= Solution
{c}
The low-temperature bond-length difference is $\epsilon_0=2.05-1.98=0.07\,\text{\AA}$. With the question's thermal criterion,
$$
E_{\mathrm{JT}}\simeq k_BT_{\mathrm{JT}}=(8.6173\times10^{-5}\,\mathrm{eV\,K^{-1}})(200\,\mathrm K)=0.01723\,\mathrm{eV}.
$$
Use $\alpha=\beta\epsilon_0$ and $E_{\mathrm{JT}}=\beta\epsilon_0^2/2$ to obtain
$$
\boxed{\alpha=\frac{2E_{\mathrm{JT}}}{\epsilon_0}\simeq0.492\,\mathrm{eV\,\text{\AA}^{-1}},\qquad\beta=\frac{2E_{\mathrm{JT}}}{\epsilon_0^2}\simeq7.03\,\mathrm{eV\,\text{\AA}^{-2}}.}
$$
In SI units these are about $7.89\times10^{-10}\,\mathrm N$ and $113\,\mathrm{N\,m^{-1}}$, respectively. They are estimates within the stipulated energy criterion; a cooperative transition also involves interactions and entropy, so its temperature is not in general exactly an isolated-ion stabilization energy divided by $k_B$.
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