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 relation between volume and temperature is given by: \[ T_1 V_1^{\gamma - 1} = T_2 V_2^{\gamma - 1} \] Where: - \( \gamma = 1.5 \) (adiabatic exponent) - \( T_1 = 27^\circ \text{C} = 300 \, \text{K} \) - \( V_1 = 800 \, \text{cc} \) - \( V_2 = 200 \, \text{cc} \) - \( T_2 \) is the final temperature. Substituting the values into the equation: \[ 300 \cdot 800^{1.5 - 1} = T_2 \cdot 200^{1.5 - 1} \] Simplifying: \[ 300 \cdot 800^{0.5} = T_2 \cdot 200^{0.5} \] \[ 300 \cdot \sqrt{800} = T_2 \cdot \sqrt{200} \] \[ 300 \cdot 28.28 = T_2 \cdot 14.14 \] \[ T_2 = \frac{300 \cdot 28.28}{14.14} \approx 500 \, \text{K} \] Thus, the final temperature of the gas is 500 K.