Online Homework

Average Kinetic Energy of a Gas

According to kinetic theory of gases, all gases are made up of large number of small particles which are in constant, random motion. The rapidly moving particles constantly collide with each other and the walls of the container. The macroscopic pressure of a gas relates directly to the average kinetic energy per molecule, according to the following formula:

`P = N(mv^2)/(3V)`

where N = number of particles; m = mass; v = average speed and V="Volume" of the container

Now, according to ideal gas law formula

`PV = nRT` ............... (2)

where n = number of moles; R = gas constant and T = temperature

or, `P = (nRT)/V` ................. (3)

From equation (1) and (3), we have

 

`N(mv^2)/(3V) = (nRT)/V`

or, `mv^2 = n/N*3RT`

or, `1/2mv^2 = n/N*3/2RT`  ............ (5) 

for n moles of a gas `n= N/N_A`

where NA = Avogadro's number

By substituting n in equation (5), we get

`E = 1/2mv^2 = 3/2*(RT)/N_A`

Now, `R/N_A = k` , Boltzmann's constant

Therefore,

`E = 3/2 kT`