# Ideal gas law and internal energy of a gas

For an ideal gas, we have \begin{align} pV = Nk_B T \end{align} Considering a monatomic ideal gas specifically, the internal energy is \begin{align} U = \frac{3}{2}Nk_BT \end{align} so that \begin{align} pV = \frac{2}{3}U\quad\quad\quad \tag{1} \end{align} If we interpret the term $$pV$$ as "the energy a gas has due to being confined to volume $$V$$ at pressure $$p$$", then $$(1)$$ seems to suggest something along the lines of \begin{align} \text{"two-thirds of the internal energy of a gas is due to its being confined to volume V"} \tag{2} \end{align} This interpretation seems suspect to me since $$\frac{3}{2}Nk_BT$$ is simply the kinetic energy of the gas, and I would expect a gas to have this same internal energy regardless of whether we confine it to a finite volume or not.

My questions are as follows. Is $$(2)$$ a correct interpretation of the relation $$(1)$$? And if not, how should we interpret $$(1)$$?

• yeah, not reliable. why not merely a numerical relationship with no confinement interpretation? Mind you, 2/3 can change if it isnt monatomic Commented Apr 30, 2023 at 0:50
• Yeah, I know that $U = cNk_B T$ more generally, but I chose to consider a monatomic gas with $c=3/2$ for the sake of concreteness. Personally, I have a hard time believing that this relationship is just a numerical coincidence without any physical meaning, but I am open to being convinced of that.
– d_b
Commented Apr 30, 2023 at 0:59
• Kinetic energy is $\tfrac 12 mv^2$. Does that mean that $mv^2$ is some sort of energy? Commented Apr 30, 2023 at 11:43
• @PhilipWood You make a good point, but the relationship between $mv^2/2$ and $mv^2$ is a trivial multiplication by a numerical factor, whereas the relationship in my post is a non-trivial one that requires some statistical mechanical calculations to derive. I think a better comparison might be something like the virial theorem, which gives a non-trivial relationship between the averaged kinetic and potential energies. Admittedly, I don't have a good intuition for the "meaning" of the virial theorem, and tend to view it simply as a mathematical relationship, so maybe the same is true here.
– d_b
Commented May 1, 2023 at 3:13

Equation (1) is correct for (monoatomic) ideal gas and says that the pressure of ideal gas with fixed energy is inversely proportional to the volume it occupies. How one interprets it is a bit subjective. You read it to say

"two-thirds of the internal energy of a gas is due to its being confined to volume $$V$$"

which sounds a bit strange because it seems to imply that the other one-third of internal energy comes from somewhere else. I read it as

internal energy is three-halves of the product $$pV$$

and take it to relate the energy of the gas to the pressure produced by molecules bouncing off the walls. If $$V$$ is increased particles take longer to transverse the length of the box, the collision rate with the walls drops, and so does pressure.

Writing you relationship as $$U=\frac 32 pV$$ for an ideal gas.

Suppose the volume is constant, how is it that increasing the pressure increases the internal energy of the gas?
To increase the pressure at constant volume heat can be added to the system which increases the average kinetic energy of the molecules and hence the internal energy.

Now suppose that the pressure is constant and the volume is increased.
How does that increase the internal energy of the gas?
To maintain the constant pressure heat can be added to the system which increases the average kinetic energy of the molecules and hence the internal energy.

The $$\frac{3}{2}$$ factor comes from dimensionality. "3" is because space has three dimensions.
If a gas initially contained in a volume $$V$$ is suddenly allowed to expand in unlimited space, its energy remains $$U$$. It is not the containment that gives it an energy. However, it is the energy combined with the molecule density $$N/V$$ that explains the pressure. The pressure is caused be molecules bumping against the walls and creating a force. The faster and the more molecules, the stronger the force. The pressure happens to be divided by 3 because the energy is distributed among three directions corresponding to the 3 dimensions.
$$p=\frac{2U/V}{3}$$