Recall that:

$H \equiv U + PV$

If we take the derivative of both sides with respect to the number of moles of species $j$:

$\left (\frac{\partial H}{\partial n_j} \right )_{P, T, n_{i\ne j}} \equiv \left (\frac{\partial U}{\partial n_j} \right )_{P, T, n_{i\ne j}} + \left (\frac{\partial (PV)}{\partial n_j} \right )_{P, T, n_{i\ne j}}$

since P is a constant

$\left (\frac{\partial H}{\partial n_j} \right )_{P, T, n_{i\ne j}} \equiv \left (\frac{\partial U}{\partial n_j} \right )_{P, T, n_{i\ne j}} + P\left (\frac{\partial V}{\partial n_j} \right )_{P, T, n_{i\ne j}}$

using the definition of partial molar properties, we note

$\bar H_i \equiv \bar U_i + P\bar V_i$

This procedure works for any of the derived property definitions (i.e., G or A also).

Using the Duhem relation, we know that we can write a total differential for $G$ in terms of $T, P,$ and $n_j$ as follows:

$dG = \left (\frac{\partial G}{\partial P} \right )_{T, n_{i}} dP + \left (\frac{\partial G}{\partial T} \right )_{P, n_{i}} dT + \sum_j \left (\frac{\partial G}{\partial n_j} \right )_{P, T, n_{i\ne j}} dn_j$

From analysis of pure substance systems, we already had written

$dG = -SdT +VdP$

so we can combine these to get (recognize that we have invoked the definition of the partial molar $\bar G_i$):

$dG = -SdT +VdP + \sum_j \bar G_j dn_j$

Recalling from our Maxwell relations, we showed that

$-S = \left (\frac{\partial G}{\partial T} \right
)_{P}$

$V = \left (\frac{\partial G}{\partial P} \right )_{T}$

Taking the derivative with respect to the number of moles of species $j$ for these relations:

$-\left (\frac{\partial S}{\partial n_j} \right ) =
\frac{\partial}{\partial n_j}\left (\frac{\partial
G}{\partial T} \right )_{P}$

$\left (\frac{\partial V}{\partial n_j} \right ) =
\frac{\partial}{\partial n_j}\left (\frac{\partial
G}{\partial P} \right )_{T}$

Invoking the definition of the partial molar Gibbs free energy, and taking either the derivative with respect to T or P of that expression we get:

$\left (\frac{\partial \bar G_i}{\partial T} \right )_{P}
= \left (\frac{\partial}{\partial T} \left [\frac{\partial
G}{\partial n_j} \right ] \right )_P$

$\left (\frac{\partial \bar G_i}{\partial P} \right )_{T} =
\left (\frac{\partial}{\partial P} \left [\frac{\partial
G}{\partial n_j} \right ] \right )_T$

Noting that the right hand sides of the top equations in the last two groups are the same (as are the bottom two), we can write:

$-\bar S_i = \left (\frac{\partial \bar G_i}{\partial T}
\right )_{P}$

$\bar V_i = \left (\frac{\partial \bar G_i}{\partial P}
\right )_{T}$

The Maxwell relations (and all other fundamental property relations) can be re-written in terms of partial molar properties. This second expression in particular is of utility in calculations as it relates $\bar G_i$ to measurable properties.