Question

A gas is suddenly expanded such that its final volume becomes 3 times its initial volume. If the specific heat at constant volume of the gas is 2R, then the ratio of initial to final pressures is nearly equal to

• A. 5
• B. 6.5
• C. 7
• D. 3.5

Hint:

A gas is suddenly expanded such that its final volume becomes 3 times its initial volume. If the specific heat at constant volume of the gas is 2R, then the ratio of initial to final pressures is nearly equal to

Answer 1

The Correct Option is A

To solve the problem, we need to understand the thermodynamics of a sudden expansion. The process is likely adiabatic since it happens suddenly, meaning no heat is exchanged with the surroundings.

The adiabatic process is governed by:
\(P_1 V_1^\gamma = P_2 V_2^\gamma\)

Given:

• Final volume, \(V_2 = 3V_1\)
• Specific heat at constant volume, \(C_v = 2R\)

 

The adiabatic index \(\gamma\) is given by:
\(\gamma = \frac{C_p}{C_v}\)

Using the relation \(C_p = C_v + R\), we have:
\(\gamma = \frac{C_v + R}{C_v} = \frac{2R + R}{2R} = \frac{3}{2}\)

Now apply the adiabatic condition:
\(P_1 V_1^{\frac{3}{2}} = P_2 (3V_1)^{\frac{3}{2}}\)

Simplifying, we get:
\(P_1 = P_2 \cdot 3^{\frac{3}{2}}\)

Hence, the ratio of initial to final pressures is:
\(\frac{P_1}{P_2} = 3^{\frac{3}{2}} = 3 \times \sqrt{3} \approx 5.2\)

Considering significant options, the closest answer is 5.

Thus, the ratio of initial to final pressures is nearly equal to 5.