Chemistry - Does the Maxwell-Boltzmann distribution apply to gases only?
Solution 1:
I'm going to go against the grain here: the Maxwell distribution does describe the distribution of molecule speeds in any (3-D) matter, regardless of phase.
Suppose we have a system of $N$ molecules with masses $m_i$, positions $\vec{r}_i$, and velocities $\vec{v}_i$ (with $i = 1, ..., N$). Assume that the total energy of this system is of the form $$ E(\vec{r}_1, \dots, \vec{r}_N; \vec{p}_1, \dots, \vec{p}_N) = U(\vec{r}_1, \dots, \vec{r}_N) + \sum_{i = 1}^N \frac{\vec{p}_i^2}{2 m_i }, $$ i.e., a potential energy depending on the molecules' positions and a kinetic energy. According to Boltzmann statistics, the probability of finding this system within a small volume of phase space $d^{3N} \vec{r} \, d^{3N} \vec{p}$ is $$ \mathcal{P}(\vec{r}_1, \dots, \vec{r}_N; \vec{p}_1, \dots, \vec{p}_N) d^{3N} \vec{r} \, d^{3N} \vec{p} \propto e^{-E/kT} d^{3N} \vec{r} \, d^{3N} \vec{p} \\= e^{-U(\vec{r}_1, \dots, \vec{r}_N)/kT} \left[ \prod_{i=1}^N e^{-p_i^2/2m_i k T}\right]d^{3N} \vec{r} \, d^{3N} \vec{p} $$ To find the probability distribution of finding molecule #1 with a particular momentum $\vec{p}_1$, we integrate over all other configuration variables (i.e., $\vec{r}_1$ through $\vec{r}_N$ and $\vec{p}_2$ through $\vec{p}_N$). Because the $E$ is the sum of a contribution from the positions and a contribution from the momenta, the Boltzmann factors can be split between these integrals, with the result that $$ \mathcal{P}(\vec{p}_1) \, d^3\vec{p}_1 \propto e^{-p_1^2/2 m_1 kT} d^3\vec{p}_1 \left[ \int e^{-U(\vec{r}_1, \dots, \vec{r}_N)/kT} d^{3N}\vec{r}\right] \left\{ \prod_{i = 2}^N \int e^{-p_i^2/2 m_i kT} d^3 \vec{p}_i \right\} $$ These integrals are nasty (especially the one over all $N$ position vectors), but they're just constants with respect to $\vec{p}_1$, which means that the can be folded into the proportionality constant: $$ \mathcal{P}(\vec{p}_1) \, d^3\vec{p}_1 \propto e^{-p_1^2/2 m_1 kT} d^3\vec{p}_1. $$ A similar logic applies to every other molecule in my system. In other words, the probability of finding a molecule with a momentum $\vec{p}$ doesn't depend at all on how they interact with each other, assuming that their interaction energy is dependent only on their collective positions. Since these molecules obey the same momentum distribution, they must also have the same velocity distribution, and in particular they obey the Maxwell speed distribution.
So the answer to the question asked in your title, "Does the Maxwell-Boltzmann distribution apply to gases only?" is "no"; it applies to all phases of matter, in the sense that it describes the distribution of particle speeds and energies. However, the question you ask at the bottom of your post, "Can I use a Maxwell-Boltzmann distribution to interpret reaction rates at the molecular level in liquids and/or solids?" may also be "no"; the connection between reaction rate and activation energy is not straightforward if the medium is dense, as was pointed out by @porphyrin in their answer.
Solution 2:
I'm willing to put myself out there and say that no, a Maxwell-Boltzmann distribution will not be able to make any statement about the ability of liquids or solids to react. Meaning also that you could not study reaction rates with a Maxwell-Boltzmann distribution for the condensed phases.
The reason why I am fairly confident this is true is because the Maxwell-Boltzmann distribution makes the assumption that it is working with an ideal gas. Making that assumption is both powerful and very limiting. The ideal gas approximation is quite good in a lot of cases, but it could not even come close to describing the behavior of a condensed phase.
If some such distribution were used to study condensed phase reaction rates, it simply wouldn't be a Maxwell-Boltzmann distribution.
I would also point out that the Maxwell-Boltzmann distribution doesn't describe all gas-phase reactions all that well. This can be inferred quite easily from the fact that the M-B distribution treats all gases as ideal, so the only possible difference between two systems is the mass of the particle and the temperature. That means also that any gas phase reaction which depends strongly on the orientation of the collision would likely be assumed by M-B to happen much more frequently than it really does.
All that to say, M-B is deeply rooted in the gas phase, and even there it falls short for certain systems, so it probably isn't possible to apply to the condensed phases.
On the other hand, it is possible that a partition function for a liquid or a solid phase could be used to study reactions in some way, and the partition function for a generic gas particle can be used to derive the M-B distribution so . . . There's that.
The problem with that is that the partition function requires you to sum the Boltzmann Factor over all states, and for a condensed phase system . . . That's basically gonna be an infinite number of states.
Anytime you approach stat. mech. or the name Boltzmann, it's wise to keep this quote by David Goodstein in mind:
“Ludwig Boltzmann, who spent much of his life studying statistical mechanics, died in 1906, by his own hand. Paul Ehrenfest, carrying on the work, died similarly in 1933. Now it is our turn to study statistical mechanics. Perhaps it will be wise to approach the subject cautiously.”
Solution 3:
In liquids the Boltzmann distribution limits reaction in what are called 'activation limited' reactions. In these reactions the two species collide but do not have sufficient energy to react because the activation energy is high, so they diffuse apart again. The number of particles with sufficient energy to react is given by the Boltzmann distribution. This falls with increase in energy so rapidly that reaction becomes a rate event, perhaps once in 100 million collisions.
Reactions with a low activation energy can react at first approach, by the Boltzmann distribution there is enough energy to surmount the activation energy to reaction. However, in these cases the reaction rate is limited by how fast the species can diffuse together, so reaction rate becomes a property of the solvent.