Bulk Modulus Formula Derivation

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.

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.

To know more on other Physics related concepts, stay tuned with BYJU’S.

Related Physics articles:


Practise This Question

Calculate the load current and load voltage in the following transformer circuit. Soyrce : 30v AC, Np = 100, Ns =100, Rs =100 Ω