Question

An ideal gas at \(27^{\circ}\)C is compressed adiabatically to \((\frac{8}{27})\) of its original volume. If \(γ = 5/3\), then the rise in temperature of a gas is

• A. \(300\) K
• B. \(675\) K
• C. \(375\) K
• D. \(27\) K

Hint:

Use T V^(gamma - 1) = constant.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
For an adiabatic change of an ideal gas, \(TV^{\gamma - 1}\) stays constant.

Step 2: Apply:
\(T_1 = 27 + 273 = 300\) K and \(V_2 = \dfrac{8}{27}V_1\), \(\gamma - 1 = \dfrac23\).
\[ \frac{T_2}{T_1} = \left(\frac{V_1}{V_2}\right)^{2/3} = \left(\frac{27}{8}\right)^{2/3} = \frac94 \]

Step 3: Final temperature and rise:
\(T_2 = 300\times\dfrac94 = 675\) K.
Rise in temperature = \(675 - 300 = 375\) K.

Step 4: Check:
Option (C). Option (B) 675 K is the final temperature itself, not the rise.

Final Answer:
T2 = 675 K, so the rise is 375 K. \[ \boxed{\text{(C) }375\ \text{K}} \]