# Volume concentration

The volume concentration ( symbol : σ ) 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 . Here the volume of a considered mixture component is related to the total volume of the mixed phase .

## Definition and characteristics

### Scope, definition

The content size volume concentration is only used as a rule when the pure substances before the mixing process and the mixed phase same physical state have, in practice, therefore, especially in gas mixtures and mixtures of liquids (subgroup of the solutions ).

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

${\ displaystyle \ sigma _ {i} = {\ frac {V_ {i}} {V}}}$

### Delimitation of volume share and volume ratio

V i is the initial volume which the pure substance i occupies before the mixing process at the same pressure and temperature as in the substance mixture. V is the actual total volume of the mixing phase after the mixing process. This is the difference to the related content quantity volume fraction φ i , there the sum of the initial volumes of all mixture components (total volume V 0 before the mixing process) is taken as a reference. With non-ideal mixtures, differences can arise between these two total volume terms and thus also the two content variables volume concentration σ i and volume fraction φ i as a result of volume reduction ( volume contraction ; σ i > φ i ; excess volume V E = V - V 0 negative) or volume increase (volume dilation ; σ i < φ i ; excess volume V E positive) during the mixing process. In practice, a sharp distinction is often not made between the two content variables, volume concentration and volume fraction, due to ignorance of the differences or because such volume changes during mixing - and thus numerical deviations between the two content variables - are often relatively small (e.g. a maximum of around 4% volume contraction in the case of mixtures of ethanol and water at room temperature; this is different, however, when applied to granular mixtures with a large grain size difference : for example, one cubic meter of coarse gravel and one cubic meter of sand mix in less than one and a half cubic meters of mixture, because the sand fills the free volumes of the gravel bed).

Another related content variable is the volume ratio ψ ij , in which the initial volume of a considered mixture component i is related to the initial volume of another considered mixture component j .

### Dimension and unit of measure

As the quotient of two dimensions of equal dimensions, the volume concentration, like the volume fraction and the volume ratio, is a dimensionless value and can assume numerical values ​​≥ 0. It can be specified as a pure decimal number without a unit of measurement , alternatively also with the addition of a fraction of the same units ( m 3 / m 3 or l / l), possibly combined with decimal prefixes (e.g. ml / l), or with auxiliary units such as Percent (% = 1/100), per mil (‰ = 1 / 1,000) or parts per million (1 ppm = 1 / 1,000,000). In this case, however, the outdated, ambiguous, no longer standard specification in percent by volume (abbreviation% by volume) should be avoided.

### Range of values

If the mixture component i is not present (i.e. when V i = 0), the minimum value σ i = 0 = 0% results . If component i is a pure substance , σ i assumes the value 1 = 100%. In contrast to the volume fraction φ i the volume concentration is σ i but not necessarily limited to a maximum value of 1 = 100%: In the event that the initial volume V i of the mixture component i is greater than the volume V of the mixed phase, the quotient can σ i = V i / V assume values ​​greater than 1. This can occur, for example, with (atypical) application of the volume concentration to the solution of a gas (e.g. ammonia NH 3 ) in water, in which a strong volume contraction occurs.

### total

The summation of the volume concentrations of all mixture components gives the ratio of the total volume V 0 before the mixing process to the actual total volume V of the mixing phase after the mixing process. This ratio corresponds to the ratio of volume concentration to volume fraction for a considered mixture component i . It is only exactly 1 for ideal mixtures and otherwise deviates from 1. In the following summary table, this is the last column formulated for a general mixture of a total of Z components (index such as a general running index for the summation concludes looked mixture component i with one):

${\ displaystyle \ sigma _ {i} {\ text {vs. }} \ varphi _ {i}}$ ${\ displaystyle V ^ {\ mathrm {E}} = V-V_ {0}}$ ${\ displaystyle \ sum _ {z = 1} ^ {Z} \ sigma _ {z} = {\ frac {V_ {0}} {V}} = {\ frac {\ sigma _ {i}} {\ varphi _ {i}}}}$
Volume contraction ${\ displaystyle \ sigma _ {i}> \ varphi _ {i}}$ ${\ displaystyle <0}$ ${\ displaystyle> 1}$
ideal mix ${\ displaystyle \ sigma _ {i} = \ varphi _ {i}}$ ${\ displaystyle = 0}$ ${\ displaystyle = 1}$
Volume dilation ${\ displaystyle \ sigma _ {i} <\ varphi _ {i}}$ ${\ displaystyle> 0}$ ${\ displaystyle <1}$

σ i = volume concentration of the mixture component
under consideration i
φ i = volume fraction of the mixture component
under consideration i
V E = excess volume
V = actual total volume of the mixing phase after the mixing process
V 0 = total volume before the mixing process (sum of the initial
volumes of all mixture components)

The fact that in ideal mixtures the sum of the volume concentrations of all mixture  components is 1 = 100% means that in this case it is sufficient to know or determine the volume concentrations of Z - 1 components (i.e. the volume concentration of one component in a two-component mixture ) , since the volume concentration of the remaining component can be calculated simply by calculating the difference to 1 = 100%.

### Temperature dependence

The value of the volume concentration for a substance mixture of a given composition is - as with all volume-related content quantities ( concentrations , volume fraction , volume ratio ) - generally temperature-dependent, so that a clear indication of the volume concentration therefore also includes the specification of the associated temperature. The reason for this is (with isobaric temperature change ) differences in the thermal expansion coefficient γ of the considered mixture component and the mixed phase. (Example values under alcohol content .) With ideal gases and their mixtures, the room expansion coefficient γ is, however, uniform (reciprocal of the absolute temperature T :) , so that the volume concentration there is not temperature-dependent. With mixtures of real gases , the temperature dependency is usually low. ${\ displaystyle \ gamma _ {\ text {ideal gas}} = {\ tfrac {1} {T}}}$

## Relationships with other salary levels

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

Since the molar volume V m of a pure substance is equal to the quotient of its molar mass and its density (at a given temperature and pressure), the terms appearing in some of the equations in the table above can be replaced accordingly:

${\ displaystyle {\ frac {M_ {i}} {\ rho _ {i}}} = V _ {\ mathrm {m}, i}}$

In the case of ideal mixtures, the values ​​of volume concentration σ i and volume fraction φ i match. In the case of mixtures of ideal gases, there is also equality with the mole fraction x i and the particle number fraction X i :

${\ displaystyle \ sigma _ {i} = \ varphi _ {i} {\ text {for ideal mixtures}}}$
${\ displaystyle \ sigma _ {i} = \ varphi _ {i} = x_ {i} = X_ {i} {\ text {for mixtures of ideal gases}}}$

## Examples

### Salary information on labels of alcoholic beverages

A prominent example of the use of volume concentrations is the indication of alcohol content on the labels of alcoholic beverages . For example, if the label of a beer bottle says “alc. 4.9% vol ”, this means that the ethanol volume concentration σ ethanol is 4.9%; 100 ml of beer therefore contains 4.9 ml of pure ethanol (reference temperature 20 ° C; see volume percentage application ).

### Mixture of alcohol and water

A mixture of equal initial volumes of pure ethanol (index i ) and water (index j ) at 20 ° C is considered. The volume ratio is therefore 1, the volume proportions of both substances are the same (50%):

${\ displaystyle V_ {i} = V_ {j} \ \ Leftrightarrow \ \ psi _ {ij} = {\ frac {V_ {i}} {V_ {j}}} = 1 \ \ Leftrightarrow \ \ varphi _ {i } = {\ frac {V_ {i}} {V_ {i} + V_ {j}}} = \ varphi _ {j} = {\ frac {V_ {j}} {V_ {i} + V_ {j} }} = 0 {,} 5 = 50 \ \%}$

With the densities of the pure substances and the resulting mixture at 20 ° C, it follows for the volume concentrations of ethanol and water (in this special case the same) at this temperature:

${\ displaystyle \ sigma _ {i} = {\ frac {\ varphi _ {i} \ cdot \ rho} {\ varphi _ {i} \ cdot \ rho _ {i} + \ varphi _ {j} \ cdot \ rho _ {j}}} = \ sigma _ {j} = {\ frac {\ varphi _ {j} \ cdot \ rho} {\ varphi _ {i} \ cdot \ rho _ {i} + \ varphi _ { j} \ cdot \ rho _ {j}}} = {\ frac {0 {,} 5 \ cdot 0 {,} 9266 \ \ mathrm {g / cm ^ {3}}} {0 {,} 5 \ cdot 0 {,} 7893 \ \ mathrm {g / cm ^ {3}} +0 {,} 5 \ cdot 0 {,} 9982 \ \ mathrm {g / cm ^ {3}}}} = 0 {,} 5184 }$

The volume concentrations are greater than the volume fractions, so there was a volume contraction during mixing. The summation of the volume concentrations gives the ratio of the total volume V 0 = V i + V j before the mixing process to the actual total volume V of the mixture after the mixing process:

${\ displaystyle \ sigma _ {i} + \ sigma _ {j} = {\ frac {V_ {0}} {V}} \ approx 1 {,} 037 \ \ {\ text {or. expressed as a reciprocal}} \ \ {\ frac {V} {V_ {0}}} \ approx 0 {,} 965}$

The volume contraction is therefore around 3.5%, i.e. H. mixing, for example, 50 ml of ethanol and 50 ml of water does not lead to 100 ml of mixture at 20 ° C, but only to about 96.5 ml.

## Individual evidence

1. a b c Standard DIN 1310 : Composition of mixed phases (gas mixtures, solutions, mixed crystals); Terms, symbols. February 1984.
2. a b P. Kurzweil: The Vieweg unit lexicon: terms, formulas and constants from natural sciences, technology and medicine . 2nd Edition. Springer Vieweg, 2000, ISBN 978-3-322-83212-2 , p. 224, 225 , doi : 10.1007 / 978-3-322-83211-5 ( lexical part [PDF; 71.3 MB]; limited preview in Google Book Search - softcover reprint 2013).
3. ^ WM Haynes: CRC Handbook of Chemistry and Physics . 96th edition. CRC Press / Taylor & Francis, Boca Raton FL 2015, ISBN 978-1-4822-6096-0 , pp. 6-7, 5-124 f., 15–43 ( limited preview in Google Book search).