Question

In an adiabatic process, the temperature reduces to \( \frac{1}{4} \)th and volume increases to 8 times. Find the adiabatic constant of the gas.

• A. \( \frac{3}{4} \)
• B. \( \frac{5}{7} \)
• C. \( \frac{5}{3} \)
• D. \( \frac{8}{5} \)

Hint:

In an adiabatic process, the temperature and volume are related by the adiabatic constant \( \gamma \), which can be determined from the changes in temperature and volume.

Answer 1

The Correct Option is C

Step 1: Relation between temperature and volume
For an ideal gas undergoing an adiabatic change, temperature varies with volume according to a power-law relation involving the adiabatic constant γ.

Step 2: Substitute the given changes
It is given that the final temperature becomes one-fourth of the initial temperature, while the volume becomes eight times the initial volume.

Using these conditions, the temperature–volume relation becomes:

1/4 = 1 / (8)^{(γ − 1)}

Step 3: Solve for γ
Rewriting the equation:

8^{(γ − 1)} = 4

Since 8 = 2³ and 4 = 2²:

3(γ − 1) = 2

γ − 1 = 2/3
γ = 5/3

Final Answer:
γ = 5/3