I am trying to find the Gibbs-energy equilibrium for the reaction $$\mathrm{A}+\mathrm{B}\rightleftharpoons\mathrm{C}.$$ According to most sources the equilibrium concentrations in an ideal solution should form a ratio $$\Delta G^0=-RT\ln\frac{[\mathrm{C}]}{[\mathrm{A}][\mathrm{B}]}.$$
I am having trouble seeing this when the stoichiometric coefficients on either side of the reaction are unequal, as in this example above.
The Gibbs energy for the solution should be $$G=n_A(\mu_A^0+RT\ln[\mathrm{A}])+n_B(\mu_B^0+RT\ln[\mathrm{B}])+n_C(\mu_C^0+RT\ln[\mathrm{C}]).$$
Assuming we started with a 1 L solution containing 1 mol of A and 1 mol of B, $\xi$ mol of each react to form $\xi$ mol of C, this equation should then be $$G=(1-\xi)(\mu_A^0+RT\ln(1-\xi))+(1-\xi)(\mu_B^0+RT\ln(1-\xi))+\xi(\mu_C^0+RT\ln\xi).$$
My understanding is that if the Gibbs free energy is at its minimum the reaction above will not be spontaneous in either direction and so the solution will be at equilibrium. The derivative of the above equation gives $$0=-(\mu_A^0+RT\ln(1-\xi))-RT-(\mu_B^0+RT\ln(1-\xi))-RT+(\mu_C^0+RT\ln\xi)+RT.$$
Some rearrangements result in $$\Delta G^0=-RT\ln\frac{[\mathrm{C}]}{[\mathrm{A}][\mathrm{B}]}+RT.$$
In other words, my expression is off by $\Delta nRT$. Having been unable to find any help on this matter on the internet, I feel I have either made some silly logical/mathematical mistake or have some fundamental misunderstanding. Please point out the error.