Variation of Kinetic Energy with temperature

[johnconnor]

Question:

Find the total kinetic energy per unit volume in a monoatomic gas at standard temperature and pressure and deduce an expression for the variation of this kinetic energy with temperature if the pressure is maintained constant. [Standard pressure = 1.01E5 Pa]

Attempt:

Pressure of a gas
$pV= \dfrac{N_A m <c>^2}{3V}V$
$\dfrac{M<c>^2}{2} = 3pV/2$
where M = N_Am
$E_k /V = 3p/2$
$E_k /V = 1.52x10^5 J/m^3$

Variation of this Ek with temperature is pressure is maintained constant:
$E_k = 3pV/2$
$E_k = 3(nRT)/2$.

Comments: For the "Variation of this Ek with temperature is pressure is maintained constant", is that all that I should show? What other comments should I include to make my working more accurate?

The guide mentioned:

You can reach the same conclusion starting from kinetic energy = 3NkT/2, provided allowance is made for the fact that the number of atoms per unit volume will change.

How so? can anyone please explain the bold sentence to me? Thank you!

 

[Infinitum]

provided allowance is made for the fact that the number of atoms per unit volume will change.

Kinetic energy given by 3kNT/2 is for one mole of the gas at a given volume. So, I think that refers to the expansion of gas with the rise in temperature. The number of molecules per unit volume at a temperature T1 will be lesser that the number of molecules per unit volume at temperature T2 < T1.