# Mass concentration

The mass concentration ( symbol according to DIN 1310 : β ; according to IUPAC : ρ or γ ), sometimes also referred to as partial density , is a so-called content quantity , i.e. a physico-chemical quantity for the quantitative description of the composition of substance mixtures / mixed phases (e.g. solutions ). Here, the mass of a considered mixture component is related to the total volume of the mixed phase .

## Definition and characteristics

The mass concentration β i is defined as the quotient of the mass m i of a considered mixture component i and the total volume V of the mixed phase:

${\ displaystyle \ beta _ {i} = {\ frac {m_ {i}} {V}}}$

V is the actual total volume of the mixing phase after the mixing process, see the explanations under volume concentration .

The mass concentration has the same dimension as the density ρ and, accordingly, also the derived SI unit kg / m 3 . In practice, the use of various decimal prefixes often modifies the mass part (e.g. g , mg, μg) and / or the volume part (e.g. dm 3 , cm 3 ) of the unit, or the unit becomes the volume part Liter l used alone or combined with a decimal prefix (e.g. ml). The following applies for the conversion: 1 g / cm 3 = 1 g / ml = 1 kg / dm 3 = 1 kg / l = 1000 g / l = 1000 kg / m 3 .

Since the mass concentration is a dimensional quantity, it must not - as is sometimes wrongly found in practice - only with dimensionless auxiliary units such as percent (% = 1/100), per mil (‰ = 1 / 1,000) or parts per million (1 ppm = 1 / 1,000,000), especially since there is then a risk of confusion e.g. B. with the mass fraction . Also other non-standard, outdated, ambiguous or misleading information such as B. percent by weight , percent by weight or the percentage symbol% ​​in combination with the addition (m / v) or (w / v) are to be avoided.

If the mixture component i is not present in the substance mixture (i.e. when m i = 0 kg), the minimum value β i = 0 kg / m 3 results . If component i is present as an unmixed pure substance , the mass concentration β i corresponds to the pure substance density ρ i .

The sum of the mass concentrations of all mixture components of a mixture of substances (in the case of solutions including the mass concentration of the solvent !) Results in the density ρ of the mixed phase, which is equal to the quotient of the total mass m (sum of the individual masses of the mixture components) and the total volume V of the mixed phase ( formulated below for a general mixture of substances made up of a total of Z components, index z as a general index for the totalization ):

${\ displaystyle \ sum _ {z = 1} ^ {Z} \ beta _ {z} = \ sum _ {z = 1} ^ {Z} {\ frac {m_ {z}} {V}} = {\ frac {m} {V}} = \ rho}$

From this closing condition it follows that the knowledge or determination of the mass concentrations of Z  - 1 components is sufficient (in the case of a two-component mixture, the mass concentration of one component), since the mass concentration of the remaining component can be calculated simply by forming the difference to the density ρ of the mixed phase (if this is known ) can be calculated.

The mass concentrations for a mixture of substances of a given composition are - like all volume-related quantities ( concentrations , volume fraction , volume ratio ) - generally dependent on the temperature (in the case of gas mixtures, possibly also on the pressure ), so that the associated temperature must be specified for a clear indication (possibly also the pressure) heard. As a rule, an increase in temperature causes an increase in the total volume V of the mixed phase ( thermal expansion ), which, with the masses remaining the same, leads to a reduction in the mass concentrations of the mixture components.

For mixtures of ideal gases it can be derived from the general gas equation that the mass concentration β i of a mixture component i is proportional to its partial pressure p i and inversely proportional to the absolute temperature T ( M i = molar mass of i ; R = universal gas constant ):

${\ displaystyle \ beta _ {i} = {\ frac {M_ {i} \ cdot p_ {i}} {R \ cdot T}}}$

## Relationships with other salary levels

The following table shows the relationships between the mass concentration β i and the other content quantities defined in DIN 1310 in the form of size equations . The formula symbols M and ρ provided with an index stand for the molar mass or density (at the same pressure and temperature as in the substance mixture) of the respective pure substance identified by the index . The symbol ρ without an index represents the density of the mixed phase. As above, the index z serves as a general index for the sums and includes i . N A is Avogadro's constant ( N A ≈ 6.022 · 10 23  mol −1 ).

Relationship between the mass concentration β i and other content quantities
Masses - ... Amount of substance - ... Particle number - ... Volume - ...
... - share Mass fraction w Amount of substance fraction x Particle number fraction X Volume fraction φ
${\ displaystyle \ beta _ {i} = w_ {i} \ cdot \ rho}$ ${\ displaystyle \ beta _ {i} = {\ frac {x_ {i} \ cdot M_ {i} \ cdot \ rho} {\ sum _ {z = 1} ^ {Z} (x_ {z} \ cdot M_ {z})}}}$ ${\ displaystyle \ beta _ {i} = {\ frac {X_ {i} \ cdot M_ {i} \ cdot \ rho} {\ sum _ {z = 1} ^ {Z} (X_ {z} \ cdot M_ {z})}}}$ ${\ displaystyle \ beta _ {i} = {\ frac {\ varphi _ {i} \ cdot \ rho _ {i} \ cdot \ rho} {\ sum _ {z = 1} ^ {Z} (\ varphi _ {z} \ cdot \ rho _ {z})}}}$
… - concentration Mass concentration β Molar concentration c Particle number concentration C Volume concentration σ
${\ displaystyle \ beta _ {i}}$ ${\ displaystyle \ beta _ {i} = c_ {i} \ cdot M_ {i}}$ ${\ displaystyle \ beta _ {i} = {\ frac {C_ {i} \ cdot M_ {i}} {N _ {\ mathrm {A}}}}}$ ${\ displaystyle \ beta _ {i} = \ sigma _ {i} \ cdot \ rho _ {i}}$
... - ratio Mass ratio ζ Molar ratio r Particle number ratio R Volume ratio ψ
${\ displaystyle \ beta _ {i} = \ zeta _ {ij} \ cdot \ beta _ {j} = {\ frac {\ rho} {\ sum _ {z = 1} ^ {Z} \ zeta _ {zi }}}}$ ${\ displaystyle \ beta _ {i} = {\ frac {M_ {i} \ cdot \ rho} {\ sum _ {z = 1} ^ {Z} (r_ {zi} \ cdot M_ {z})}} }$ ${\ displaystyle \ beta _ {i} = {\ frac {M_ {i} \ cdot \ rho} {\ sum _ {z = 1} ^ {Z} (R_ {zi} \ cdot M_ {z})}} }$ ${\ displaystyle \ beta _ {i} = {\ frac {\ rho _ {i} \ cdot \ rho} {\ sum _ {z = 1} ^ {Z} (\ psi _ {zi} \ cdot \ rho _ {z})}}}$
Quotient
amount of substance / mass
Molality b
${\ displaystyle \ beta _ {i} = b_ {i} \ cdot M_ {i} \ cdot \ beta _ {j}}$ ( i = solute, j = solvent)
specific amount of partial substances q
${\ displaystyle \ beta _ {i} = q_ {i} \ cdot M_ {i} \ cdot \ rho}$

In the table above in the equations in the mole fraction x and Teilchenzahlanteil X occurring denominator - Terme are equal to the average molar mass of the material mixture and can be replaced in accordance with: ${\ displaystyle {\ overline {M}}}$

${\ displaystyle \ sum _ {z = 1} ^ {Z} (x_ {z} \ cdot M_ {z}) = \ sum _ {z = 1} ^ {Z} (X_ {z} \ cdot M_ {z }) = {\ overline {M}}}$

## Examples

### Calcium intake through mineral water

Given is a mass concentration of calcium (present in an aqueous solution in the form of calcium ions Ca 2+ ) in mineral water of 140 mg / l. Find the amount of calcium that you add to your body when consuming 1.5 liters of mineral water. By transforming the above definition equation and inserting the numerical values ​​and units, the result is:

${\ displaystyle m _ {\ mathrm {Ca}} = \ beta _ {\ mathrm {Ca}} \ cdot V = 140 \ \ mathrm {mg / l} \ cdot 1 {,} 5 \ \ mathrm {l} = 210 \ \ mathrm {mg}}$

### Solution of sodium chloride in water

A solution of sodium chloride (common salt) NaCl in water H 2 O with the mass fractions w NaCl = 0.03 = 3% and correspondingly w H 2 O = 1 -  w NaCl = 0.97 = 97% is considered. With the density ρ of this solution at 20 ° C, the mass concentrations of NaCl or H 2 O at this temperature follow :

${\ displaystyle \ beta _ {\ mathrm {NaCl}} = w _ {\ mathrm {NaCl}} \ cdot \ rho = 0 {,} 03 \ cdot 1019 {,} 6 \ \ mathrm {g / l} = 30 { ,} 6 \ \ mathrm {g / l}}$
${\ displaystyle \ beta _ {\ mathrm {H_ {2} O}} = w _ {\ mathrm {H_ {2} O}} \ cdot \ rho = 0 {,} 97 \ cdot 1019 {,} 6 \ \ mathrm {g / l} = \ rho - \ beta _ {\ mathrm {NaCl}} = 1019 {,} 6 \ \ mathrm {g / l} -30 {,} 6 \ \ mathrm {g / l} = 989 { ,} 0 \ \ mathrm {g / l}}$

### Nitrogen and oxygen in air

Air as the gas mixture of the earth's atmosphere contains the two main components nitrogen (particles: N 2 molecules) and oxygen (particles: O 2 molecules). When viewed approximately as a mixture of ideal gases , the usually tabulated mean volume fractions of the individual gases in dry air at sea level (N 2 : approx. 78.1%; O 2 : approx. 20.9%) are to be equated with the volume concentrations σ , so:

${\ displaystyle \ sigma _ {\ mathrm {N_ {2}}} \ approx 0 {,} 781 = 78 {,} 1 \ \% \ qquad \ sigma _ {\ mathrm {O_ {2}}} \ approx 0 {,} 209 = 20 {,} 9 \ \%}$

With the aid of the pure substance densities of nitrogen and oxygen for a certain temperature T and a certain pressure p , for example for standard conditions (temperature 273.15  K = 0  ° C ; pressure 101,325  Pa = 1.01325  bar ), the mass concentrations β of nitrogen can be derived from this and determine oxygen under the given boundary conditions:

${\ displaystyle \ beta _ {\ mathrm {N_ {2}}} = \ sigma _ {\ mathrm {N_ {2}}} \ cdot \ rho _ {\ mathrm {N_ {2}}} \ approx 0 {, } 781 \ cdot 1 {,} 250 \ \ mathrm {kg / m ^ {3}} \ approx 0 {,} 976 \ \ mathrm {kg / m ^ {3}}}$
${\ displaystyle \ beta _ {\ mathrm {O_ {2}}} = \ sigma _ {\ mathrm {O_ {2}}} \ cdot \ rho _ {\ mathrm {O_ {2}}} \ approx 0 {, } 209 \ cdot 1 {,} 429 \ \ mathrm {kg / m ^ {3}} \ approx 0 {,} 299 \ \ mathrm {kg / m ^ {3}}}$

In reality the air is not completely dry; Due to the water vapor as an additional component in the mixture, the mass concentrations of nitrogen and oxygen are somewhat lower.

## Individual evidence

1. a b c Standard DIN 1310 : Composition of mixed phases (gas mixtures, solutions, mixed crystals); Terms, symbols. February 1984 (with a note that the formula symbol β is used instead of the previously standardized symbol in order to avoid confusion with the density).${\ displaystyle \ varrho}$
2. a b c d P. Kurzweil: The Vieweg unit lexicon: terms, formulas and constants from natural sciences, technology and medicine . 2nd Edition. Springer Vieweg, 2013, ISBN 978-3-322-83212-2 , p. 49, 224, 225, 262 , doi : 10.1007 / 978-3-322-83211-5 ( limited preview in the Google Book Search - softcover reprint of the 2nd edition 2000). - ( Lexical part PDF; 71.3 MB).
3. UR Kunze, G. Schwedt: Fundamentals of qualitative and quantitative analysis . 4th edition. Thieme, Stuttgart [a. a.] 1996, ISBN 3-13-585804-9 , pp. 71 (there footnote: ρ * to distinguish it from the density ρ , β i is suggested as a new symbol ).
4. K.-H. Lautenschläger: Pocket Book of Chemistry . 18th edition. Harri Deutsch, Frankfurt am Main 2001, ISBN 3-8171-1654-3 , p. 51 .
5. a b c G. Jander, KF Jahr, R. Martens-Menzel, G. Schulze, J. Simon: Measure analysis: theory and practice of titrations with chemical and physical indications . 18th edition. De Gruyter, Berlin / Boston 2012, ISBN 978-3-11-024898-2 , pp. 54 , doi : 10.1515 / 9783110248999 ( limited preview in Google Book search).
6. a b entry on mass concentration . In: IUPAC Compendium of Chemical Terminology (the “Gold Book”) . doi : 10.1351 / goldbook.M03713 Version: 2.3.3.
7. ^ WM Haynes: CRC Handbook of Chemistry and Physics . 95th edition. CRC Press / Taylor & Francis, Boca Raton (FL) 2014, ISBN 978-1-4822-0867-2 , pp. 5–142 ( limited preview in Google Book Search).