A closed auditorium of volume 27100 m^3 is filled with 2210 people at the beginning of a show, and the air in the space is at a temperature of 292 K and a pressure of 1.013·10^5 Pa. If there were no ventilation, by how much would the temperature (in K) of the air rise during the 1.5-h show if each person metabolizes at a rate of 73.1 W?
The capacity do to work
To determine how much the temperature of the air would rise during the show, we need to calculate the total heat generated by the metabolisms of the people in the closed auditorium.
First, let's calculate the total heat generated by the metabolisms of the people. We can use the formula:
Q = P * t
where Q is the heat generated, P is the power (73.1 W in this case), and t is the duration of the show (1.5 hours).
Q = 73.1 W * 1.5 h
Now, let's calculate the energy required to raise the temperature of the air in the closed auditorium using the formula:
ΔQ = m * c * ΔT
where ΔQ is the change in heat, m is the mass of air in the auditorium, c is the specific heat capacity of air, and ΔT is the change in temperature.
To find the mass of air, we can use the ideal gas law:
PV = nRT
where P is the pressure (1.013·10^5 Pa), V is the volume (27100 m^3), n is the number of moles of air, R is the ideal gas constant (8.314 J/(mol·K)), and T is the initial temperature (292 K).
Rearranging the equation, we get:
n = (PV) / (RT)
Now, let's calculate the number of moles of air:
n = (1.013·10^5 Pa * 27100 m^3) / (8.314 J/(mol·K) * 292 K)
Once we have the number of moles of air, we can calculate the mass of air using the molar mass of air (approximately 29 g/mol).
Next, let's calculate the change in heat:
ΔQ = m * c * ΔT
We can rearrange this equation to:
ΔT = ΔQ / (m * c)
Finally, let's substitute the values into the equation and calculate the change in temperature. The specific heat capacity of air is approximately 1005 J/(kg·K).
Remember to convert the mass of air to kilograms to match the units of specific heat capacity:
ΔT = Q / ((m * 1005 J/(kg·K))
After performing the calculations, you should be able to find the change in temperature.