Unizor - Physics4Teens - Energy - Molecular Movement Problems
Channel: Zor Shekhtman on UNIZOR Education
Date: 2019-06-03

Unizor - Physics through a Looking Glass of Mathematician

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)Etot /V

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

Substituting the values for pressure and volume, we obtain
Etot = (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 Etot=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 = kB·N = const
where
kB = 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/(kB·T)
Substituting the values,
N = 105·48/(1.381·10−23·293) = 0.12·1028
It's a lot!