# Motivation

I ask this question precisely because I am looking for a fundamental, quantitative explanation of the limitations of the ideal gas law and when it should not be used.

Note, I have found the following answers https://physics.stackexchange.com/a/91727/59023, https://physics.stackexchange.com/a/43701/59023, and https://physics.stackexchange.com/a/17364/59023 but in each case there was not a definitive or quantitative limitation prescribed. The answers mention the limitations but it is not clear whether those limitations invalidate the use of $P = n \ k_{B} \ T$ [i.e., $n$ is the number density, $k_{B}$ the Boltzmann constant, and $T$ a scalar temperature (well, one could make it a pseudotensor if they prefer)].

My primary motivation in asking this question derives from a conversation I had with a colleague about the differences between equilibrium and nonequilibrium gases.

# Example

For a specific example, consider the nearly collisionless (well, weakly collisional at best) plasma in the solar wind. Here all the interactions are due to long-range forces (i.e., Coulomb collisions and/or interactions with fluctuating electromagnetic fields), which as several of the answers above eluded, is a condition that the ideal gas law assumes to be negligible. Further, the nearly collisionless, nonequilibrium state of this ionized kinetic gas coupled with the ubiquitous observations of non-Maxwellian, anisotropic velocity distributions (e.g., see What is the correct relativistic distribution function? and references therein) causes me to doubt the applicability of ideal gas to these plasmas.

However, I know of several examples in physics where the assumptions used to derive relationships/approximations like the ideal gas law are not rendered invalid simply because one or more of the assumptions do not hold.

# Clarifications

To be clear, I am not concerned with:

• the kinetic definition of pressure; or
• the kinetic definition of number density; or
• about how one defines a kinetic temperature since we can define what is meant by $T$ as @AccidentalFourierTransform eludes to in comments to the question at Derivation of ideal gas law.

# Questions

• I am curious how poor is the approximation $P = n \ k_{B} \ T$ for the solar wind plasma?
• Note: I am not concerned about the multiple species or components so much as the underlying reasons justifying when we can assume $P = n \ k_{B} \ T$.
• Should we consider a different equation of state for the system to derive a new/different relationship between pressure, density, and temperature?
• Which is more critical to the relevance of the validity of $P = n \ k_{B} \ T$: the nonequilibrium or nearly collisionless state of the plasma?
• There is no point at which you might say "Ah, the ideal gas law miserably fails starting at this point", which seems to be what you're after. There's no quantitative point like that. It just becomes a worse and worse approximation. So you as a physicist need to decide for yourself if the ideal gas law is good enough or not. If you need to calculate something to only 20% error, perhaps the ideal gas law works. If you need to calculate to better than 1%, perhaps it does not, and you will have to use some kind of numerical model that is better than the ideal gas law. – Spaderdabomb Aug 5 '16 at 23:09
• @Spaderdabomb - Perhaps my issue is that it seems to be one of these assumed relationships even though none of the assumptions required for its derivation hold in a collisionless plasma. I am trying to determine whether it is even meaningful to use the approximation, since what does pressure mean in the context of a collisionless gas? – honeste_vivere Aug 5 '16 at 23:14
• Well the pressure is the amount of force per unit area those "collisionless" particles would exert on an impenetrable wall. This is why you can have a pressure from a collisionless gas. So even if you assume the plasma to be collissionless, and thus use the ideal gas law to model it, you can still have a pressure resulting from the plasma. – Spaderdabomb Aug 5 '16 at 23:17
• I think the point is that, if the gas is collisionless, it is not in equilibrium in any real sense. – Jon Custer Aug 6 '16 at 0:02
• You can get a sense of the failure modes by examining the assumptions that go into the derivation of the law through kinetic theory, but you've clearly already poked around that avenue. What do you find unclear about it? The law loses applicability gracefully with increasing density (both the approximate-non-interaction and the negligible volume assumptions start to fail) and can get into trouble by exciting internal state of the particles which can be be graceful or not depending on the nature of the excitation. The van der Waals gas covers the former situation pretty well, but not plasma. – dmckee Aug 6 '16 at 2:25