Ideal solution

In chemistry, an ideal solution or ideal mixture is a solution in which the enthalpy of solution (or "enthalpy of mixing") is zero; the closer to zero the enthalpy of solution is, the more "ideal" the behavior of the solution becomes. Equivalently, an ideal mixture is one in which the activity coefficients (which measure deviation from ideality) are equal to one.

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 can not 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 (U_AB) and like neighbors U_AA and U_BB must be of the same average strength i.e. 2U_AB=U_AA+ U_BB 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. U_AB=U_AA=U_BB, 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 ideal. Since the interaction energies between A and B are the same, it follows that there is no 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

An ideal mixture is defined as a mixture that satisfies:

$f_{i}=x_{i}f_{i}^{*}$

where $f_{i}$ is the fugacity of component $i$ and $f_{i}^{*}$ is the fugacity of $i$ as a pure substance.

Since the definition of fugacity in a pure substance is:

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

Where $g^{\mathrm {gas} }(T,p^{u})$ is the molar free energy of an ideal gas at a temperature $T$ and a reference pressure $p^{u}$ which might be taken as $P^{0}$ or the pressure of the mix to ease operations.

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.

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 ang 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

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

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

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.

See also

• Activity coefficient
• Entropy of mixing
• Margules function
• Regular solution