# Mixed Gas Absorbtion

Consider a gas misture that contains two type of atom, A and B. The gas is in equilibrium at temperature $T$. If on the surface of the gas container there are M sites that can absorb an atom gas, and if i know $m_A$, $m_B$ and the partial pressure $P_A$, $P_B$ how can I compute the fraction of sites that adsorb A or B. The bond energy of absorbed atom are respectly $-\epsilon_a$ and $-\epsilon_b$ ($\epsilon_a,\ \epsilon_b >0$).

My attempt:

From grand-canonical partition function: \begin{equation} Z(z,V,T) = (1+e^{\beta(\epsilon_a+\mu)}+e^{\beta(\epsilon_b+\mu)})^M \end{equation} I can compute the mean number of absorbed atom per site: \begin{equation} \frac{\langle N_{ad}\rangle }{M}= z\frac{\partial}{\partial z}\log Z(z,V,T) = \frac{e^{\beta(\epsilon_a+\mu)}+e^{\beta(\epsilon_b+\mu)}}{1+e^{\beta(\epsilon_a+\mu)}+e^{\beta(\epsilon_b+\mu)}} \end{equation} Now I can't figure out how compute the fraction of absorbed atom for each type: \begin{equation} \frac{\langle N_{ad}^A\rangle}{\langle N\rangle} \end{equation} where: \begin{equation} \langle N_{ad}^A\rangle = \frac{e^{\beta(\epsilon_a+\mu)}}{1+e^{\beta(\epsilon_a+\mu)}+e^{\beta(\epsilon_b+\mu)}} \end{equation}