Editing Ideal solution (section)

=== Volume ===
If we differentiate this last equation with respect to <math>p</math> at <math>T</math> constant we get:
:<math>\left(\frac{\partial g(T,P)}{\partial P}\right)_{T}=RT\left(\frac{\partial \ln f}{\partial P}\right)_{T}.</math>
Since we know from the Gibbs potential equation that:
:<math>\left(\frac{\partial g(T,P)}{\partial P}\right)_{T}=v</math>

with the molar volume <math>v</math>, these last two equations put together give:
:<math>\left(\frac{\partial \ln f}{\partial P}\right)_{T}=\frac{v}{RT}.</math>

Since all this, done as a pure substance, is valid in an ideal mix just adding the subscript <math>i</math> to all the [[intensive variable]]s and changing <math>v</math> to <math>\bar{v_i}</math>, with optional overbar, standing for [[partial molar volume]]:

:<math>\left(\frac{\partial \ln f_i}{\partial P}\right)_{T,x_i}=\frac{\bar{v_i}}{RT}.</math>

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

:<math>v_i^* = \bar{v}_i</math>
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:
:<math>V = \sum_i V_i^*.</math>