# Amount of substance concentration

Physical size
Surname Amount of substance concentration
Formula symbol ${\ displaystyle c}$
Size and
unit system
unit dimension
SI mol · m -3 L −3 · N
Remarks
Usual unit: 1 mol / l = 1 mol / dm 3  = 1000 mol / m 3

The molar concentration ( symbol : c ), outdated also known as molarity , is a so-called content quantity according to DIN 1310 , i.e. a physical-chemical quantity for the quantitative description of the composition of substance mixtures / mixed phases (e.g. solutions ). Here the amount of substance of a considered mixture component is related to the total volume of the mixed phase .

## Definition and characteristics

The molar concentration c i is defined as the quotient of the molar amount n i of a considered mixture component i and the total volume V of the mixed phase:

${\ displaystyle c_ {i} = {\ frac {n_ {i}} {V}}}$

As an alternative to the index notation used here, a notation is also used in which the mixture component i considered is placed in brackets after the symbol c , using the example of sulfuric acid : c (H 2 SO 4 ). A notation that used to be common in the past consisted in putting the mixture component under consideration in square brackets: [H 2 SO 4 ].

The " particles " i on which the concept of the amount of substance is based must be specified; they can be material elementary objects such as atoms , molecules , ions or formula units . Here it is often assumed that the mixture component i considered is present exclusively in the form of these particles. B. an equilibrium concentration established in a solution by dissociation or reaction with the solvent , unless this is explicitly considered. In addition to specifying the species i, a complete specification of the molar concentration also includes the specification of the temperature and possibly the pressure (see below), in the case of solutions also the solvent (if no solvent is specified in the case of solutions, it is usually an aqueous solution ).

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

If the mixture of substances is not homogeneous , the above definition only provides an average concentration of the substance amount; deviating values ​​can then occur in partial volumes of the mixture of substances.

The derived SI unit of the molar concentration is mol / m 3 , in practice the unit mol / l (= mol / dm 3 ) is common, if necessary the molar unit part is also combined with decimal prefixes ( e.g. mmol / l ).

If the mixture component i is not present in the substance mixture (i.e. if n i = 0 mol), the minimum value c i = 0 mol / m 3 results . If component i is present as an unmixed pure substance , a molar concentration c i can also be specified for this , see the example below for water.

The total amount of substance concentration of the substance mixture is obtained by adding up the substance concentrations of all individual mixture components.

The DIN 1310 standard does not recommend shortening the term “substance concentration” for size c to the word “concentration” . Nor should the designations of the other concentration quantities mass concentration β , particle number concentration C and volume concentration σ be shortened to the word concentration.

The terms “molarity” (not to be confused with the independent content quantity molality !), “Molar concentration” or “molar concentration” for the substance concentration and the term “molar” or the symbol “ᴍ” (to be distinguished from the symbol M for the SI -Decimal prefix "Mega" and the symbol M for the molar mass !) For the unit mol / l do not conform to the standard, but are still frequently used. "Molar" should be avoided in this context because, on the one hand, an adjective as a unit designation falls outside the scope of the SI and, on the other hand, according to the general rule, a molar quantity is a quantity related to the amount of substance , i.e. a size quotient in which the quantity of substance is Denominator is (for example molar mass, molar volume , molar enthalpy of reaction , molar heat capacity ). In the case of the amount of substance concentration, however, the amount of substance is in the numerator, rather it is a volume-related quantity. Instead of obsolete forms of representation such as B. in the case of hydrochloric acid ( aqueous solution of hydrogen chloride HCl) the spelling "2.5 molar hydrochloric acid" or "2.5 ᴍ HCl" should indicate the molar concentration using an equation such as "hydrochloric acid, c HCl  = 2.5 mol / l "or using a corresponding short form such as" HCl, 2.5 mol / l ".

## Temperature and pressure dependency

The substance concentrations for a substance mixture of a given composition are - like all volume-related quantities ( concentrations , volume fraction , volume ratio ) - dependent on the temperature , in the case of gas mixtures also on the pressure (in the case of liquid and solid mixed phases, the pressure dependence can generally be neglected), so that a clear one Therefore, the specification of the associated temperature (possibly also the pressure) should be included. As a rule, an increase in temperature causes an increase in the total volume V of the mixed phase ( thermal expansion ), which leads to a reduction in the molar concentrations of the mixture components if the amounts of substance remain the same. Conversely, cooling usually causes a volume contraction of the mixed phase and thus an increase in the molar concentrations of the mixture components.

Since the change from room temperature to refrigerator temperature or vice versa can falsify the molar concentration of a solution, the temperature influence on the molar concentration should be taken into account in laboratory practice, as the effect is not negligible with regard to the desired accuracy. The extent depends on the one hand on the size of the temperature difference to be considered and on the other hand on the size of the spatial expansion coefficient γ of the mixed phase (the latter is significantly greater with solvents such as acetone , ethanol or methanol than with water ). An exact correction calculation for the temperature dependency of the substance concentration is usually not carried out, since the spatial expansion coefficient γ of the mixed phase with its own temperature dependency must be known exactly. However, a rough estimate can e.g. B. for dilute solutions and not too great differences between reference temperature T 1 and comparison temperature T 2 can be carried out as follows (use of the temperature-independent coefficient of expansion γ of the solvent):

${\ displaystyle c_ {i, T_ {2}} \ approx {\ frac {c_ {i, T_ {1}}} {1+ \ gamma \ cdot (T_ {2} -T_ {1})}}}$

From this it follows, for example, that when a dilute solution of the substance i cools down from 30 ° C to 10 ° C, its molar concentration c i is only about 0.5% for the solvent water or about 2-3% for the solvents acetone, Ethanol or methanol increases. By maintaining a constant temperature when working with solutions, the problem can be avoided in the first place. If this cannot be guaranteed, if there are high demands on accuracy, mass-related content values ​​such as B. the specific partial substance amount can be avoided.

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

${\ displaystyle c_ {i} = {\ frac {p_ {i}} {R \ cdot T}}}$

## Relationships with other salary levels

The following table shows the relationships between the molar concentration c i and the other content values ​​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. The index z serves as a general index for the sums (consideration of a general mixture of substances from a total of Z components) and includes i . N A is Avogadro's constant ( N A ≈ 6.022 · 10 23  mol −1 ).

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

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: ${\ displaystyle {\ overline {M}}}$

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

## Sample calculations

### Dilute sulfuric acid

Given is a dilute aqueous solution of sulfuric acid H 2 SO 4 at 20 ° C. which, in a volume V of 3 liters, contains a mass m of pure sulfuric acid of 235.392 grams. From this information, the mass concentration β of the sulfuric acid can first be calculated:

${\ displaystyle \ beta _ {\ mathrm {H_ {2} SO_ {4}}} = {\ frac {m _ {\ mathrm {H_ {2} SO_ {4}}}} {V}} = {\ frac { 235 {,} 392 \ \ mathrm {g}} {3 \ \ mathrm {l}}} = 78 {,} 464 \ \ mathrm {g \ cdot l ^ {- 1}}}$

With the help of the molar mass M of sulfuric acid (98.08 g / mol), it follows that the solution has a molar concentration of H 2 SO 4 of 0.8 mol / l (non-standard specification: “0.8 molar (0 , 8 ᴍ) aqueous sulfuric acid solution "):

${\ displaystyle c _ {\ mathrm {H_ {2} SO_ {4}}} = {\ frac {\ beta _ {\ mathrm {H_ {2} SO_ {4}}}} {M _ {\ mathrm {H_ {2 } SO_ {4}}}}} = {\ frac {78 {,} 464 \ \ mathrm {g \ cdot l ^ {- 1}}} {98 {,} 08 \ \ mathrm {g \ cdot mol ^ { -1}}}} = 0 {,} 800 \ \ mathrm {mol \ cdot l ^ {- 1}}}$

### Cane sugar solution

A dilute aqueous solution of cane sugar ( sucrose C 12 H 22 O 11 ) at 20 ° C. with a particle number N of sucrose molecules of 3.6132 · 10 18 in a solution volume V of 2  cubic centimeters (milliliters) is given. From this information, the particle number concentration C of the sucrose can first be calculated:

${\ displaystyle C _ {\ mathrm {sucrose}} = {\ frac {N _ {\ mathrm {sucrose}}} {V}} = {\ frac {3 {,} 6132 \ cdot 10 ^ {18}} {0 { ,} 002 \ \ mathrm {l}}} = 1 {,} 8066 \ cdot 10 ^ {21} \ \ mathrm {l ^ {- 1}}}$

Using Avogadro's constant N A , it follows that the solution has a molar concentration of sucrose of 0.003 mol / l (non-standard specification: "3 millimolar (3 mᴍ) aqueous sucrose solution"):

${\ displaystyle c _ {\ mathrm {sucrose}} = {\ frac {C _ {\ mathrm {sucrose}}} {N _ {\ mathrm {A}}}} = {\ frac {1 {,} 8066 \ cdot 10 ^ {21} \ \ mathrm {l ^ {- 1}}} {6 {,} 022 \ cdot 10 ^ {23} \ \ mathrm {mol ^ {- 1}}}} = 3 {,} 00 \ cdot 10 ^ {- 3} \ \ mathrm {mol \ cdot l ^ {- 1}}}$

### Pure water

A substance concentration c can also be assigned to a pure substance such as pure water H 2 O. Since the mass fraction w in this case is 1 = 100%, with the known density ρ of water at 20 ° C and the molar mass M of water (18.015 g / mol) for the molar concentration c of water at 20 ° C:

${\ displaystyle c _ {\ mathrm {H_ {2} O}} = {\ frac {w _ {\ mathrm {H_ {2} O}} \ cdot \ rho} {M _ {\ mathrm {H_ {2} O}} }} = {\ frac {1 \ cdot 998 {,} 2 \ \ mathrm {g \ cdot l ^ {- 1}}} {18 {,} 015 \ \ mathrm {g \ cdot mol ^ {- 1}} }} = 55 {,} 41 \ \ mathrm {mol \ cdot l ^ {- 1}}}$