# Change in Free Energy in Gas Reaction

I try to estimate whether particles form pairs or not.

What I know is the difference in the potential Energy $\Delta U_\mathrm{pot}$ which occurs when both particles form a pair.

When I have a system with $N$ particles, at constant volume and constant temperature the formation of pairs, should reduce the entropy of the system. So to see whether the pair formation occurs spontatnous I want to calculate the Gibbs Free Energy.

I assume that the particles behave like an ideal gas, but each particle consists of $n$ atoms has $(6n - 6)$ degrees of freedom.

The change in the gibbs free energy from system 1 ($N$ single particles, each consists of $n$ atoms) to system 2 (particles from $0.5N$ pairs, each pair consists of $2n$ atoms) I have:

$$\Delta G = \Delta U + V\Delta p - T\Delta S$$

The internal energy I can calculate from the equipartition theorem: \begin{align} \Delta U &= U_2 - U_1\\ &= U_{\mathrm{kin}, 2} - U_{\mathrm{kin}, 1} + \Delta U_\mathrm{pot}\\ &= 0.5 N \cdot (6n - 3)kT - N \cdot (3n - 3)kT + \Delta U_\mathrm{pot}\\ &= N(3n - 1.5)kT - N(3n - 3)kT + \Delta U_\mathrm{pot}\\ &= 1.5 N kT + \Delta U_\mathrm{pot} \end{align}

$V\Delta p$ I get from the ideal gas law \begin{align} V\Delta p &= \Delta N k T\\ V\Delta p &= 0.5 N k T \end{align}

Now my problem is $T \Delta S$. All I find is the the Sackur–Tetrode equation $$\frac{S}{kN} = \ln\left(\frac{V}{N\Lambda^3}\right)+\frac{5}{2}$$ which describes only the change the entropy of a monatomic ideal gas ($\Lambda$ is the thermal wavelength).

How can I calculate the entropy difference for the case of multiatomic ideal gas?

• Are you saying that you are assuming that the reaction goes to completion? Are you familiar with the concept of free energies of formation? Of chemical potential? Do you know how to determine the partial molar free energy of a species in an ideal gas mixture? – Chet Miller Jan 3 '18 at 0:15
• Yes, I want to compare both extreme cases, and therefore am assuming completion. I am aware of free energies of formation or the chemical potential, but these are hypothetical species, and therefore there exist no tabulated standard values for the compounds. – thepith Jan 5 '18 at 14:14
• If I would know the free energy of an multiatomic ideal gas (as a function of the number of degrees of freedom), I could of course directly calculate the change in the free energy. How does the molar free energy of a mixture help me, when I am considering the cases of full conversion? – thepith Jan 5 '18 at 14:22