# Ideal solution

In chemistry, an ideal solution or ideal mixture is a solution with thermodynamic properties analogous to those of a mixture of ideal gases. The enthalpy of solution (or "enthalpy of mixing") is zero[1] as is the volume change on mixing; the closer to zero the enthalpy of solution is, the more "ideal" the behavior of the solution becomes. The vapor pressure of the solution obeys Raoult's law, and the activity coefficient of each component (which measures deviation from ideality) is equal to one.[2]

The concept of an ideal solution is fundamental to chemical thermodynamics and its applications, such as the use of colligative properties.

## Physical origin

Ideality of solutions is analogous to ideality for gases, with the important difference that intermolecular interactions in liquids are strong and cannot simply be neglected as they can for ideal gases. Instead we assume that the mean strength of the interactions are the same between all the molecules of the solution.

More formally, for a mix of molecules of A and B, the interactions between unlike neighbors (UAB) and like neighbors UAA and UBB must be of the same average strength, i.e., 2 UAB = UAA + UBB and the longer-range interactions must be nil (or at least indistinguishable). If the molecular forces are the same between AA, AB and BB, i.e., UAB = UAA = UBB, then the solution is automatically ideal.

If the molecules are almost identical chemically, e.g., 1-butanol and 2-butanol, then the solution will be almost ideal. Since the interaction energies between A and B are almost equal, it follows that there is a very small overall energy (enthalpy) change when the substances are mixed. The more dissimilar the nature of A and B, the more strongly the solution is expected to deviate from ideality.

## 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 (i) obeys Raoult's law $p_i=x_ip_i^*$ for all compositions. Here $p_i$ is the vapor pressure of component i above the solution, $x_i$ is its mole fraction and $p_i^*$ is the vapor pressure of the pure substance i at the same temperature.[3][4][5]

This definition depends on vapor pressures which are a directly measurable property, at least for volatile components. The thermodynamic properties may then be obtained from the chemical potential μ (or partial molar Gibbs energy g) of each component, which is assumed to be given by the ideal gas formula

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

The reference pressure $p^u$ may be taken as $P^0$ = 1 bar, or as the pressure of the mix to ease operations.

On substituting the value of $p_i$ from Raoult's law,

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

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 $f_i=x_if_i^*$,

Here $f_i$ is the fugacity of component $i$ in solution and $f_i^*$ is the fugacity of $i$ as a pure substance.[6][7] Since the fugacity is defined by the equation

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

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.[8]

## Thermodynamic properties

### Volume

If we differentiate this last equation with respect to $P$ at $T$ constant we get:

$\left(\frac{\partial g(T,P)}{\partial P}\right)_{T}=RT\left(\frac{\partial \ln f}{\partial P}\right)_{T}$

but we know from the Gibbs potential equation that:

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

These last two equations put together give:

$\left(\frac{\partial \ln f}{\partial P}\right)_{T}=\frac{v}{RT}$

Since all this, done as a pure substance is valid in a mix just adding the subscript $i$ to all the intensive variables and changing $v$ to $\bar{v_i}$, standing for Partial molar volume.

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

Applying the first equation of this section to this last equation we get

$v_i^*=\bar{v_i}$

which means that in an ideal mix the volume is the addition of the volumes of its components.

### Enthalpy and heat capacity

Proceeding in a similar way but derivative with respect of $T$ we get to a similar result with enthalpies

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

derivative with respect to T and remembering that $\left( \frac{\partial \frac{g}{T}}{\partial T}\right)_P=-\frac{h}{T^2}$ we get:

$-\frac{\bar{h_i}-h_i^\mathrm{gas}}{R}=-\frac{h_i^*-h_i^\mathrm{gas}}{R}$

which in turn is $\bar{h_i}=h_i^*$.

Meaning that the enthalpy of the mix is equal to the sum of its components.

Since $\bar{u_i}=\bar{h_i}-p\bar{v_i}$ and $u_i^*=h_i^*-pv_i^*$:

$u_i^*=\bar{u_i}$

It is also easily verifiable that

$C_{pi}^*=\bar{C_{pi}}$

### Entropy of mixing

Finally since

$\bar{g_i}=\mu _i=g_i^\mathrm{gas}+RT\ln \frac{f_i}{p^u}=g_i^\mathrm{gas}+RT\ln \frac{f_i^*}{p^u}+RT\ln x_i=\mu _i^*+ RT\ln x_i$

Which means that

$\Delta g_{i,\mathrm{mix}}=RT\ln x_i$

and since

$G=\sum_i x_i{g_i}$

then

$\Delta G_\mathrm{mix}=RT\sum_i{x_i\ln x_i}$

At last we can calculate the entropy of mixing since $g_i^*=h_i^*-Ts_i^*$ and $\bar{g_i}=\bar{h_i}-T\bar{s_i}$

$\Delta s_{i,\mathrm{mix}}=-R\sum _i \ln x_i$
$\Delta S_\mathrm{mix}=-R\sum _i x_i\ln x_i$

## Consequences

Solvent-Solute interactions are similar to solute-solute and solvent-solvent interactions

Since the enthalpy of mixing (solution) is zero, 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

$\Delta G_{\mathrm{m,mix}} = RT \sum_i x_i \ln x_i$

or for a two component solution

$\Delta G_{\mathrm{m,mix}} = RT (x_A \ln x_A + x_B \ln x_B)$

where m denotes molar, i.e., change in Gibbs free energy per mole of solution, and $x_i$ is the mole fraction of component $i$.

Note that this free energy of mixing is always negative (since each $x_i$ is positive and each $\ln x_i$ must be negative), i.e., ideal solutions are always completely miscible.

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

$\Delta G_{\mathrm{m,mix}} = \sum_i x_i \Delta\mu_{i,\mathrm{mix}}$

where $\Delta\mu_{i,\mathrm{mix}}=RT\ln x_i$ is the change in chemical potential of $i$ on mixing.

If the chemical potential of pure liquid $i$ is denoted $\mu_i^*$, then the chemical potential of $i$ in an ideal solution is

$\mu_i = \mu_i^* + \Delta \mu_{i,\mathrm{mix}} = \mu_i^* + RT \ln x_i$

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

$\ P_{i}=(P_{i})_{pure} x_i$

where

$(P_i)_{pure}\,$ is the equilibrium vapor pressure of the pure component
$x_i\,$ is the mole fraction of the component in solution

It can also be shown that volumes are strictly additive for ideal solutions.

## 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.