Question

A closed auditorium of volume \(2.50 \cdot 10^{4} \mathrm{~m}^{3}\) is filled with 2000 people at the beginning of a show, and the air in the space is at a temperature of \(293 \mathrm{~K}\) and a pressure of \(1.013 \cdot 10^{5} \mathrm{~Pa}\). If there were no ventilation, by how much would the temperature of the air rise during the 2.00 -h show if each person metabolizes at a rate of 70.0 W?

Short answer

Question: Calculate the rise in temperature inside a closed auditorium after a show that lasts 2 hours, given the following information: volume = 2.50 × 10^4 m^3, initial temperature = 293 K, initial pressure = 1.013 × 10^5 Pa, number of people = 2000, and metabolic rate per person = 70 W. Answer: The rise in temperature inside the closed auditorium after the 2-hour show is approximately 2.4 K (or 2.4°C).

Step-by-step solution

Calculate the total energy released by people

To find out the total energy generated by all the people inside the auditorium, we need to multiply the metabolic rate of each person (70 W) by the total number of people (2000) and by the duration of the show in seconds (2 h). Total energy (Q) = Number of people × Metabolic rate × Time duration Total energy (Q) = 2000 × 70 W × 2 * 3600 s

Calculate the number of moles of air in the auditorium

We need to find the number of moles of air in the auditorium. To do this, we will use the Ideal Gas Law, which is: PV = nRT where P is pressure, V is volume, n is the number of moles, R is the gas constant, and T is the temperature. Given pressure P = 1.013 × 10^5 Pa, volume V = 2.50 × 10^4 m^3, and temperature T = 293 K, we can solve for n as follows: n = PV / RT We also need the gas constant R. For air, we can use R = 8.314 J/(mol K).

Calculate the change in temperature

We will now use the total energy released by the people and the number of moles of air to determine the change in temperature. We will use the following formula for the energy generated in terms of the change in temperature (ΔT): Q = n × C_p × ΔT where C_p is the molar heat capacity of air at constant pressure, which is 29.10 J/(mol K). Divide both sides by n × C_p to solve for ΔT: ΔT = Q/(n × C_p)

Calculate the final values and present the result

Now, plug in the values obtained in steps 1, 2, and 3 into the equation for ΔT and calculate the final result. After calculating, express the result as the increase in temperature during the 2-hour show. Remember to present your final answer clearly with units.

Key concepts

Ideal Gas Law

The Ideal Gas Law is essential for understanding various physical situations, including the behavior of gases in enclosed spaces like our auditorium example. This law provides a clear relationship between pressure (P), volume (V), number of moles (n), temperature (T), and the universal gas constant (R). The formula expressed as
\[ PV = nRT \]
can be used to determine the amount of gas, in moles, within a certain volume and under a specific set of conditions.
In our exercise, this formula allowed us to find the number of moles of air in the auditorium. Once we had the pressure, volume, and temperature, we could rearrange the formula to solve for 'n', thus:
\[ n = \frac{PV}{RT} \]
This step is vital since it provides us with the input we need to understand how energy transfer—like the metabolic heat produced by the people—would affect the air temperature inside the auditorium.

Molar Heat Capacity

The concept of molar heat capacity is crucial when it comes to understanding how substances absorb or release heat. Defined as the amount of heat energy required to raise the temperature of one mole of a substance by one degree Celsius, it has the units J/(mol K). In terms of gases, we often refer to the molar heat capacity at constant pressure, denoted as \( C_p \), as gases can expand when heated at constant pressure.
In the context of our auditorium, we need to know the molar heat capacity of air to calculate how much the temperature would increase when the people inside release heat into the air as a result of their metabolic activity. Using the determined number of moles of air (from applying the Ideal Gas Law) and the total energy generated by the occupants (from knowing their metabolic rate), the formula
\[ Q = n \times C_p \times \Delta T \]
allows us to solve for the change in temperature, \( \Delta T \), after rearranging it to
\[ \Delta T = \frac{Q}{n \times C_p} \].

Metabolic Rate Energy Transfer

Metabolic rate refers to the rate at which organisms consume energy to maintain basic bodily functions and perform activities. It's measured in watts, where 1 watt is equal to 1 joule of energy per second. When thinking about humans in an auditorium, each person's metabolic rate creates heat as a byproduct, and this becomes a source of energy transfer into the surrounding environment.
For our exercise, calculating the total energy transfer was key to determining how much the air temperature would rise. By multiplying the metabolic rate of a single person by the number of people present and the duration of the show, we obtained the total energy released in the auditorium. Understanding this concept allows us to quantify the impact of human presence on a closed environment's temperature, something particularly relevant in the design of HVAC systems and the study of thermal comfort in buildings.