Volume ratio

This article describes the volume ratio in mixtures of substances. For the term of the same name in magnetic resonance imaging, see the corresponding section in Diffusivity .

The volume ratio ( Symbol : ψ ) according to DIN 1310 , a physico-chemical quantity for quantitative description of the composition of mixtures / mixed phase , a so-called content size . It gives the ratio of the volumes of two considered mixture components to one another.

Definition and characteristics

The volume ratio ψ _ij is defined as the value of the quotient of the volume V _{i of} a considered mixture component i and the volume V _{j of} one of the other considered mixture components j :

${\displaystyle \psi _{ij}={\frac {V_{i}}{V_{j}}}}$

V _i and V _j are those initial volumes which the pure substances i and j occupy before the mixing process at the same pressure and temperature as in the substance mixture. The content size "volume ratio" 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.

To avoid ambiguity when specifying volume ratios, the numerator component and denominator component must always be specified, e.g. B. by the specified index notation. Interchanging the numerator and denominator components leads to the reciprocal value . In multicomponent mixtures, a corresponding number of volume ratios can be formulated: with a total of Z components, Z ² pieces, if the respective reciprocal values ​​and trivial volume ratios count as well ( variation with repetition ), otherwise pieces ( combination without repetition ). ${\displaystyle \psi _{ji}={\tfrac {1}{\psi _{ij}}}={\tfrac {V_{j}}{V_{i}}}}$ ${\displaystyle \psi _{ii}={\tfrac {V_{i}}{V_{i}}}=1}$ ${\displaystyle {\tbinom {Z}{2}}}$

In contrast to 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 , for the volume fraction φ _i the sum of the initial volumes of all mixture components is used as a reference, for the volume concentration σ _i the actual Final volume of the mixed phase, which in the case of non-ideal mixtures due to volume reduction ( volume contraction ) or volume increase (volume dilatation) during the mixing process can differ from the sum of the initial volumes of all mixture components (see excess volume ).

As the quotient of two dimensions of the same size, the volume ratio, like the volume fraction and the volume concentration, is a value of the dimension number 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, non-standard, but nonetheless frequently found indication of volume percent (abbreviation: vol .-%) should be avoided. If the mixture  component i is not present (i.e. if V _i = 0), the minimum value is ψ _ij  = 0. If the mixture component j is not present ( V _j  = 0, for example if there is no mixture but a pure substance i ), the volume ratio ψ _{ij is} not defined .

The value of the volume ratio for a mixture of substances of a given composition is - as with all other volume-related content parameters ( concentrations including volume concentration , volume fraction ) - generally temperature-dependent, so that a clear indication of the volume ratio therefore also includes the specification of the associated temperature. The reason for this is (with isobaric temperature change ) differences in the thermal expansion coefficients γ of the two mixture components considered. In the case of ideal gases, however , the spatial expansion coefficient γ is uniform (reciprocal of the absolute temperature T :) , so that in the case of mixtures of ideal gases, the volume ratio is not temperature-dependent. With mixtures of real gases , the temperature dependency is usually low. ${\displaystyle \gamma _{\text{ideales Gas}}={\tfrac {1}{T}}}$

Relationships with other salary levels

The following table shows the relationships between the volume ratio ψ _ij and the other content values ​​defined in DIN 1310 in the form of size equations . M _i and M _j stand for the respective molar masses , ρ _i and ρ _j for the respective densities of the pure substances i and j (at the same pressure and the same temperature as in the substance mixture).

Relationships of the volume ratio ψ _ij with other salary quantities

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

Summing for all mixture components, the volume ratios ψ _zi to a fixed component of the mixture i , one obtains the reciprocal of the volume fraction of the fixed component of the mixture i (mixture of a total of Z components index such as a general running index for the summation , integration of the trivial volume ratio in the sum) : ${\displaystyle \psi _{ii}={\tfrac {V_{i}}{V_{i}}}=1}$

${\displaystyle \sum _{z=1}^{Z}\psi _{zi}=\sum _{z=1}^{Z}{\frac {V_{z}}{V_{i}}}={\frac {1}{\varphi _{i}}}}$

Since the molar volume V _{m of} a pure substance is equal to the quotient of its molar mass M and density ρ (at a given temperature and pressure), the terms that appear several times in the same way in the table above (ratio of molar masses multiplied by the inverse ratio of densities ) can also be replaced by the ratio of the molar volumes:

${\displaystyle {\frac {M_{i}}{M_{j}}}\cdot {\frac {\rho _{j}}{\rho _{i}}}={\frac {V_{\mathrm {m} ,i}}{V_{\mathrm {m} ,j}}}}$

If the mixture components i and j are ideal gases , the molar volumes are the same and their ratio is consequently equal to one. It follows from the above table that for mixtures of ideal gases the values ​​of the volume ratio ψ _ij and the molar ratio r _ij or the particle number ratio R _{ij are the} same:

${\displaystyle \psi _{ij}=r_{ij}=R_{ij}{\text{ für ideale Gase }}i,j}$

Example: Mixture of alcohol and water

If you produce a mixture of equal masses of pure ethanol and water , both substances have a mass fraction w of 0.5 = 50% in the resulting mixture , the mass ratio ζ is 1. With the densities ρ of the pure substances at 20 ° C follows for the value of the volume ratio at 20 ° C:

${\displaystyle \psi _{\mathrm {Ethanol/Wasser} }=\zeta _{\mathrm {Ethanol/Wasser} }\cdot {\frac {\rho _{\mathrm {Wasser} }}{\rho _{\mathrm {Ethanol} }}}=1\cdot {\frac {0{,}998\ \mathrm {g\cdot cm^{-3}} }{0{,}789\ \mathrm {g\cdot cm^{-3}} }}=1{,}26}$

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, 2013, ISBN 978-3-322-83212-2 , p. 225, 419 , 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).