Question

An ideal gas with an adiabatic exponent 1.5, initially at 27°C is compressed adiabatically from 800 cc to 200 cc. The final temperature of the gas is:

• A. 700 K
• B. 500 K
• C. 250 K
• D. 600 K

Hint:

In adiabatic compression or expansion, use the relation \( T_1 V_1^{\gamma - 1} = T_2 V_2^{\gamma - 1} \) to calculate the final temperature.

Answer 1

The Correct Option is B

For an adiabatic process, the relationship between volume and temperature is defined by: \[T_1 V_1^{\gamma - 1} = T_2 V_2^{\gamma - 1}\] Given parameters: - \( \gamma = 1.5 \) (adiabatic exponent) - \( T_1 = 27^\circ \text{C} = 300 \, \text{K} \) (initial temperature) - \( V_1 = 800 \, \text{cc} \) (initial volume) - \( V_2 = 200 \, \text{cc} \) (final volume) - \( T_2 \) represents the final temperature. Substituting the provided values into the equation: \[300 \cdot 800^{1.5 - 1} = T_2 \cdot 200^{1.5 - 1}\] Simplification yields: \[300 \cdot 800^{0.5} = T_2 \cdot 200^{0.5}\] \[300 \cdot \sqrt{800} = T_2 \cdot \sqrt{200}\] \[300 \cdot 28.28 = T_2 \cdot 14.14\] Solving for \( T_2 \): \[T_2 = \frac{300 \cdot 28.28}{14.14} \approx 500 \, \text{K}\] The final temperature of the gas is approximately 500 K.