Ideal solution

Formal definition

Different related definitions of an ideal solution have been proposed. The simplest definition is that an ideal solution is a solution for which each component obeys Raoult's law ${\displaystyle p_i=x_i{p}_i^{\ast }}$ for all compositions. Here ${\displaystyle p_i}$ is the vapor pressure of component ${\displaystyle i}$ above the solution, ${\displaystyle x_i}$ is its mole fraction and ${\displaystyle p_i^{\ast }}$ is the vapor pressure of the pure substance ${\displaystyle i}$ at the same temperature.

This definition depends on vapor pressure, which is a directly measurable property, at least for volatile components. The thermodynamic properties may then be obtained from the chemical potential μ (which is the partial molar Gibbs energy g) of each component. If the vapor is an ideal gas,

${\displaystyle \mu (T,p_i)=g(T,p_i)=g^{\mathrm{u}}(T,p^u)+RT\ln \frac{p_i}{p^u}.}$

The reference pressure ${\displaystyle p^u}$ may be taken as ${\displaystyle P^o}$ = 1 bar, or as the pressure of the mix, whichever is simpler.

On substituting the value of ${\displaystyle p_i}$ from Raoult's law,

${\displaystyle \mu (T,p_i)=g^{\mathrm{u}}(T,p^u)+RT\ln \frac{p_i^{\ast }}{p^u}+RT\ln x_i={\mu }_i^{\ast }+RT\ln x_i.}$

This equation for the chemical potential can be used as an alternate definition for an ideal solution.

However, the vapor above the solution may not actually behave as a mixture of ideal gases. Some authors therefore define an ideal solution as one for which each component obeys the fugacity analogue of Raoult's law ${\displaystyle f_i=x_i{f}_i^{\ast }}$. Here ${\displaystyle f_i}$ is the fugacity of component ${\displaystyle i}$ in solution and ${\displaystyle f_i^{\ast }}$ is the fugacity of ${\displaystyle i}$ as a pure substance. Since the fugacity is defined by the equation

${\displaystyle \mu (T,P)=g(T,P)=g^{\mathrm{u}}(T,p^u)+RT\ln \frac{f_i}{p^u}}$

this definition leads to ideal values of the chemical potential and other thermodynamic properties even when the component vapors above the solution are not ideal gases. An equivalent statement uses thermodynamic activity instead of fugacity.

Thermodynamic properties

Volume

If we differentiate this last equation with respect to ${\displaystyle p}$ at ${\displaystyle T}$ constant we get:

${\displaystyle {\left(\frac{\mathrm{\partial }g(T,P)}{\mathrm{\partial }P}\right)}_T=RT{\left(\frac{\mathrm{\partial }\ln f}{\mathrm{\partial }P}\right)}_T.}$

Since we know from the Gibbs potential equation that:

${\displaystyle {\left(\frac{\mathrm{\partial }g(T,P)}{\mathrm{\partial }P}\right)}_T=v}$

with the molar volume ${\displaystyle v}$, these last two equations put together give:

${\displaystyle {\left(\frac{\mathrm{\partial }\ln f}{\mathrm{\partial }P}\right)}_T=\frac{v}{RT}.}$

Since all this, done as a pure substance, is valid in an ideal mix just adding the subscript ${\displaystyle i}$ to all the intensive variables and changing ${\displaystyle v}$ to ${\displaystyle \overline{v_i}}$, with optional overbar, standing for partial molar volume:

${\displaystyle {\left(\frac{\mathrm{\partial }\ln f_i}{\mathrm{\partial }P}\right)}_{T,x_i}=\frac{\overline{v_i}}{RT}.}$

Applying the first equation of this section to this last equation we find:

${\displaystyle v_i^{\ast }={\overline{v}}_i}$

which means that the partial molar volumes in an ideal mix are independent of composition. Consequently, the total volume is the sum of the volumes of the components in their pure forms:

${\displaystyle V=\sum _i{V}_i^{\ast }.}$

Enthalpy and heat capacity

Proceeding in a similar way but taking the derivative with respect to ${\displaystyle T}$ we get a similar result for molar enthalpies:

${\displaystyle \frac{g(T,P)-g^{\mathrm{g}\mathrm{a}\mathrm{s}}(T,p^u)}{RT}=\ln \frac{f}{p^u}.}$

Remembering that ${\displaystyle {\left(\frac{\mathrm{\partial }\frac{g}{T}}{\mathrm{\partial }T}\right)}_P=-\frac{h}{T^2}}$ we get:

${\displaystyle -\frac{\overline{h_i}-h_i^{\mathrm{g}\mathrm{a}\mathrm{s}}}{R}=-\frac{h_i^{\ast }-h_i^{\mathrm{g}\mathrm{a}\mathrm{s}}}{R}}$

which in turn means that ${\displaystyle \overline{h_i}=h_i^{\ast }}$ and that the enthalpy of the mix is equal to the sum of its component enthalpies.

Since ${\displaystyle \overline{u_i}=\overline{h_i}-p\overline{v_i}}$ and ${\displaystyle u_i^{\ast }=h_i^{\ast }-p{v}_i^{\ast }}$, similarly

${\displaystyle u_i^{\ast }=\overline{u_i}.}$

It is also easily verifiable that

${\displaystyle C_{pi}^{\ast }=\overline{C_{pi}}.}$

Entropy of mixing

Finally since

${\displaystyle \overline{g_i}={\mu }_i=g_i^{\mathrm{g}\mathrm{a}\mathrm{s}}+RT\ln \frac{f_i}{p^u}=g_i^{\mathrm{g}\mathrm{a}\mathrm{s}}+RT\ln \frac{f_i^{\ast }}{p^u}+RT\ln x_i={\mu }_i^{\ast }+RT\ln x_i}$

we find that

${\displaystyle \mathrm{\Delta }{g}_{i,\mathrm{m}\mathrm{i}\mathrm{x}}=RT\ln x_i.}$

Since the Gibbs free energy per mole of the mixture ${\displaystyle G_m}$ is $${\displaystyle G_m=\sum _i{x}_i{g}_i}$$ then

${\displaystyle \mathrm{\Delta }{G}_{\mathrm{m},\mathrm{m}\mathrm{i}\mathrm{x}}=RT\sum _i{x}_i\ln x_i.}$

At last we can calculate the molar entropy of mixing since ${\displaystyle g_i^{\ast }=h_i^{\ast }-T{s}_i^{\ast }}$ and ${\displaystyle \overline{g_i}=\overline{h_i}-T\overline{s_i}}$

${\displaystyle \mathrm{\Delta }{s}_{i,\mathrm{m}\mathrm{i}\mathrm{x}}=-R\sum _i\ln x_i}$

${\displaystyle \mathrm{\Delta }{S}_{\mathrm{m},\mathrm{m}\mathrm{i}\mathrm{x}}=-R\sum _i{x}_i\ln x_i.}$

Consequences

Solvent–solute interactions are the same as solute–solute and solvent–solvent interactions, on average. Consequently, the enthalpy of mixing (solution) is zero and the change in Gibbs free energy on mixing is determined solely by the entropy of mixing. Hence the molar Gibbs free energy of mixing is

${\displaystyle \mathrm{\Delta }{G}_{\mathrm{m},\mathrm{m}\mathrm{i}\mathrm{x}}=RT\sum _i{x}_i\ln x_i}$

or for a two-component ideal solution

${\displaystyle \mathrm{\Delta }{G}_{\mathrm{m},\mathrm{m}\mathrm{i}\mathrm{x}}=RT(x_A\ln x_A+x_B\ln x_B)}$

where m denotes molar, i.e., change in Gibbs free energy per mole of solution, and ${\displaystyle x_i}$ is the mole fraction of component ${\displaystyle i}$. Note that this free energy of mixing is always negative (since each ${\displaystyle x_i\in [0,1]}$, each ${\displaystyle \ln x_i}$ or its limit for ${\displaystyle x_i\to 0}$ must be negative (infinite)), i.e., ideal solutions are miscible at any composition and no phase separation will occur.

The equation above can be expressed in terms of chemical potentials of the individual components

${\displaystyle \mathrm{\Delta }{G}_{\mathrm{m},\mathrm{m}\mathrm{i}\mathrm{x}}=\sum _i{x}_i\mathrm{\Delta }{\mu }_{i,\mathrm{m}\mathrm{i}\mathrm{x}}}$

where ${\displaystyle \mathrm{\Delta }{\mu }_{i,\mathrm{m}\mathrm{i}\mathrm{x}}=RT\ln x_i}$ is the change in chemical potential of ${\displaystyle i}$ on mixing. If the chemical potential of pure liquid ${\displaystyle i}$ is denoted ${\displaystyle {\mu }_i^{\ast }}$, then the chemical potential of ${\displaystyle i}$ in an ideal solution is

${\displaystyle {\mu }_i={\mu }_i^{\ast }+RT\ln x_i.}$

Any component ${\displaystyle i}$ of an ideal solution obeys Raoult's Law over the entire composition range:

${\displaystyle p_i=(p_i{)}_{\text{pure}}{x}_i}$

where ${\displaystyle (p_i{)}_{\text{pure}}}$ is the equilibrium vapor pressure of pure component ${\displaystyle i}$ and ${\displaystyle x_i}$is the mole fraction of component ${\displaystyle i}$ in solution.

Non-ideality

Deviations from ideality can be described by the use of Margules functions or activity coefficients. A single Margules parameter may be sufficient to describe the properties of the solution if the deviations from ideality are modest; such solutions are termed regular.

In contrast to ideal solutions, where volumes are strictly additive and mixing is always complete, the volume of a non-ideal solution is not, in general, the simple sum of the volumes of the component pure liquids and solubility is not guaranteed over the whole composition range. By measurement of densities, thermodynamic activity of components can be determined.