Bulk modulus and compressibility

Squeezed from every side

A stone at the bottom of a lake is pressed equally on every face. It keeps its shape, but its volume shrinks a tiny bit.

The stress here is the extra pressure $p$; the strain is the fractional change of volume, $\dfrac{\Delta V}{V}$. Their ratio is the bulk modulus:

The minus sign is there because the volume falls ($\Delta V$ is negative) when the pressure rises, so $B$ comes out positive.

Compressibility

The compressibility is $k = \dfrac{1}{B}$: the fraction of volume lost for each pascal. Water's is $4.5 \times 10^{-10}\ \text{Pa}^{-1}$, tiny, which is why we usually treat water as incompressible.

You should find that water loses under $2\%$ of its volume at $4\ \text{km}$, and steel almost nothing.

Gases

Squeeze a gas slowly, so its temperature stays the same: $pV$ is constant, and its bulk modulus works out to be just its pressure, $B = p$. At $10^{5}\ \text{Pa}$, that is about $20\,000$ times smaller than water's. Gases are easy to squeeze.

Watch out. Solids, liquids and gases all have a bulk modulus. Only solids have a shear modulus.

A worked example

An aluminium sphere of volume $400\ \text{cm}^3$ is lowered to where the extra pressure is $3.0 \times 10^{7}\ \text{Pa}$. How much does it shrink? ($B = 7.5 \times 10^{10}\ \text{Pa}$.)

1. Fraction. $\dfrac{\Delta V}{V} = \dfrac{3.0 \times 10^{7}\ \text{Pa}}{7.5 \times 10^{10}\ \text{Pa}} = 4.0 \times 10^{-4}$.
2. Volume. $\Delta V = 400\ \text{cm}^3 \times 4.0 \times 10^{-4} = 0.16\ \text{cm}^3$.

Think first. The fraction comes first; it is the same for a big and a small sphere of the same metal.

Practise with new numbers

Try it in Python

The program finds how much sea water is squeezed at several depths.

Remember these

• $B = -\dfrac{p}{\Delta V/V}$; compressibility $k = \dfrac{1}{B}$.
• Water: $B = 2.2\ \text{GPa}$; nearly incompressible.
• A gas at a steady temperature has $B = p$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A pressure of $3.7 \times 10^{7}\ \text{Pa}$ acts on glass ($B = 37\ \text{GPa}$). By what fraction does its volume fall?

2. Why is there a minus sign in the definition of $B$?

3. What is the bulk modulus of air at a pressure of $1.0 \times 10^{5}\ \text{Pa}$, squeezed slowly?

4. Which has the larger compressibility, water or steel?