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 = length * breadth * height

volume of gym = 50 * 45 * 12

volume of gym = 27000 m^3

density = mass / volume

1.3 = mass / 27000

mass = 35100 kg

let the number of particles be n so,

n * 4.8 * 10^-26 = 35100

number of particles n = 7.3125 * 10^29

number of moles = mass / molar mass

molar mass of air = 28.97 gm / mol

number of moles = 35100 * 10^3 /  28.97

number of moles = 1211598.2 moles

specific heat at constant volume of air = 0.718 kJ/kg K

heat = mass * specific heat * change temperature

heat = 35100 * 0.718 * 4

heat required to raise temperature by 4 degree = 100807.2 kJ

heat must be radiated by each person = 100807.2 / 450

heat must be radiated by each person = 224.016 kJ

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