The initial pressure and volume of an ideal gas are P$_0$ and V$_0$. The final pressure of the gas when the gas is suddenly compressed to volume $\frac{V_0}{4}$ will be :

(Given $\gamma$ = ratio of specific heats at constant pressure and at constant volume)

When the gas is compressed suddenly, it undergoes an adiabatic process where no heat is exchanged with the surroundings.<br/><br/> Therefore, we can use the adiabatic equation of state to relate the initial and final pressure and volume of the gas: $P_0V_0^\gamma=P_fV_f^\gamma$ where $P_f$ and $V_f$ are the final pressure and volume of the gas, respectively.<br/><br/> Since the gas is compressed to $\frac{V_0}{4}$, we have: $V_f=\frac{V_0}{4}$ Substituting this into the adiabatic equation of state, we get: $$P_f=P_0\left(\frac{V_0}{V_f}\right)^\gamma=P_0\left(\frac{4}{1}\right)^\gamma=4^\gamma P_0$$<br/><br/> Therefore, the final pressure of the gas when it is suddenly compressed to $\frac{V_0}{4}$ is $4^\gamma P_0$.