Imagine that 501 people are present in a movie theater of volume 8.00 ×10^3 m^3 that is sealed s

step 1 The value of R is 8 .31 jules per mole calvins. The volume is given as 8 ,000 meters cubes. The

Imagine that 501 people are present in a movie theater of...

Imagine that 501 people are present in a movie theater of volume $8.00 \times 10^{3} \mathrm{m}^{3}$ that is sealed shut so no air can escape. Each person gives off heat at an average rate of $110 \mathrm{W} .$ By how much will the temperature of the air have increased during a 2.0 -h movie? The initial pressure is $1.01 \times 10^{5} \mathrm{Pa}$ and the initial temperature is $20.0^{\circ} \mathrm{C} .$ Assume that all the heat output of the people goes into heating the air (a diatomic gas).

Key Concepts

Ideal Gas Law

The ideal gas law relates pressure, volume, temperature, and the number of moles of a gas. It provides a way to compute the number of moles or mass of a gas in a given volume under specified conditions, which is essential for applying the specific heat capacity to determine the temperature change caused by a given energy input.

Diatomic Gas Characteristics

Diatomic gases, which are composed of two atoms, have specific internal degrees of freedom that affect their energy storage capacity. Their specific heat at constant volume is influenced by translational and rotational motions, and understanding these properties is important when using the specific heat value to calculate temperature changes.

Specific Heat Capacity of Gases

Specific heat capacity is the amount of heat required to raise the temperature of a unit mass (or mole) of a substance by one degree. For gases, particularly at constant volume, the specific heat capacity determines how much energy is needed to increase the temperature, which is central to calculating temperature changes from an added energy amount.

Conversion of Power to Energy

Power is the rate at which work is done or energy is transferred. When a power output is sustained over a period of time, you multiply the power by the time duration to obtain the total energy produced. This conversion is crucial when dealing with continuous energy emissions that result in temperature changes in a system.

Energy Conservation

This concept discusses how energy is neither created nor destroyed, only transferred or transformed. In a closed system, any energy added (such as heat from people) increases the internal energy of the system. This fundamental principle is used to calculate changes in temperature when energy is introduced into a system with a known heat capacity.

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