What is the most entropic ideal-gas binary mixture?

5.4 What is the most entropic ideal-gas binary mixture?

For a binary mixture we have \(n_A + n_B = n\), \(x_A = n_A / n\), and \(x_B = n_B / n = 1 - x_A\). Thus Eq. (5.5) translates into

\begin{equation*} \begin{split} \Delta_{mix} S_m^{id} &=\;\; - R \left( x_A \ln x_A + x_B \ln x_B \right) = - R \left( x_A \ln x_A + (1-x_A) \ln (1-x_A) \right)\\ \mbox{max} \;\Rightarrow 0 = \frac{d}{dx_A} \Delta_{mix} S_m^{id} &=\;\; - R \left( \ln x_A + 1 - \ln (1-x_A) - 1 \right) = - R \ln \left( \frac{x_A}{1-x_A}\right)\\ \end{split} \label{eq:DmixSm_ideal} \end{equation*}

(5.8)

This is only fulfilled for \(x_A = x_B = 0.5\) which is the only maximum. This result is quite intuitive because a 50% mixture allows for the most efficient disorder of particle arrangement. Since

\begin{equation*} \Delta_{mix} G_m^{id} = - T \Delta_{mix} S_m^{id} \label{eq:DmixGm_ideal} \end{equation*}

(5.9)

the minimum of the Gibbs potential is found for \(x_A = x_B = 0.5\) as well and it is more pronounced for higher temperature.