# Ideal gas equation vs Kinetic theory equation

From the ideal gas equation: $$PV = NkT$$, since Pressure times Volume = Energy, my understanding is that the total (internal) energy of $$N$$ molecules of a gas $$= NkT$$.

However from the kinetic theory equation: Average kinetic energy per molecule of gas = $$\frac{3}{2} kT$$ and hence the total kinetic energy for $$N$$ molecules = $$\frac{3}{2} NkT$$.

Since potential energy is considered non-existent in ideal gases, kinetic energy = internal energy. However the 2 formulae lead to different results. What is the reason for this, what am I missing?

• Pressure x volume = energy is not a hard and fast rule. In this case, its only correct up to a factor of 2/3. May 9 '19 at 15:51
• Also note that rotational nor vibrational degrees of freedom, contributing to the gas molar heat capacity, do not contribute in pressure not mechanical work via p and V. May 9 '19 at 15:55
• Who says that the internal energy of an ideal gas (or its kinetic energy) is supposed to be PV? May 9 '19 at 17:05
• If its not then what does the PV represent? May 9 '19 at 18:41
• It merely represents part of the equation of state PV=nRT, unrelated to the internal energy. Why to you think it is related to internal energy...simply because it has the same units?? May 9 '19 at 19:25

Actually its incorrect to think that $$PV=energy$$, just because they have same units.
Strictly speacking, its defined that $$dW=PdV$$ where $$dW$$ is work done in change in volume of $$dV$$.