left: Tables on chemistry. Verlag Handwerk und Technik, Hamburg, 1991, ISBN 3-582-01234-4 , p. 62 (tables with limit conductivities of ions).
• Giulio Milazzo: Electrochemistry. Springer-Verlag, Vienna 1951 (extensive tables with limit conductivities, ion mobilities / migration speeds of ions).
• Temperature coefficient alpha . In: K. Rommel: The small conductivity primer. Self-published by WTW GmbH Weilheim, 1980, pp. 31–34 (theory, tables and diagrams on temperature coefficients of electrolytic conductivity and ion mobility).
• Conductivity and temperature. In: Conductivity Primer, An Introduction to Conductometry. Self-published by WTW GmbH Weilheim, 1993, pp. 11–12 (polynomial model for calculating the non-linear temperature profile of conductivity and ion mobility).

Individual evidence

1. ^ Giulio Milazzo: Electrochemistry. Springer Verlag, Vienna 1952.
2. ↑ Ulrich Hübschmann, Erwin Links: Tables on chemistry. Verlag Handwerk und Technik, Hamburg 1991, ISBN 3-582-01234-4 , p. 62.
3. ↑ ^{a b} Rommel: The small conductivity primer. WTW Weilheim, self-published, 1980, p. 21.
4. ↑ Hans Keune: chimica, a knowledge store. Volume II, VEB German publishing house for basic industry Leipzig, 1972, p. 148, Gl.8.56.
5. ↑ Hans Keune: chimica, a knowledge store. Volume II, VEB German publishing house for primary industry Leipzig, 1972, p. 148, Gl.8.57.
6. ↑ Hübschmann, Links: Tables on chemistry. Verlag Handwerk und Technik, Hamburg, 1991, ISBN 3-582-01234-4 , p. 61.
7. ↑ K.Rommel: The small conductivity primer, self-published by WTW GmbH Weilheim, BRD, 1980, table of equivalent conductivity values, p. 16
8. ^ Rolf Kaltofen: Book of tables of chemistry. (thick version), Verlag für Grundstoffindustrie Leipzig, GDR, 1974, substance concentrations of acids and saline solutions pp. 282–285.
9. ↑ Hübschmann, Links: Tables on chemistry. Verlag Handwerk und Technik, Hamburg, BRD, 1991, Grenzleitbaren p. 61-62, p. 32-33 densities and mass concentrations of solutions.
10. ↑ Fratscher, Picht: Material data and characteristics of process engineering. Verlag für Grundstofftindustrie Leipzig, GDR / FRG, 1979/1993, data on potassium hydroxide p. 41.
11. ↑ beer Werth: Tabellenbuch chemical engineering. Europa Lehrmittel, 2005, p. 91, conductivity of anhydrous sulfuric acid and nitric acid.
12. ↑ K.Rommel: The small conductivity primer, self-published by WTW GmbH Weilheim, FRG, 1980, article “Temperaturko coefficient alpha”, pp. 31–34
13. ↑ Conductivity primer - An introduction to conductometry, self-published by WTW GmbH Weilheim, BRD, 1993, article "Conductivity and temperature" pp. 11-12
14. ^ ^{A b} Karl-Heinz Näser, Dieter Lempe, Otfried Regen: Physical chemistry for technicians and engineers. VEB Deutscher Verlag für Grundstofftindustrie Leipzig, GDR 1990, p. 334.
15. ↑ K.Rommel: The small conductivity primer, self-published by WTW GmbH Weilheim, BRD, 1980, p. 34, table 5 with temperature coefficients of water.
16. ↑ K.Rommel: The small conductivity primer, self-published by WTW GmbH Weilheim, FRG, 1980, table 6, p. 34.
17. ^ Karl-Heinz Näser, Dieter Lempe, Otfried Regen: Physical chemistry for technicians and engineers , Verlag für Grundstoffindindustrie Leipzig, GDR, 1990, ISBN 3-342-00545-9 , Waldensche rule p. 343.
18. ^ Giulio Milazzo: Electrochemistry. Springer Verlag Vienna 1952, p. 52.
19. ^ Giulio Milazzo: Electrochemistry. Springer Verlag Vienna 1952, pp. 48–52.
20. ^ Giulio Milazzo: Electrochemistry. Springer Verlag, Vienna 1951, specific conductivity of sodium chloride melt at 801 and 850 ° C, p. 53, table 15.
21. ↑ Hübschmann, Links: Tables on chemistry. Verlag Handwerk und Technik, Hamburg, FRG 1991, conductivity of sodium chloride melt at 900 ° C, p. 61.
22. ↑ beer Werth: Tabellenbuch chemical engineering. Europa Lehrmittel, 2005, p. 91: specific conductivity of sodium chloride melt at 900 ° C; P. 148: Equivalent conductivities of aqueous sodium chloride solution at 19 ° C and various concentrations.
23. ^ Karl-Heinz Näser, Dieter Lempe, Otfried Regen: Physical chemistry for technicians and engineers , Verlag für Grundstoffindustrie Leipzig, GDR, 1990, ISBN 3-342-00545-9 , p. 327.
24. ↑ Udo R. Kunze, Georg Schwedt: Fundamentals of qualitative and quantitative analysis. Thieme Verlag, Stuttgart 1996, ISBN 3-13-585804-9 , p. 268.