I am having some confusion with something my professor has written and would like to have this issue cleared up.

I'll start from the beginning, I understand all this, I'll mention where it stops making sense. So, to determine the excess Gibbs free energy we first look at the definition of Gibbs free energy:

$$ G = \sum_{\alpha} \mu_{\alpha}N_{\alpha} $$ All good...expanding this out: $$ G = \sum_{\alpha} N_{\alpha}[\mu^o_{\alpha} + RT\ln(x_{\alpha} \gamma_{\alpha})] $$ All good, let's split this up into ideal and 'excess' non ideal bits: $$ G = \underbrace{\sum_{\alpha} N_{\alpha}[\mu^o_{\alpha} + RT\ln(x_{\alpha})]}_{\text{Ideal}} + \underbrace{RT\sum_{\alpha} N_{\alpha}\ln( \gamma_{\alpha})}_{\text{Excess}} $$ Ok so now there's a function for excess Gibbs free energy, writing this in partial molar form (dividing both sides by total number of mols): $$ g^E= RT\sum_{\alpha} x_{\alpha}\ln( \gamma_{\alpha}) $$ Writing this for two components: $$ \frac{g^E}{RT}= x_1\ln(\gamma_{1}) + x_2\ln(\gamma_{2}) $$

Ok, so next move. We're given an excess function as $Ax_1x_2$..fine, I know where that's from but then he goes and writes:

\begin{equation} g^E = \frac{An_1n_2}{n_1 + n_2} \end{equation} I am convinced he has just made this up, if we look at the correct function $Ax_1x_2$, surely the excess energy will be given by:

$$ g^E = \frac{An_1n_2}{(n_1+n_2)^2} $$ Because $x_1 = \frac{n_1}{n_1+n_2}$ and similar for $n_2$.

Could someone please advice where he got this from. Thank you.


I think I figured out my issue, he doesn't know what he's doing.

The notation $g^E$ is actually used for total Gibbs excess energy, for example:

$$ g^E = RTn_1\ln(\gamma_1) + RTn_2\ln(\gamma_2) $$

Therefore it makes sense to write something like: $$ \frac{\partial g^E}{\partial n_i} = RT\ln(\gamma_i) $$

Furthermore, if the PARTIAL MOLAR Gibbs free energy is given as $Ax_1x_2$ then the TOTAL MOLAR Gibbs free energy is simply: $$ g^E = G^E\times N = \frac{n_1n_2}{(n_1+n_2)^2} \times (n_1 + n_2) = \frac{n_1n_2}{(n_1+n_2)} $$


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