You are sitting in a gymnasium watching your local high school basketball team.

Question

You are sitting in a gymnasium watching your local high school basketball team. Assume there are 450 people in the gym. If the gym has (constant) dimensions 50m long, 45m wide, 12m tall, find the following:

A.) The number of air particles in the gym (mass of single air particle is 4.8x10-26 kg, density of air is 1.3 kg/m3), assuming none leak out from the walls/doors/etc.

B.) The number of moles of air in the gym.

C.) The total heat required to raise the temperature by 4 degrees Celsius assuming no het is lost through the walls. Hint: 1st law of thermodynamics; how much work is done when volume is constant?

D.) How many joules of heat must each person in the gym radiate away in order for the temperature to rise by that 4 degrees Celsius?

Explanation / Answer

Volume of gym = 50 * 45 * 12 = 27000 m3

density of air = 1.3 kg / m3

So mass of air in gym = density*volume = 1.3 * 27000 = 35100 kg

Mass of 1 particle =  4.8x10-26 kg

A) So total number of particles = total mass / mass of 1 particle

= 35100 / 4.8x10-26 = 7.3125 * 1029

B) Number of moles of air in gym = total number of air particles/ NA

= 7.3125 * 1029 / (6.023 * 1023) = 1.214*106 moles

C) Since volume remains constant, the work done = 0

So heat required to raise temperature by 4 degree = increase in internal energy on increase of temperature by 4 degree

Cv for air = 718 J/Kg-K

total heat required = n Cv * change in temperature = 1.214*106 * 718 * 4 = 3.48688*109 J

D)  joules of heat must each person should radiate = total heat / number of person

=  3.48688*109 / 450 = 7.7486 * 106 J

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