# Does the pressure-volume-temperature realation of gases also work on solids?

The ratio between the pressure-volume product and the temperature of a system remains constant.

My question is: Does this law also works on solids? If not, Is there any kind of equivalent law?

• what is the pressure of a solid? – pentane Jul 7 '18 at 21:24
• The pressure in a pressurised fluid tank in which the solid is immersed? – Philip Wood Jul 7 '18 at 22:18
• @pentane, I suppose that pressure in a solid would be the amount of force you are producing on it in order to compress it. – Ender Look Jul 8 '18 at 2:22
• @PhilipWood, I was not talking exactly about that but could also be. I'm not sure if my idea is correct, but wouldn't the pressure in a solid be the amount of force exerted on the solid (in order to compress it). – Ender Look Jul 8 '18 at 2:24
• @Ender Look. If you exert "a force" on a solid you will cause the solid to accelerate! To exert a pressure requires a combination of forces pushing at right angles on the faces of the solid. – Philip Wood Jul 8 '18 at 7:30

For small changes in the state of an ideal gas you can derive from $pV=nRT$ (by taking logs and then taking differentials) the relationship$$\text{d} \ln V=\frac{1}{T}\text{d}T-\frac{1}{p}\text{d}p.$$This has the same form as Chester Miller's relationship for a solid, so you can regard $\frac{1}{T}$ as his $\alpha$, and $\frac{1}{p}$ as his $\beta$ for a gas!
So, to answer the question, the $p, V, T$ relationship for gases does not work for solids, but in differential form the same law does apply to gases and solids. But note the comment of my2cts about relative magnitudes: $\alpha$ and $\beta$ are much smaller for a solid than for a gas. This implies that the magnitudes of small changes $\text{d}T$ and $\text{d}p$ over which $\alpha$ and $\beta$ are approximately constant is much smaller for a gas than for a solid.
$$dlnV=\alpha dT-\beta dP$$ where $\alpha$ is the coefficient of volume thermal expansion and $\beta$ is the bulk compressibility (both functions of temperature and pressure).