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 1.013·105 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?
To calculate the temperature rise of the air during the show, we need to consider the heat generated by each person and the volume of the auditorium.
First, let's calculate the total heat generated by all the people in the auditorium. The total heat generated is equal to the power (Watts) times the time (seconds) each person is in the auditorium. Since the time is given in hours, we need to convert it to seconds.
Time = 1.5 hours = 1.5 * 60 * 60 seconds = 5400 seconds
Total heat generated = Power per person * Number of people * Time
Total heat generated = 73.1 W * 2210 * 5400 seconds
Next, we need to calculate the specific heat capacity of air. The specific heat capacity (c) of air is approximately 1005 J/(kg·K).
To calculate the mass of air in the auditorium, we need to know the density of air. The density of air at the given pressure (1.013·105 Pa) and temperature (292 K) can be calculated using the ideal gas law. The ideal gas law states that PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature.
R = 8.314 J/(mol·K) (ideal gas constant)
n = PV / RT
n = (1.013·105 Pa * 27100 m^3) / (8.314 J/(mol·K) * 292 K)
Now, we need to calculate the molecular weight of air. The molecular weight of air is approximately 28.97 g/mol.
Mass of air = n * molecular weight of air
Mass of air = n * 28.97 g/mol
To convert the mass of air to kilograms:
Mass of air = (n * 28.97 g/mol) / 1000
The total heat generated by the people is equal to the heat gained by the air in the auditorium. Using this equation, we can calculate the temperature rise (ΔT) of the air:
ΔT = Total heat generated / (mass of air * specific heat capacity of air)
Finally, substitute the values into the equation to calculate the temperature rise of the air.