# How fast do molecules move in objects?

I guess it depends on the heat or the type of the material but can you give some examples or formulas to calculate it ?

The best example would be the average speed of the air molecules (all types in the air) at room temperature or water molecules at human body temperature.

It depends on the mass of the molecule in question. Here's a quick, back-of-the-envelope answer. In a body at thermal equilibrium, every energy mode has the same average amount of energy, $\frac12kT$, where $T$ is temperature and $k$ is Boltzmann's constant. One of the energy modes is the translational kinetic energy of a molecule in some direction $x$, $\frac12mv_x^2$. We can solve

$$\frac12kT=\frac12mv_x^2$$

to find

$$v_x=\sqrt{\frac{kT}m}$$

and then plug in $k=1.38×10^{-23}\rm{m^2 kg s^{-2} K^{-1}}$, $T=300\rm{K}$, and for $m_{\rm{N}_2}=2×14\rm{u}=2×14×1.66×10^{−27} \rm{kg}=4.65×10^{−26} \rm{kg}$ to get

$$v_x=298\rm{m/s}=667mph.$$

The molecule is also moving in the $y$ and $z$ axes, so the answer depends on what exactly you mean by average speed: mean spead vs. root-mean-square speed.

This ignores rotational and vibrational degrees of freedom. Similar calculations may be performed for other substances.

• Thank you this is very helpful. By average speed, I meant the actual distance traveled in a specific period of time as we describe the speed of a car without thinking about axes. Ignoring rotation is fine for me but can you explain the vibration of gas molecules ? I thought they didn't vibrate by themselves (without colliding others) since they are not connected unlike liquid and solids molecules.
– Xtro
Jul 27 '14 at 18:31
• Then you probably want $\langle v\rangle=\sqrt{\frac{8kT}{\pi m}}$=476m/s=1065mph. All molecules vibrate: the distance between the atoms in the molecule oscillates like a spring. Also see en.wikipedia.org/wiki/Maxwell%E2%80%93Boltzmann_distribution and en.wikipedia.org/wiki/Molecular_vibration Jul 28 '14 at 16:25
• Oh yeah you are right. When I was thinking about molecules, I forgot to think about atoms in them. Of-course they vibrate :) Can we say an hydrogen atom also vibrates by itself without colliding anything else ? Maybe because interaction between the proton and electron or maybe the interaction between quarks ?
– Xtro
Jul 28 '14 at 18:11
• It's harder to apply classical concepts to electrons. See physics.stackexchange.com/questions/20187/… for a back-of-the envelope discussion. Quark vibrations should exist within a proton, but a proton is a very messy system because of the self-interaction of gluons. Jul 28 '14 at 20:49