# Formulas for compressibility of solids

I am taking a course in mechanics this semester, as well as a course in reservoir physics. Both courses have sections devoted to pressure/compressibility of solids, but the formulas look slightly different, so I wondered if they really mean the same or not.

In my mechanics class I am told that:

$$\Delta P = B \left(\frac{- \Delta V}{V_0}\right)$$

Where $B$ is a constant known as the bulk modulus of a given material.

In my reservoir physics class I am presented with the formula:

$$c = -\frac{1}{V} \left(\frac{\partial V}{\partial p}\right)_{T}$$

Where $c$ is referred to as isothermal compressibility.

So my question is - are these formulas basically the same, where $c = \frac{1}{B}$? And if they are not the same, can someone please explain the difference to me? I would really appreciate if someone could help me with this!

## 1 Answer

Yes, the bulk modulus $B$ is the inverse of the isothermal compressibility $c$, $$B = \frac {1}{c}.$$ See e.g. Wikipedia.

The "bulk modulus" is more typical terminology in mechanics where we don't care about heat much and where the typical assumption is that the temperature is kept fixed (because mechanical engines start to malfunction if their temperature goes awry); the bulk modulus is "isothermal" because of the choice of the discipline, mechanics.

In thermodynamics, one speaks about compressibility – which is terminology reminiscent of gases which are "easy" in thermodynamics – and the adjective "isothermal" is very important in thermodynamics because thermodynamics is all about the differences between different ways how the heat may propagate or not propagate (in thermodynamics, we really want the temperature to change etc., it's pretty much the point of the discipline, so things are often non-isothermal).

• Awesome! Thanks a lot for clearing this up. I really appreciate it! – user12277 Sep 16 '12 at 19:00