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:


\begin{equation}
\Delta G = \Delta U + V\Delta p - T\Delta S
\end{equation}


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
\begin{equation}
\frac{S}{kN} = \ln\left(\frac{V}{N\Lambda^3}\right)+\frac{5}{2}
\end{equation}
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?


