Unizor - Physics4Teens - Energy - Ideal Gas Kinetics - Problems

Notes to a video lecture on http://www.unizor.com

Problem 1

How much kinetic energy have all the molecules in the room?
How fast should an average size car move to have this amount of kinetic energy?

Assume the following:
(a) the room dimensions are 4x4x3 meters (that is, V=48m³);
(b) normal atmospheric pressure is 100,000 Pascals (that is, p=100,000N/m²);
(c) a mass of an average size car is 2,000 kg (that is, M=2,000kg).

Solution

From the lecture on Kinetics of Ideal Gas we know the relationship between the pressure on the walls of a reservoir, volume of a reservoir and total kinetic energy of gas inside this reservoir:
p = (2/3)E_tot /V

From this we derive a formula for total kinetic energy:
E_tot = (3/2)·p·V

Substituting the values for pressure and volume, we obtain
E_tot = (3/2)·100,000·48 =
= 7,200,000(joules)

The kinetic energy of a car is
E = M·v²/2
Therefore, given the kinetic energy and mass, we can determine the car's speed:
v = √2·E/M
Substituting calculated above E_tot=7,200,000(J) for E and the value for mass M=2,000(kg), we obtain
v = √2·7,200,000/2,000 ≅
≅ 85(m/sec) ≅
≅ 306(km/hour) ≅
≅ 190(miles/hour)


Problem 2

Given the temperature, pressure and volume of the air in a room, determine the number of gas molecules in it.

Assume the following:
(a) the room dimensions are 4x4x3 meters (that is, V=48m³);
(b) normal atmospheric pressure is 100,000 Pascals (that is, p=100,000=10⁵N/m²);
(c) temperature is 20°C (that is, T=20+273=293°K).

Solution
Recall the combined law of ideal gas
p·V/T = k_B·N = const
where
k_B = 1.381·10⁻²³ (J/°K) is Boltzmann's constant and
N is the number of gas molecules in a reservoir.
From this we derive the number of molecules
N = p·V/(k_B·T)
Substituting the values,
N = 10⁵·48/(1.381·10⁻²³·293) = 0.12·10²⁸
It's a lot!