# Bulk Modulus Formula Derivation

## Bulk Modulus

Bulk modulus of a substance is defined as the ratio of infinitesimal pressure increase to a decrease of the volume. Bulk modulus is meaningful only for a fluid. It is denoted as either K or B. The reciprocal of bulk modulus is compressibility of a substance and this is the relation between bulk modulus and compressibility.

The mathematical representation of bulk modulus is given as follows:

For bulk modulus K > 0;

$K=-V\frac{dP}{dV}$

Where,

P: pressure

V: volume

$\frac{dP}{dV}$ : derivative of pressure with respect to volume

### Bulk modulus for unit mass

$K=\rho \frac{dP}{d\rho }$

Where,

⍴: density

$\frac{dP}{dV}$ : derivative of pressure with respect to density

Bulk modulus in thermodynamics with constant temperature and constant entropy is given as follows:

Isentropic bulk modulus Ks;

$K_{s}=\gamma p$

Isothermal bulk modulus KT;

$K_{T}= p$

Where,

p: pressure

γ: heat capacity ratio

Following is the table of bulk modulus of a few common materials:

 Material Bulk modulus in GPa Diamond 443 Rubber 1.5 to 2 Steel 160

## Bulk Modulus Formula Derivation

As we know the ratio between change in pressure to change in volumetric strain is dependent on bulk modulus of the material, following is the derivation showing the relationship:

$\frac{-\delta V}{V}=\frac{\delta p}{K}$ (negative sign to indicate that with increase in pressure, volume decreases)

Where,

δV: change in volume

δp: change in pressure

V: actual volume

K: bulk modulus

$K=-V\frac{dp}{dV}$ (as δp tends to 0) (eq.1)

$V=\frac{1}{\rho }$ (unit mass of the substance) (eq.2)

$Vd\rho +\rho dV=0$ (after differentiation)

$dV=-(\frac{V}{\rho })d\rho$ (eq.3)

$K=\frac{-Vdp}{-(\frac{V}{\rho })d\rho }$ (after substituting eq.3 in eq.1)

$∴ K=\rho \frac{dP}{d\rho }$

Thus, above is the derivation of bulk modulus which is mainly applicable to liquids as gases are highly compressible which makes K vary.

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