# Energy of a gas

## Main Question or Discussion Point

For gases at room temperature and pressure, at a given time, why are most of the energy states empty? (I am analysing the situation using the principles of quantum physics, i.e. wavefunctions, quantum states, etc.) Is this valid for all gases? Are we dealing with ideal gases only?

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alxm
Because there's an infinite number of states?

Please confirm if I am correct on the following:

According to the principles of classical physics, the number of energy states would be infinite becuase energy is a continuous variable. Am I right?

And, according to the principles of quantum physics, the number of energy states would also be infinite because the principal quantum number (which is a variable of the energy eigenvalue) is any positive integer from 1 to infinity. Am I right? I believe this is what you were referring to through your reply.

If I am right, I would then like to ask this question:

Even if there is an infinite number of energy states, why do most of the energy states have to be empty? For instance, the probability distribution for the occupancy of particles might be skewed to one end.

And in the case of a flat probability distribution, the curve should be zero, shouldn't it?

alxm
According to the principles of classical physics, the number of energy states would be infinite becuase energy is a continuous variable. Am I right?

And, according to the principles of quantum physics, the number of energy states would also be infinite because the principal quantum number (which is a variable of the energy eigenvalue) is any positive integer from 1 to infinity. Am I right? I believe this is what you were referring to through your reply.
Well, you have an infinite number of bound states for a particle in a potential well, but you also have a continuum of states for a free particle. A free particle can, after all, have any kinetic energy, even in QM. So, the kinetic energy of your gas particles is continuous, to begin with.
Even if there is an infinite number of energy states, why do most of the energy states have to be empty? For instance, the probability distribution for the occupancy of particles might be skewed to one end.
Well, you have a finite amount of energy to distribute. In equilibrium, it'd be distributed as the Boltzmann distribution. If the energy has some other 'uneven' distribution, then you're not in equilibrium. The energy will (eventually) redistribute itself, lowering the entropy. The entropy of a system is essentially a measure of the distribution of energy, which is an irreversible process. A rod that's been heated at one end will have its temperature even out eventually, but rod at thermal equilibrium won't spontaneously become hotter at one end. (Or more specifically, not to the extent that any work could be extracted from that. AKA "Maxwell's demon")

you have an infinite number of bound states for a particle in a potential well,
Does a particle have kinetic energy inside a potential well? What is the value of its potential energy inside the well?

but you also have a continuum of states for a free particle.
I guess, because you have a continuum of states for a free particle in the special case of zero potential, the gap between energy levels drops in some arbitrary manner with decreasing potential?

A free particle can, after all, have any kinetic energy, even in QM.
But the kinetic energy can't be negative, even in QM, right?

So, the kinetic energy of your gas particles is continuous, to begin with.
But that's when the particle is free. What about the more complex case of a classical gaseous particle in some arbitrary potential?

Well, you have a finite amount of energy to distribute. In equilibrium, it'd be distributed as the Boltzmann distribution.
Shouldn't we say Fermi-Dirac/ Bose-Einstein distribution as we are dealing in energy levels and therefore analysing in the quantum world?

If the energy has some other 'uneven' distribution, then you're not in equilibrium.
By 'uneven', do you mean a curve that deviates from the Boltzmann distribution?

The energy will (eventually) redistribute itself, lowering the entropy.
But as far as I know, the entropy of any system increases.