Question

The ratio of specific heat capacities at constant pressure to that at constant volume for a given mass of a gas is \(\frac{5}{2}\). If the percentage increase in volume of the gas while undergoing an adiabatic change is \(\frac{3}{2}\), then the percentage decrease in pressure will be:

• A. \( \frac{15}{4} \)
• B. \( \frac{3}{5} \)
• C. \( \frac{4}{5} \)
• D. \( \frac{5}{3} \)

Hint:

In adiabatic processes, use \( \frac{\Delta P}{P} = -\gamma \frac{\Delta V}{V} \) for quick percentage change calculations.

Answer 1

The Correct Option is A

Step 1: Identify adiabatic relation.
For an adiabatic process:
\[ PV^\gamma = \text{constant} \]
where \( \gamma = \frac{C_p}{C_v} \).

Step 2: Given value of \(\gamma\).
\[ \gamma = \frac{5}{2} \]

Step 3: Use logarithmic form for small changes.
\[ \frac{\Delta P}{P} + \gamma \frac{\Delta V}{V} = 0 \]
\[ \frac{\Delta P}{P} = -\gamma \frac{\Delta V}{V} \]

Step 4: Substitute given change in volume.
Percentage increase in volume = \( \frac{3}{2} = 1.5 \)
\[ \frac{\Delta P}{P} = -\frac{5}{2} \times \frac{3}{2} \]

Step 5: Calculate percentage decrease.
\[ \frac{\Delta P}{P} = -\frac{15}{4} \]

Step 6: Interpret result.
Negative sign indicates decrease in pressure. Magnitude of decrease:
\[ \frac{15}{4} \]

Step 7: Final conclusion.
\[ \boxed{\frac{15}{4}} \] Hence, correct answer is option (A).