At equilibrium , there is no change in chemical potential for the system: \[\sum_i n_i d\mu_i = 0 \label{eq1} \] This is the Gibbs-Duhem relationship and it places a compositional constraint upon any changes in the chemical potential in a mixture at constant temperature and pressure for a given composition. This result is easily derived when one considers that \(\mu_i\) represents the partial molar Gibbs function for component \(i\). And as with other partial molar quantities: \[ G_\text{tot} = \sum_i n_i \mu_i \nonumber \] Taking the derivative of both sides yields: \[ dG_\text{tot} = \sum_i n_i d \mu_i + \sum_i \mu_i d n_i \nonumber \] But \(dG\) can also be expressed as: \[dG = Vdp - sdT + \sum_i \mu_i d n_i \nonumber \] Setting these two expressions equal to one another: \[ \sum_i n_i d \mu_i + \cancel{ \sum_i \mu_i d n_i } = Vdp - sdT + \cancel{ \sum_i \mu_i d n_i} \nonumber \] And after canceling terms, one gets: \[ \sum_i n_i d \mu_i = Vdp - sdT \label{eq41} \] For a system at constant temperature and pressure: \[V~dp - s~dT = 0 \label{eq42} \] Substituting Equation \ref{eq42} into \ref{eq41} results in the Gibbs-Duhem equation (Equation \ref{eq1}). This expression relates how the chemical potential can change for a given composition while the system maintains equilibrium. Gibbs- Duhem for Binary Systems For a binary system consisting of components two components, \(A\) and \(B\): \[ n_Bd\mu_B + n_Ad\mu_A = 0 \nonumber \] Rearranging: \[ d\mu_B = -\dfrac{n_A}{n_B} d\mu_A \nonumber \] Consider a Gibbs free energy that only includes \(μ_n\) conjugate variables as we obtained it from our scaling experiment at \(T\) and \(P\) constant: \[G = \mu_An_A + \mu_Bn_B \nonumber \] Consider a change in \(G\): \[ \begin{align*} dG &= d(\mu_An_A) + d(\mu_Bn_B) \\[4pt] &= n_Ad\mu_A+\mu_Adn_A + n_Bd\mu_B+\mu_Bdn_B \end{align*} \] However, if we simply write out a change in \(G\) due to the number of moles we have: \[dG = \mu_A dn_A +\mu_B dn_B \nonumber \] Consequently the other terms must add up to zero: \[0 = n_Ad\mu_A+ n_Bd\mu_B \nonumber \] \[d\mu_A= - \dfrac{n_B}{n_A}d\mu_B \nonumber \] \[d\mu_A= - \dfrac{\chi_B}{\chi_A}d\mu_B \nonumber \] In the last step we have simply divided both denominator and numerator by the total number of moles. This expression is the Gibbs-Duhem equation for a 2-component system. It relates the change in one thermodynamic potential (\(d\mu_A\)) to the other (\(d\mu_B\)). The Gibbs-Duhem equation relates the change in one thermodynamic potential (\(d\mu_A\)) to the other (\(d\mu_B\)). Gibbs-Duhem in the Ideal Case In the ideal case we have: \[\mu_B = \mu^*_B + RT \ln x_B \nonumber \] Gibbs-Duhem gives: \[d\mu_A = - \dfrac{\chi_B}{\chi_A} d\mu_B \nonumber \] As: \[d\mu_B = 0 + \dfrac{RT}{\chi_B} \nonumber \] with \(x_B\) being the only active variable at constant temperature, we get: \[d\mu_A = - \dfrac{\chi_B}{\chi_A} \dfrac{RT}{\chi_B} = \dfrac{RT}{\chi_A} \nonumber \] If we now wish to find \(\mu_A\) we need to integrate \(d\mu_A\), e.g. form pure 1 to \(\chi_A\). This produces: \[\mu_A = \mu^*_A + RT \ln \chi_A \nonumber \] This demonstrates that Raoult's law can only hold over the whole range for one component if it also holds for the other over the whole range.