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I am reading these lecture notes on statistical mechanics. On page $10$, the author asserts that the average kinetic energy of particles floating in water is given by $$\frac{3RT}{2N_A}, $$ where $R$ is the ideal gas constant, $T$ is temperature, and $N_A$ is Avogadro's number.

Why is this formula applicable? I know that it works for gases, but particles floating in water are not a gas.

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The ideal gas constant $R$ is a conversion factor between energy and temperature. It is actually a factor that describes the SI system of units (or any other system of units) rather than a factor that describes gasses. Since we use the same units of energy and temperature for gasses and for liquids the conversion factor, $R$, is the same

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  • $\begingroup$ Thanks. Could you please clarify why the constant 3/2 remains the same also? As you say, dimensional analysis permits one to guess everything up to a constant. But why the constant is the same as a monatomic gas is unclear to me. $\endgroup$
    – alligator
    Commented Feb 7, 2022 at 15:08
  • $\begingroup$ (In particular, the derivation in the notes appears to be for a particle moving in a 2-d slide of water. But the 3 corresponds to three spatial degrees of freedom. How do we reconcile that?) $\endgroup$
    – alligator
    Commented Feb 7, 2022 at 15:24
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    $\begingroup$ The 3/2 is actually probably not a good approximation for a molecule like water. The professor appears to just be using it for convenience. In an ideal gas there are 3 degrees of freedom (x,y,z translation). In a molecule like water there are those 3 plus 3 vibrational modes and 3 rotational modes. Depending on the temperature some of those may be frozen out. I wouldn't focus on the 3/2, it will not generally be present. $\endgroup$
    – Dale
    Commented Feb 7, 2022 at 15:43
  • $\begingroup$ I think the 3/2 is for the particle moving in water (e.g. dust or dye) and not the water itself. Do you think it's a better approximation in that case? $\endgroup$
    – alligator
    Commented Feb 7, 2022 at 17:16
  • $\begingroup$ @alligator that is certainly possible, I didn't read it too carefully. Whether or not that is a good approximation would depend on the chemical makeup of the dye/dust, but it could indeed be valid for some dyes I assume. In any case, I wouldn't worry too much about the 3/2, it is subject to change and often will be determined empirically $\endgroup$
    – Dale
    Commented Feb 7, 2022 at 17:20

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