Thermodynamics problem -Does this approach seem right?

[TroyElliott]

Homework Statement

A dilute gas consisting of N hydrogen atoms in equilibrium at temperature T and pressure P. A fraction of the atoms combine to form diatomic hydrogen. For $N_{s}$ single atoms and $N_{d}$ diatomic molecules, the free energy of the system is

$$G = N_{s}k_{b}T\ln{(\frac{N_{s}}{N_{s}+N_{d}}\frac{P}{P^{0}_{s}})}+N_{d}k_{b}T\ln{(\frac{N_{d}}{N_{s}+N_{d}}\frac{P}{P^{0}_{d}})} - \epsilon N_{d}.$$

Here $\epsilon$ is the binding energy of the $H_{2}$ molecule, and $P^{0}_{s}$ and $P^{0}_{d}$ are functions of only temperature. Find the relation between $N_{s}$ and $N_{d}$ in equilibrium.

Homework Equations

$\Delta G = 0$
$G_{initial} = N_{s}k_{b}T\ln{(\frac{P}{P^{0}_{s}})}?$

The Attempt at a Solution

$\Delta G = G_{final}-G_{initial} = 0.$ I assume that the $G_{final}$ is the formula for the free energy given in the problem statement. Is it right to assume that $G_{initial}$ is given by setting $N_{d}$ to zero? I am assuming that initially all the hydrogen was just made up of single atoms and the free energy formula above would still be valid for this. Does this seem like the right approach? The algebra gets messy and I haven't been able to get a clean relationship between $N_{s}$ and $N_{d}$.

Thanks!

 

[Chestermiller]

No. I don't agree with your approach. You have that, in any state, $$2N_d+N_s=N$$ You need to find the value of $N_s$ that minimizes G.