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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 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 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? $\endgroup$ – Chet Miller May 9 at 17:05
  • $\begingroup$ If its not then what does the PV represent? $\endgroup$ – IK-_-IK May 9 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?? $\endgroup$ – Chet Miller May 9 at 19:25

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