= Solution
The <stellar adiabatic exponent> $\Gamma_2$ is defined at fixed <specific entropy> and composition by
$$
\boxed{\frac{\Gamma_2}{\Gamma_2-1}=\left(\frac{\partial\log P}{\partial\log T}\right)_s,\qquad \nabla_{\rm ad}=\frac{\Gamma_2-1}{\Gamma_2}.}
$$
Combining parts (iii) and (iv), the <radiative temperature gradient> of the constant-opacity <grey atmosphere> is
$$
\nabla_{\rm rad}=\frac{d\log T/d\tau}{d\log P/d\tau}=\frac{3\tau}{8+12\tau}\longrightarrow\frac14.
$$
For $1<\Gamma_2<4/3$, the <adiabatic temperature gradient> is strictly below $1/4$, and the <Schwarzschild criterion> is eventually violated. If $\Gamma_2$ is constant, solving for the crossing gives
$$
\boxed{\tau>\frac{8(\Gamma_2-1)}{12-9\Gamma_2}\quad\Longrightarrow\quad\text{convective instability}.}
$$
At $\Gamma_2=4/3$ the radiative gradient approaches marginality from below, so there is no finite crossing in this model. If $\Gamma_2$ varies, the conclusion requires its subcritical value to persist in the deep layers. Uniform composition is needed for this use of the <Schwarzschild criterion>; a composition gradient requires the <Ledoux criterion>.
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