One mole of an ideal gas goes through the process $p = p_0 \left[1 - \alpha \left(\frac{V}{V_0}\right)^3\right]$, where $p$ and $V$ are pressure and volume, $p_0$, $V_0$ and $\alpha$ are constants. If the maximum attainable temperature of the gas is $\frac{3}{4} \cdot \frac{p_0 V_0}{R}$, then the value of $\alpha$ is