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: Adiabatic process relation.
For an adiabatic process, the relation between pressure, volume, and temperature is given by: \[ T V^{\gamma - 1} = \text{constant} \] where \( \gamma \) is the adiabatic constant. Step 2: Apply the given data.
The temperature reduces to \( \frac{1}{4} \)th and the volume increases by 8 times. Using the adiabatic equation, we get: \[ \left( \frac{1}{4} \right) (8)^{\gamma - 1} = 1 \] Solving for \( \gamma \), we get \( \gamma = \frac{5}{3} \). Step 3: Conclusion.
Thus, the adiabatic constant \( \gamma = \frac{5}{3} \). Final Answer: \[ \boxed{\frac{5}{3}} \]