JEE MAIN - Physics (2023 - 13th April Evening Shift - No. 4)
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)
P$$_0$$(4)$$^{\frac{1}{\gamma}}$$
P$$_0$$
4P$$_0$$
P$$_0$$(4)$$^{\gamma}$$
Explanation
When the gas is compressed suddenly, it undergoes an adiabatic process where no heat is exchanged with the surroundings.
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.
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: $$P_f=P_0\left(\frac{V_0}{V_f}\right)^\gamma=P_0\left(\frac{4}{1}\right)^\gamma=4^\gamma P_0$$
Therefore, the final pressure of the gas when it is suddenly compressed to $$\frac{V_0}{4}$$ is $$4^\gamma P_0$$.
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.
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: $$P_f=P_0\left(\frac{V_0}{V_f}\right)^\gamma=P_0\left(\frac{4}{1}\right)^\gamma=4^\gamma P_0$$
Therefore, the final pressure of the gas when it is suddenly compressed to $$\frac{V_0}{4}$$ is $$4^\gamma P_0$$.
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