Question

A piston–cylinder device initially contains $1 \, m^3$ of air at $200 \, kPa$ and $25^\circ C$. Air expands at constant pressure while a heater of $250 \, W$ is switched on for 10 minutes. There is a heat loss of $4 \, kJ$. Assuming air as an ideal gas, the final temperature of air is $____________________$ $^\circ C$ (rounded off to one decimal place). Given: $R = 0.287 \, kJ/kgK$, $c_p = 1.005 \, kJ/kgK$.

Hint:

In constant pressure heating, always use $Q = m c_p \Delta T$. Don’t forget to subtract heat losses to get $Q_{net}$.

Answer 1

Correct Answer: 85

Step 1: Mass of air. Using ideal gas law: \[ m = \frac{PV}{RT} \] \[ = \frac{200 \times 1}{0.287 \times (25 + 273)} = \frac{200}{85.77} = 2.33 \, kg \]
Step 2: Net heat added. Heater power: $250 \, W = 0.25 \, kJ/s$ for 600 s $\Rightarrow$ \[ Q_{heater} = 0.25 \times 600 = 150 \, kJ \] \[ Q_{net} = 150 - 4 = 146 \, kJ \]
Step 3: Energy balance. At constant pressure: \[ Q_{net} = m c_p (T_2 - T_1) \] \[ 146 = 2.33 \times 1.005 (T_2 - 298) \] \[ 146 = 2.343 (T_2 - 298) \] \[ T_2 - 298 = \frac{146}{2.343} \approx 62.4 \] \[ T_2 = 360.4 \, K \]
Step 4: Convert to Celsius. \[ T_2 = 360.4 - 273.15 = 87.2^\circ C \] Final Answer: \[ \boxed{87.2^\circ C} \]