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I am attempting to generate the probability for the specific speed of a molecule using the Maxwell-Boltzmann distribution, but I cannot decide on which temperature I should use in the equation.

For example, I have a very, very cold liquid (let's say 70 K) at room temperature (we'll call that 296.15 K), but would I use the temperature of the liquid in the equation, or should I use the temperature of the air around the liquid?

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    $\begingroup$ from wiki (MB distribution) "Particle in this context refers to either gaseous atoms or molecules, and the system of particles is assumed to have reached thermodynamic equilibrium." Therefore, liquid with different temperature than environment is not very suitable for Maxwell-Boltzmann distribution, though it might be possible for a very crude estimates (temperature gradually varing from 70K to 296.15K) $\endgroup$
    – user26143
    Commented Mar 1, 2014 at 9:35
  • $\begingroup$ I thought that Maxwell Boltzmann distributions were applicable in systems in equilibrium. In this case, you certainly are not in equilibrium. The calculation would bequite complex. I dont think you can calculate that with just temperature. I second what user26143 said. However, please let me know if you find something that counters this notion. I am interested in it. $\endgroup$
    – CoffeeIsLife
    Commented Mar 2, 2014 at 5:06
  • $\begingroup$ I was indeed working on a crude estimation, and the liquid is certainly not going to make it all the way up to room temperature under the circumstances that I'm in $\endgroup$
    – nmagerko
    Commented Mar 3, 2014 at 3:42
  • $\begingroup$ The problem is what is your system? If you are considering the liquid itself, then you direct apply 70K to the equation. But if you include the room, then you have an inequilibrium system and you cannot do it. $\endgroup$
    – Xiaoge Su
    Commented May 30, 2014 at 3:43

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If you are assuming a pseudo-steady state condition (where the bulk of the liquid is in thermal equilibrium with itself for the timescale of interest), and you want to study the bulk molecules of the liquid, then you want to use the temperature of the liquid.

Now, if you are going to be studying the surface of the liquid, using a Maxwell-Boltzmann distribution is much more questionable - it definitely isn't strictly correct, at least on time scales of any substantial length, but you might be able to get away with it for rough approximations.

If you gave some more details as to the specific application, I could probably give you a better answer.

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