Question

The initial pressure and volume of an ideal gas are P₀ and V₀. The final pressure of the gas when the gas is suddenly compressed to volume \(\frac{V_0}{4}\) will be(Given γ = ratio of specific heats at constant pressure and at constant volume)

• A. \(P_0\)
• B. \(4P_0\)
• C. \(P_0(4)^y\)
• D. \(P_0(4)^{\frac{1}{y}}\)

Answer 1

The Correct Option is C

As the gas is suddenly compressed, the process is adiabatic. The equation for the gas in an adiabatic process is: \[ P V^\gamma = \text{constant}. \] For the initial state: \[ P_0 V_0^\gamma = P_2 \left(\frac{V_0}{4}\right)^\gamma. \] Rearranging to solve for \( P_2 \): \[ P_2 = P_0 \cdot \frac{V_0^\gamma}{\left(\frac{V_0}{4}\right)^\gamma}. \] Simplify the denominator: \[ P_2 = P_0 \cdot \frac{V_0^\gamma}{\frac{V_0^\gamma}{4^\gamma}} = P_0 \cdot 4^\gamma. \] Thus, the final pressure is: \[ P_2 = P_0 (4)^\gamma. \] Hence, the correct answer is \( \boxed{P_0 (4)^\gamma} \).