Volume fraction

The volume fraction ( symbol : φ ), formerly also called volume fraction , is a so-called content quantity according to DIN 1310 , i.e. a physico-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 sum of the initial volumes of all mixture components .

Definition and characteristics

Scope, definition

The content size volume fraction 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 following table is in the size equations distinguish between

the simple case of a binary mixture ( Z = 2, two-substance mixture of components i and j , for example the solution of a single substance i in a solvent j ) and

the generally applicable formulation for a mixture of substances made up of a total of Z components (index z as a general index for the sums , includes i and possibly j ).

	binary mixture ( Z = 2)	general mixture ( Z components)
definition	${\ displaystyle \ varphi _ {i} = {\ frac {V_ {i}} {V_ {i} + V_ {j}}}}$	${\ displaystyle \ varphi _ {i} = {\ frac {V_ {i}} {V_ {0}}} \ \ {\ text {with}} \ \ V_ {0} = \ sum _ {z = 1} ^ {Z} V_ {z}}$
Range of values	${\ displaystyle 0 \ leq \ varphi _ {i} \ leq 1}$
Sum criterion	${\ displaystyle \ varphi _ {i} + \ varphi _ {j} = 1 \ \ Rightarrow \ \ varphi _ {j} = 1- \ varphi _ {i}}$	${\ displaystyle \ sum _ {z = 1} ^ {Z} \ varphi _ {z} = 1 \ \ Rightarrow \ \ varphi _ {j} = 1- \ sum \ limits _ {z = 1, z \ neq j } ^ {Z} \ varphi _ {z}}$

The volume fraction φ _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 ₀ before the mixing process. The latter is the sum of the initial volumes of all mixture components ( including i ) of the mixture.

As the quotient of two dimensions of equal dimensions, the volume fraction, like the volume concentration and the volume ratio, is a dimensionless value and can be specified as in the table above by a pure decimal number without a unit of measure , 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 thousand (‰ = 1 / 1,000) or parts per million (1 ppm = 1 / 1,000 .000). In this case, however, the outdated, ambiguous, no longer standardized specification in percentages by volume (abbreviation: percentages by volume) must be avoided; instead, the intended content must be clearly stated. For example, instead of “35.2% by volume”, the following should be formulated today: “The volume proportion of the mixture component i is 35.2%” or in the form of an equation: “ φ _i = 35.2%”. Nevertheless, information in percent by volume is still quite common, e.g. B. when specifying the composition of gas mixtures or specifying explosion limits .

The volume fraction φ _{i of} a considered mixture component i can assume numerical values between 0 = 0% (component i is not contained in the mixture) and 1 = 100% (component i is present as a pure substance ).

The volume proportions of all components of a mixture add up to 1 = 100%. From this it follows that the knowledge or determination of the volume fractions of Z - 1 components is sufficient (in the case of a two-substance mixture, the volume fraction of one component), since the volume fraction of the remaining component can be calculated simply by forming the difference to 1 = 100%.

Differentiation of volume concentration 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. The total volume V ₀ before the mixing process is the sum of the initial volumes of all mixing components. This is the difference to the related content quantity volume concentration σ _i , there the actual total volume V of the mixed phase after the mixing process is taken as a reference. With non-ideal mixtures, differences can arise between these two terms of total volume and thus also the two content quantities volume fraction φ _i and volume concentration σ _i as a result of volume reduction ( volume contraction ; φ _i < σ _i ; excess volume V ^E = V - V ₀ negative) or volume increase (volume dilatation ; φ _i > σ _i ; excess volume V ^E positive) during the mixing process. In practice, a sharp distinction is often not made between the two content parameters volume fraction and volume concentration, due to ignorance of the differences or because such volume changes during mixing - and thus numerical deviations between the two content parameters - are often relatively small (e.g. a maximum of around 4% volume contraction for mixtures of ethanol and water at room temperature). Larger deviations can occur with mixtures involving porous or granular materials.

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 .

The ratio of volume concentration to volume fraction for a mixture component i under consideration is equal to the ratio of total volume V ₀ before the mixing process to the actual total volume V of the mixing phase after the mixing process and is equal to the sum of the volume concentrations of all mixture components. It is only exactly 1 for ideal mixtures and otherwise deviates from 1, see the last column in the overview table below:

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

φ _i = volume fraction of the mixture component under consideration i
σ _i = volume concentration 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)

Temperature dependence

The value of the volume fraction for a mixture of substances of a given composition is - as with all other volume-related content variables ( concentrations including volume concentration , volume ratio ) - generally temperature-dependent, so that a clear indication of the volume fraction 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 mixture components . With ideal gases and their mixtures, however, the room expansion coefficient γ is uniform (reciprocal of the absolute temperature T :) , so that there the volume fraction is not temperature-dependent. With mixtures of real gases , the temperature dependency is usually low. Salary parameters such as B. the mass fraction w , which do not include the volume concept, are advantageously not dependent on the temperature. ${\ displaystyle \ gamma _ {\ text {ideal gas}} = {\ tfrac {1} {T}}}$

Relationships with other salary levels

The following table shows the relationships between the volume fraction φ _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. 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 ).

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

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}}

With ideal mixtures, the values of the volume fraction φ _i and the volume concentration σ _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 \ varphi _ {i} = \ sigma _ {i} {\ text {for ideal mixtures}}}

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

Usage, example

The volume fraction is used in various specialist areas, especially in chemistry , but also, for example, in mineralogy and petrology . Here, the volume fraction is used to describe the composition of rocks or minerals ( mixed crystal ), especially because it is comparatively easy to measure the volume of the individual components when optically recording thin sections .

A calculation example for the difference between the volume fraction φ _i and the volume concentration σ _i in non-ideal mixtures of ethanol and water can be found in the article Volume concentration .

Individual evidence

↑ ^a ^b ^c Standard DIN 1310 : Composition of mixed phases (gas mixtures, solutions, mixed crystals); Terms, symbols. February 1984.

↑ Standard DIN EN ISO 80000-9 : Quantities and units - Part 9: Physical chemistry and molecular physics. August 2013. Section 3: Terms, symbols and definitions , table entry no. 9–15.

↑ ^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. 34, 225, 307 , doi : 10.1007 / 978-3-322-83211-5 ( lexical part as PDF file, 71.3 MB ; limited preview in Google Book Search - first edition: 2000, 978-3-322- 83211-5 (e-book)).

↑ ^a ^b K. Schwister, V. Leven: Process engineering for engineers: text and exercise book . 2nd Edition. Fachbuchverlag Leipzig in Carl-Hanser-Verlag, Munich 2014, ISBN 978-3-446-44214-6 , p. 21, 90 ( limited preview in Google Book Search - 978-3-446-44001-2 (e-book)).

[DIN_1310-1] Standard DIN 1310 : Composition of mixed phases (gas mixtures, solutions, mixed crystals); Terms, symbols. February 1984.

[DIN_EN_ISO_80000-9-2] Standard DIN EN ISO 80000-9 : Quantities and units - Part 9: Physical chemistry and molecular physics. August 2013. Section 3: Terms, symbols and definitions , table entry no. 9–15.

[Kurzweil-3] 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. 34, 225, 307 , doi : 10.1007 / 978-3-322-83211-5 ( lexical part as PDF file, 71.3 MB ; limited preview in Google Book Search - first edition: 2000, 978-3-322- 83211-5 (e-book)).