Partial molar size

A partial molar quantity indicates how much one mole of the -th substance in a mixture contributes to a thermodynamic property of the mixture. ${\displaystyle E_{i}}$ ${\displaystyle i}$ ${\displaystyle E}$

If there is a homogeneous phase , which is a mixture of substances, then an extensive thermodynamic property of the phase results as the sum of the contributions of the individual substances. For example, the total internal energy of the phase is the sum of the partial molar internal energies of the substances contained in the mixture, multiplied by the respective amount of substance present : ${\displaystyle r}$ ${\displaystyle E}$ ${\displaystyle U}$ ${\displaystyle U_{i}}$ ${\displaystyle n_{i}}$

${\displaystyle U=\sum _{i=1}^{r}n_{i}U_{i}}$.

Likewise, the volume of the phase is the sum of the partial molar volumes of the substances contained in the mixture, multiplied by the respective amount of substance : ${\displaystyle V}$ ${\displaystyle V_{i}}$ ${\displaystyle n_{i}}$

${\displaystyle V=\sum _{i=1}^{r}n_{i}V_{i}}$,

and so on for any extensive properties.

Due to the interactions of the substances with one another, the partial molar contribution of a substance usually also depends on the proportions of all other substances in the mixture, for example it is written in detail

${\displaystyle {U=n_{1}\,U_{1}(T,p,x_{2},x_{3},\dots ,x_{r})+n_{2}\,U_{2}(T,p,x_{1},x_{3},\dots ,x_{r})+\ldots +n_{r}\,U_{r}(T,p,x_{1},x_{2},\dots ,x_{r-1})}}$

with the amount of substance . ${\displaystyle x_{i}}$

Introductory example

The molar volume of pure water is 18 cm³ per mole . If one mole of water is added to a given volume of water, the volume increases by 18 cm³. The volumes of the two combined amounts of water are additive.

However, if you add the mole of water to a given (large) volume of alcohol , you only get a volume increase of 14 cm³. In this case the two volumes are not additive. The mole of water only contributes 14 cm³ to the total volume of the mixture, so the partial molar volume of water in (almost) pure alcohol is 14 cm³ / mol.

If the mole of water is added to a mixture of water and alcohol, other values ​​result for the partial molar volume of the water, depending on the mixing ratio. With pure water 18 cm³ / mol are reached again. The partial molar volume of the alcohol in the mixture, like that of water, is also dependent on the mixture.

When mixed with alcohol, the water makes a smaller volume contribution because the volume occupied by the water molecules depends on their environment. In a mixture with a large amount of alcohol, each water molecule is completely surrounded by alcohol molecules rather than other water molecules. This denser arrangement leads to a smaller space requirement.

In general, mixing effects ( volume contraction , heat of mixing , etc.) are due to the fact that the mean interactions of the water and alcohol molecules with one another in the mixture are different from the mean interactions of the water molecules with one another in pure water and the mean interactions between the alcohol molecules with one another in pure alcohol. The partial molar sizes allow a quantitative treatment of mixing effects.

definition

As described at the beginning, an extensive property of a mixture can be described as the sum of the molar partial sizes of the mixture components weighted with the respective amounts of substance . However, establishing this relationship is not sufficient for a clear definition of the partial molar quantities involved , since there are an infinite number of possibilities to assemble the desired sum from individual summands. The cumulative condition could at best define one of the unambiguously if the others have already been determined. The definition of must therefore be done in a different way. ${\displaystyle E}$ ${\displaystyle E_{i}}$ ${\displaystyle E_{i}}$ ${\displaystyle E}$ ${\displaystyle E_{i}}$ ${\displaystyle E_{i}}$

The partial molar size of the -th substance of the mixture belonging to an extensive thermodynamic variable of a mixture is defined by ${\displaystyle E}$ ${\displaystyle E_{i}}$ ${\displaystyle i}$

${\displaystyle E_{i}=\left({\frac {\partial E}{\partial n_{i}}}\right)_{T,\,p,\,n_{j\neq i}}}$.

So it is equal to the infinitesimal change in size that results from adding an infinitesimal amount of the -th substance, divided by the number of moles added, if the temperature, pressure and the number of moles of all other substances are kept constant during the addition. ${\displaystyle E}$ ${\displaystyle i}$

For the mentioned partial molar volume, for example, is

${\displaystyle V_{i}=\left({\frac {\partial V}{\partial n_{i}}}\right)_{T,\,p,\,n_{j\neq i}}}$.

If the phase contains only one substance, the partial molar sizes are identical to the molar sizes .

Transition to finite proportions

The definition considers the infinitesimal change in property caused by adding an infinitesimal amount of substance . The transition to the finite molar contributions sought by the constituents present in the mixture initially appears to require integration. This would also be difficult because each of them can depend in a complicated way on all the amounts of substance present , which in turn can change in a complicated way when the system is gradually brought together. It turns out, however, that the relationship between the and the is surprisingly simple. ${\displaystyle \mathrm {d} E_{i}}$ ${\displaystyle E}$ ${\displaystyle \mathrm {d} E_{i}}$ ${\displaystyle n_{1},\dots ,n_{r}}$ ${\displaystyle \mathrm {d} E_{i}}$ ${\displaystyle E_{i}}$

The starting point of the considerations is the differential of , which is understood as a function of the variables temperature, pressure and the number of moles of the substances contained. For a given state the phase is general ${\displaystyle E}$ ${\displaystyle (T,p,n_{1},\dots ,n_{r})}$

${\displaystyle {\begin{alignedat}{4}\mathrm {d} E(T,p,n_{1},\dots ,n_{r})&=\left({\frac {\partial E}{\partial T}}\right)_{p,\,n_{1},\dots ,n_{r}}\,\mathrm {d} T+\left({\frac {\partial E}{\partial p}}\right)_{T,\,n_{1},\dots ,n_{r}}\,\mathrm {d} p+&\left({\frac {\partial E}{\partial n_{1}}}\right)_{T,\,p,\,n_{j\neq 1}}\,\mathrm {d} n_{1}&+\ldots +&\left({\frac {\partial E}{\partial n_{r}}}\right)_{T,\,p,\,n_{j\neq r}}\,\mathrm {d} n_{r}\\&=\left({\frac {\partial E}{\partial T}}\right)_{p,\,n_{1},\dots ,n_{r}}\,\mathrm {d} T+\left({\frac {\partial E}{\partial p}}\right)_{T,\,n_{1},\dots ,n_{r}}\,\mathrm {d} p+&E_{1}\qquad \quad \mathrm {d} n_{1}&+\ldots +&E_{r}\qquad \quad \mathrm {d} n_{r}\end{alignedat}}}$.

Now imagine the size of the phase multiplied. During this process the temperature and the pressure remain unchanged ( , ) because they are intense quantities. The differential that describes the change in with such an increase is therefore ${\displaystyle \mathrm {d} T=0}$ ${\displaystyle \mathrm {d} p=0}$ ${\displaystyle E}$

${\displaystyle \mathrm {d} E=E_{1}\,\mathrm {d} n_{1}+\ldots +E_{r}\,\mathrm {d} n_{r}}$.

Since the partial molar sizes are also intensive sizes, they also remain unchanged. The direct integration of the differential therefore provides for the change in the extensive one associated with the enlargement : ${\displaystyle E_{1},\dots ,E_{r}}$ ${\displaystyle E}$

${\displaystyle {\begin{aligned}\Delta E=\int \mathrm {d} E&=\int E_{1}\,\mathrm {d} n_{1}+\ldots +\int E_{r}\,\mathrm {d} n_{r}\\&=E_{1}\int \mathrm {d} n_{1}+\ldots +E_{r}\int \mathrm {d} n_{r}\\&=E_{1}\,\Delta n_{1}+\ldots +E_{r}\,\Delta n_{r}\end{aligned}}}$,

because they can be drawn in front of the respective integrals as constant quantities. ${\displaystyle E_{1},\dots ,E_{r}}$

Done the mental multiplication by a factor , then the numerical values take the extensive quantities to the fold and we have to ${\displaystyle k}$ ${\displaystyle E,n_{1},\dots ,n_{r}}$ ${\displaystyle k}$

${\displaystyle \Delta E=k\,E-E=(k-1)\,E}$,

${\displaystyle \Delta n_{i}=k\,n_{i}-n_{i}=(k-1)\,n_{i}}$.

Substituting into the previous equation leads to

${\displaystyle (k-1)E=E_{1}(k-1)n_{1}+\ldots +E_{r}(k-1)n_{r}}$

and thus after shortening

${\displaystyle E=n_{1}E_{1}+\ldots +n_{r}E_{r}}$.

This is the desired expression for the property of the phase in the state : It is simply the sum of the partial molar sizes of the substances involved, multiplied by the respective amounts of substance. This relationship was used at the beginning of this article as an introductory explanation of the partial molar sizes and is derived here from the actual definition. ${\displaystyle E}$ ${\displaystyle (T,p,n_{1},\dots ,n_{r})}$ ${\displaystyle E_{i}}$

Both and those are only determined up to one constant. They must all be calculated with reference to the same reference state. ${\displaystyle E}$ ${\displaystyle E_{i}}$

Division by the total amount of substance in the phase gives the molar : ${\displaystyle n}$ ${\displaystyle E_{m}}$

${\displaystyle E_{m}={\frac {E}{n}}={\frac {n_{1}}{n}}E_{1}+\ldots +{\frac {n_{r}}{n}}E_{r}=x_{1}E_{1}+\ldots +x_{r}E_{r}}$

with the amount of substance . ${\displaystyle x_{i}=n_{i}/n}$

example

Let there be two miscible substances and , which in the pure state have the molar enthalpies and . Let their partial molar enthalpies in a mixture containing moles of substance and moles of substance be and . The total enthalpy of the two substances in the unmixed state is (because the enthalpies and extensive quantities) ${\displaystyle 1}$ ${\displaystyle 2}$ ${\displaystyle h_{1}}$ ${\displaystyle h_{2}}$ ${\displaystyle n_{1}}$ ${\displaystyle 1}$ ${\displaystyle n_{2}}$ ${\displaystyle 2}$ ${\displaystyle H_{1}}$ ${\displaystyle H_{2}}$ ${\displaystyle n_{1}h_{1}}$ ${\displaystyle n_{2}h_{2}}$

${\displaystyle n_{1}h_{1}+n_{2}h_{2}}$.

The enthalpy of the finished mixture is (according to the definition of the partial molar enthalpies and ) ${\displaystyle H_{1}}$ ${\displaystyle H_{2}}$

${\displaystyle n_{1}H_{1}+n_{2}H_{2}}$.

The enthalpy absorbed or released during the mixing process is the mixing enthalpy : ${\displaystyle \Delta H}$

${\displaystyle {\begin{aligned}\Delta H&=(n_{1}H_{1}+n_{2}H_{2})-(n_{1}h_{1}+n_{2}h_{2})\\&=n_{1}(H_{1}-h_{1})+n_{2}(H_{2}-h_{2})\end{aligned}}}$.

Division by the amount of substance in the mixed system gives the molar enthalpy of mixing : ${\displaystyle n_{1}+n_{2}}$

${\displaystyle \Delta h=x_{1}(H_{1}-h_{1})+x_{2}(H_{2}-h_{2})}$,

where and are the respective molar proportions. ${\displaystyle x_{1}}$ ${\displaystyle x_{2}}$

Which and which depend on temperature and pressure, they also depend on the proportions of the amount of substance in the mixture. With knowledge of these parameters, the molar enthalpy of mixing of the mixture can be calculated immediately, i.e. it can be predicted whether the mixing process will release or consume the heat of the mixture. ${\displaystyle h_{i}}$ ${\displaystyle H_{i}}$ ${\displaystyle H_{i}}$

Chemical potential

The partial molar volume and the partial molar enthalpy have already been mentioned as frequently occurring examples. Another often encountered partial molar quantity is the chemical potential .

As explained in the article on Gibbs energy , the Gibbs energy differential is as a function of its natural variables , and given by ${\displaystyle T}$ ${\displaystyle p}$ ${\displaystyle n_{1},\dots ,n_{r}}$

${\displaystyle {\mathrm {d} G(T,p,n_{1},\dots ,n_{r})=\left({\frac {\partial G}{\partial T}}\right)_{p,n_{1},\dots ,n_{r}}\mathrm {d} T+\left({\frac {\partial G}{\partial p}}\right)_{T,n_{1},\dots ,n_{r}}\mathrm {d} p\,+\,\sum _{i=1}^{r}\left({\frac {\partial G}{\partial n_{i}}}\right)_{T,p,n_{j\neq i}}\mathrm {d} n_{i}}}$.

The partial derivative occurring in each summand of the last term

${\displaystyle \left({\frac {\partial G}{\partial n_{i}}}\right)_{T,p,n_{j\neq i}}=:\mu _{i}}$

is also referred to as the chemical potential of the -th substance. Comparison with the definition of partial molar quantities shows that it is also the partial molar Gibbs energy of the th substance. It immediately follows that the Gibbs energy of a mixture is the sum of the chemical potentials multiplied by the respective amounts of substance: ${\displaystyle \mu _{i}}$ ${\displaystyle i}$ ${\displaystyle i}$

${\displaystyle G=n_{1}\mu _{1}+\ldots +n_{r}\mu _{r}}$,

and that the molar Gibbs energy of a mixture is the sum of the chemical potentials multiplied by the respective amount of substance:

${\displaystyle {\frac {G}{n}}={\frac {n_{1}}{n}}\mu _{1}+\ldots +{\frac {n_{r}}{n}}\mu _{r}=x_{1}\mu _{1}+\ldots +x_{r}\mu _{r}}$.

Among the thermodynamic potentials, these relationships only apply to the Gibbs energy, because it has the variables , and natural variables as the only potential , which are also used in the definition of the partial molar quantities. ${\displaystyle T}$ ${\displaystyle p}$ ${\displaystyle n_{1},\dots ,n_{r}}$

notation

The usual spelling as capital letters with an index for the respective substance,, corresponds to the recommendation of the IUPAC and allows the differentiation of the total size of the mixture as well as the molar property of the -th pure substance. But if a clarification is necessary, the IUPAC recommends the notation overline: . ${\displaystyle E_{i}}$ ${\displaystyle E}$ ${\displaystyle E_{\mathrm {m,i} }}$ ${\displaystyle i}$ ${\displaystyle {\overline {E}}_{i}}$

See also

• Excess volume

Web links

• Video: Partial molar sizes and Gibbs-Duhem relationship - How does a mixture change when 1 mol of component A is added? . Jakob Günter Lauth (SciFox) 2013, made available by the Technical Information Library (TIB), doi : 10.5446 / 15675 .

Individual evidence

1. ^ PW Atkins: Physical Chemistry. VCH, Weinheim 1990, 2nd repr. 1st edition, ISBN 3-527-25913-9 , p. 168
2. ↑ PW Atkins, J. de Paula: Physical chemistry. 5th edition, Wiley-VCH, Weinheim 2013, ISBN 978-3-527-33247-2 , p. 164
3. ^ PW Atkins: Physical Chemistry. VCH, Weinheim 1990, 2nd repr. 1st edition, ISBN 3-527-25913-9 , p. 176 (here specializing in water and alcohol)
4. ↑ ^{a b c d} K. Denbigh: The Principles of Chemical Equilibrium. 4th ed., Cambridge University Press, Cambridge 1981, ISBN 0-521-28150-4 , p. 101
5. ↑ Entry on partial molar quantity . In: IUPAC Compendium of Chemical Terminology (the “Gold Book”) . doi : 10.1351 / goldbook.P04418 Version: 2.3.3.
6. ↑ K. Denbigh: The Principles of Chemical Equilibrium. 4th ed., Cambridge University Press, Cambridge 1981, ISBN 0-521-28150-4 , pp. 101, 93 (The considerations there for U have been transferred to E.)
7. ↑ ^{a b c} K. Denbigh: The Principles of Chemical Equilibrium. 4th ed., Cambridge University Press, Cambridge 1981, ISBN 0-521-28150-4 , p. 102
8. ↑ K. Denbigh: The Principles of Chemical Equilibrium. 4th ed., Cambridge University Press, Cambridge 1981, ISBN 0-521-28150-4 , p. 104
9. ↑ entry to chemical potential, μ _B . In: IUPAC Compendium of Chemical Terminology (the “Gold Book”) . doi : 10.1351 / goldbook.C01032 Version: 2.3.3.
10. ↑ ^{a b} IUPAC, K.-H. Homann (Hrsg.), M. Hausmann (transl.): Quantities, units and symbols in physical chemistry. VCH, Weinheim 1996, ISBN 3-527-29326-4 , p. 52