Chapter 1, Problem 8P

A group of 35 students attend a class in a room that measures 11 m by 8 m by 3 m. Each student takes up about 0.075 ${\text{m}}^{\text{3}}$ and gives out about 80 W of heat $\left(1\text{ W}=1\text{ J/s}\right)$. Calculate the air temperature rise during the first 20 minutes of the class if the room is completely sealed and insulated. Assume the heat capacity, $C_v$, for air is 0.718 kJ/(kg K). Assume air is an ideal gas at 20°C and 101.325 kPa. Note that the heat absorbed by the air Q is related to the mass of the air m, the heat capacity, and the change in temperature by the following relationship:

$Q=m{\displaystyle {\int }_{T_1}^{T_2}{C}_{v}dT=m{C}_v\left(T_2-T_1\right)}$

The mass of air can be obtained from the ideal gas law:

$PV=\frac{m}{\text{MwT}}RT$

where P is the gas pressure, V is the volume of the gas, Mwt is the molecular weight of the gas (for air, 28.97 kg/kmol), and R is the ideal gas constant $\left[\text{8}{\text{.314 kPa m}}^{\text{3}}\text{/}\left(\text{kmol K}\right)\right]$.