# Does the term “diatomic ideal gas” make any sense?

As per the kinetic theory of ideal gases, ideal gases are considered to behave as point particles that occupy negligible volume, and exert no intermolecular attractions. At the same time, diatomic ideal gas molecules, by definition, must be extended in shape, in order for it to have the moments of inertia about their axes (and contributing to the rotational kinetic energy of the gas). These two don't seem to relate, even theoretically.

So, my question is Does "diatomic ideal gas" make any sense, even theoretically?

• The diatomic molecules are still pretty much point particles for low-density gases (like at atmospheric pressure). – Nat May 20 '17 at 4:05

Of course it does.

It helps a little bit to compare the ideal gas to a model that does take note of the size of the molecules and the forces the exert on one another. The van der Waals gas has explicit parameters for both behaviors. Compare the equations of state for these two models \begin{align} Pv &= k_B T \tag{ideal gas} \\ \left(P + \frac{a}{v^2}\right)(v - b) &= k_B T \tag{van der Waals gas} \;, \end{align} where $a$ represent the net attraction between molecules, $b$ represent the volume actually occupied by a molecule, and I have written $v = V/N$ for the average volume available to any given molecule (so that $b/v$ is the fraction of the volume occupied by molecules a thing that is assumed to vanish in the ideal case).

This more complicated model exhibits the ability to condense into a liquid, a behavior that the ideal gas model does not duplicate. However, at low density and high temperature the van der Waals gas has the same heat capacity as the ideal gas.

On the other hand a non-monatomic ideal gas simple gets the addition of rotational and vibrational internal modes, and continues to use the same equation of state, though the new assumptions affect the heat capacity at all temperatures, but doesn't allow the system to exhibit condensation. If you take into account the effect of quantum mechanics on these internal modes the heat capacity exhibits steps in data on real gases that the monoatomic ideal gas does not faithfully report.

So, the van der Waals gas and introducing internal molecular modes represent two different way to introduce more complicated physics to the simplest gas model.

Either way we are extending a useful but incomplete model to allow it to capture more observed behavior than the naive version.

• But doesn't diatomic imply the molecule isnt a point particle, and thus cannot be ideal? – Pritt Balagopal May 20 '17 at 3:54
• It doesn't have to be a "point particle" it just has to be small enough that you can ignore its volume. In the language of the van der Waals parameters $b/v$ needs to be smaller than error term of the the highest precision prediction you expect to make using the model. Don't get hung up on the language used by a single author in defining common models like that; read around a bit. In this case you'll find descriptions involving phrases like "molecules are small enough that taken together they represent a negligible fraction of the working volume" and similar verbiage. – dmckee May 20 '17 at 3:58

Mathematically one can represent an "ideal" electric dipole or magnetic dipole, which is equivalent. You can have a point size while still maintaining a moment (electrical, magnetic, inertial)

• How is that possible? As far as my knowledge is concerned, dipole moment is defined as: $$\mu = \text{Charge} × \text{Distance}$$ As long as dipoles are considered, they must be extended, no? – Pritt Balagopal May 23 '17 at 3:21
• For ideal dipole we take $charge \to \infty$ and $distance \to 0$ while keeping the product constant. In this case the approx formula for electric field etc. of a dipole will become the formula for the actual field etc. of an ideal dipole. – aditya_stack 1 hour ago