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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?

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    $\begingroup$ Pressure x volume = energy is not a hard and fast rule. In this case, its only correct up to a factor of 2/3. $\endgroup$
    – jacob1729
    May 9 '19 at 15:51
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    $\begingroup$ 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. $\endgroup$
    – Poutnik
    May 9 '19 at 15:55
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    $\begingroup$ Who says that the internal energy of an ideal gas (or its kinetic energy) is supposed to be PV? $\endgroup$ May 9 '19 at 17:05
  • $\begingroup$ If its not then what does the PV represent? $\endgroup$
    – IK-_-IK
    May 9 '19 at 18:41
  • $\begingroup$ 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?? $\endgroup$ May 9 '19 at 19:25
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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 $dV$.

So your upper derivation of kinetic energy from ideal gas equation is incorrect. The kinetic energy will be found using, equipartion law and degree of freedom.

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