# Could one estimate the size of air-molecules based on analysis of the sound?

Say one knows air is composed of molecules (atoms). Could one estimate the size of molecules (atoms) by analysing the sound properties as one perceives them (clearness, speed, etc.)?

Yes, you can measure the molecular mass by measuring the speed of the sound and using the equation for the speed of sound in an ideal gas:

$$v = \sqrt{\frac{\gamma k T}{m}}$$

where $\gamma$ is the adiabatic index, $k$ is Boltzmann's constant, $T$ is the temperature and $m$ is the mass of a gas molecule. This works because most gases at normal pressures are close to ideal. This gives you the mass of the molecule not its size, but if you know the density of the liquid or solid phase of the gas you can divide by the density to get the approximate volume of the molecule.

• What's $\gamma$?
– rob
May 11 '14 at 8:14
• @rob see link that JohnRennie wrote(speed of sound in an ideal gas), Wikipedia page says this: "$\gamma$ is the adiabatic index also known as the isentropic expansion factor. It is the ratio of specific heats of a gas at a constant-pressure to a gas at a constant-volume($C_p/C_v$)"
– G B
May 11 '14 at 8:18
• I can see that this gives a good estimate of the mass of a molecule. But at normal STP conditions, isn't the average separation between molecules about 10 molecular diameters? (Air is about 1000 times less dense that water). So wouldn't this give an estimate 10 times too big? May 11 '14 at 10:07
• @mmesser314: you'd need to use the density of the liquid or solid (as I said in my answer). In practice just use the density of water and you'll be within a factor of two of the correct size. May 11 '14 at 10:32
• @John Rennie: Ah, right. Careless reading. May 11 '14 at 10:39