# PV=nRT for solids?

1. May 29, 2008

### Mechanic

PV=nRT for solids??

For a perfect gas PV = nRT. This is a very handy little equation that allows determination of temperature given pressure, volume, etc. for gasses, but is there some equivalent equation that relates temperature, pressure and volume for solids?
Thanks

2. May 29, 2008

### John Creighto

Although the ideal gas law can be derived from statistical physics for many real gases it is just a linearization. I’d probably write something like:

PoVo=Z R To
Z=(Zo+(dZ/dTo)dTo+(dZ/dVo)dVo+(dZ/dPo)dPo )

Where R is the ideal gas constant
V is the molar volume

The Wikipedia article on compressibility may give more insight.
http://en.wikipedia.org/wiki/Compressibility

3. May 29, 2008

### Staff: Mentor

No.

Or rather - there is no equation of practical value. Just like with solutions. You need a lot of tables with experimental coefficients to use these equations.

4. May 29, 2008

### John Creighto

Yeah but these things are often liberalized. For instance the spring coefficient or the coefficient of thermal expansion.

5. May 29, 2008

### Staff: Mentor

Yes, they are linearized - but still you have to find coefficients for your material in experiment (or tables).

Ideal gas equation gives reasonably good results regardless of the gas used for the wide range of PT - using only one universal constant. There is no one universal constant for solids and liquids nor one universal equation that'll not use experimentally determined parameter. I bet that's the answer OP was looking for. Could be I am wrong.

6. May 29, 2008

### Mechanic

Maybe I should start differently...according to the Ideal Gas law, for a given amount of gas at constant volume the temperature will increase if P is increased. Is the same thing true of a solid?

7. May 29, 2008

### FredGarvin

No. However you do need to state the entire situation in which you are looking. For example, if you have a piece of steel that is sitting by itself and you raise the temperature, there will be no increase in stress due to the delta T. However, if you have that same piece in intimate contact and constrained and add a delta T to it, you will most certainly induce stresses.

In what I think is the spirit of your question, the answer is it is not applicable. That is why it is referred to as the ideal GAS law.

8. May 29, 2008

### Alienjoey

No, because solids are considered to have a "definite" volume, since their particles are tightly packed together.

Here's an example: If you cook a steak in the oven at, for example, three times room temperature, does the steak expand to three times its size? With solids and liquids, the densities must be used to convert to moles, while temperature and pressure have little effect on the volume.

9. May 29, 2008

### Staff: Mentor

That's an interesting question. I wonder... In most (if not all) cases when you increase pressure volume will decrease. That means some work was done on the object - this work is PdV and it most likely increased object temperature (even if the process is isothermic work was done, just the heat was transferred outisde). So in a way it is the same for solids as for gas, although you will need very high pressures for this effect to be observable.

Last edited: May 29, 2008
10. May 29, 2008

### John Creighto

Maybe at zero k but the vibration of the molecules causes phonons to travel within the solid which behave like an ideal gas. The vibration of the molecules (related to temperature) determines the average distance between molecules related to volume.

11. May 29, 2008

### Mapes

$$\left(\frac{\partial T}{\partial P}\right)_V=-\left(\frac{\partial V}{\partial P}\right)_T\left(\frac{\partial V}{\partial T}\right)_P^{-1}=\beta/\alpha=(\mathrm{compressibility})/ (\mathrm{volumetric}\,\mathrm{thermal} \,\mathrm{expansion}\,\mathrm{coefficient})\neq 0$$