# Is there an equation to calculate the average speed of liquid molecules?

I seem to remember from first year physics that we can calculate the RMS speed of a stationary, ideal gas with $v=\sqrt{\frac{3RT}{M}}$. Does a similar equation exist for liquids?

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The difficulty in answering your question is what you mean by the velocity of the particles in a liquid.

In a gas the mean free path is long compared to the size of the particles, so the motion of the particles is well described as simple translational motion in a straight line with occasional changes of direction due to collisions. Contrast this with a solid. If by velocity you mean the movement over distance large compared to the particle size then the velocity of the particles is virtually zero. However if you calculate the energy per particle you find that there is indeed $\tfrac{3}{2}kT$ associated with the translational motion of each particle. So the particles appear to be moving ... only they aren't.

The reason for this is that in a solid the particles sit within a potential well due to their interactions with neighbouring particles, and they oscillate about their mean position. For isotropic solids like metals you get $\tfrac{3}{2}kT$ associated with the kinetic energy and another $\tfrac{3}{2}kT$ associated with the potential energy so the total energy is about $3kT$ per particle. This gives a molar specific heat of around 25J/mol, and indeed a large range of solids have heat capacities of about this value.

So the average particle velocity in a solid is either the same as a gas, or zero, depending on what you mean by average velocity.

In a liquid we have a similar situation, but with some crucial differences. The particles still sit in a potential well due to interactions with neighbours, but now there can be long range motion as well. However the long range motion is much, much slower than the particle velocity. Specifically, the long range (long means > 1 particle diameter!) velocity is proportional to the self-diffusion coefficent.

So the average particle velocity in a liquid is either the same as a gas, or proportional to the self-diffusion coefficient, depending on what you mean by average velocity.

The self diffusion coefficient is given by:

$$D = g a^2 \nu \tag{1}$$

where $g$ is a geometric factor, $a$ is the distance between particles in the liquid and $\nu$ is the number of times per second a particle changes places with a neighbour. We can (very roughly) get a value for the velocity because if $nu$ is the number of times a particle swaps places per second, and each swap requires a movement of about 1 particle spacing, $a$, then the velocity is about $a\nu$. Rearranging (1) gives:

$$v = a\nu = \frac{D}{g a}$$

As an example the self-diffusion coefficient of water is $2.3 \times 10^{-9}$ m$^2$/sec, and the diameter of a water molecule is around $3.1 \times 10^{-10}$ m (I got this from the molar density). I have no idea what value to use for $g$, so let's use $1$ and accept that at best well get an order of magnitude estimate of the velocity. Putting in these values gives a value for $v$ of about $10$ m/sec.

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By the equipartition theorem, sure enough the same equation works for any state of matter. In fact, molecular dynamics simulations use this as the very definition to compute temperature and apply a thermostat to a system (some keywords off the top of my head: Nose-Hoover, Andersen, Berendsen, velocity rescale). Furthermore, from what I remember, the equipartition theorem can be in a very straightforward way generalized into nonequilibrium systems (but this is a bit off topic).

What is different in fluids, then, is of course how the temperature relates to other state variables, i.e. the equation of state. You might use the van der Waals equation as a starting point to quantify systems with interactions. For (much) more sophisticated theories I might suggest the books Theory of Simple Liquids by Hansen and MacDonald or Theory of Molecular Fluids by Gray, Gubbins and Joslin.

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