Question

An ideal gas at $27^\circ\text{C}$ is compressed adiabatically to $\left(\frac{8}{27}\right)$ of its original volume. If the ratio of specific heats, $\gamma = \frac{5}{3}$ then the rise in temperature of the gas is

• A. $500\text{ K}$
• B. $125\text{ K}$
• C. $250\text{ K}$
• D. $375\text{ K}$

Hint:

Always read carefully to see whether a question asks for the final temperature ($675\text{ K}$) or the rise in temperature ($675 - 300 = 375\text{ K}$). Under time pressure, it is easy to accidentally pick option (A) because $500\text{ K}$ looks like a round number, or misread the final temperature value!

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
An ideal gas at 27°C is adiabatically compressed to 8/27 of its original volume (γ = 5/3); find the temperature rise ΔT.

Step 2: Key Formula or Approach:
Adiabatic relation: T₁V₁^(γ–1) = T₂V₂^(γ–1). Convert T₁ to Kelvin (27+273=300 K). γ–1 = 2/3.

Step 3: Detailed Explanation:
T₂/T₁ = (V₁/V₂)^(2/3) = (27/8)^(2/3) = [(3/2)³]^(2/3) = 9/4. T₂ = 300 × 9/4 = 675 K. ΔT = 675 – 300 = 375 K.

Step 4: Final Answer:
Temperature rise is 375 K, matching option (D).