Question

What is the relation between compressibility and bulk modulus?

Answer 1

Hint:Elasticity of a material is defined as that property, due to which a body regains its initial shape and size. When external force acts on the body, internal restoring forces regain its original configuration. The internal restoring force acting per unit area is Stress. And the ratio of change in dimensions to original dimensions is defined as the Strain (\[\varepsilon \]).

Complete answer:
Hooke’s Law: It states that within the proportional limits, the stress developed by an external force is directly proportional to the strain.
\[\sigma \propto \varepsilon \]
The proportionality constant in Hooke's law is called the modulus of elasticity. We have three types of modulus: Young’s modulus, Bulk modulus, Modulus of rigidity. Young’s modulus is the ratio of longitudinal stress and strain.

Bulk Modulus: When a force acts on a body such that the body is uniformly compressed from all the sides resulting in change of volume of the body. Then the ratio of uniform compression (P) to the volumetric strain is bulk modulus of elasticity (B).
\[B = - \dfrac{P}{{\dfrac{{\Delta V}}{V}}}\]
S.I. the unit of bulk modulus is \[N{m^{ - 2}}\].

Compressibility (K) is numerically equal to reciprocal of bulk modulus. The ability to reduce the size of a material by action of pressure on it. Compressibility of gases are higher than liquids and solids.
\[K = \dfrac{1}{B}\]
Unit of compressibility is \[{N^{ - 1}}{m^2}\].

Note: Third modulus of elasticity is modulus of rigidity, the ratio of tangential stress to shear strain. Water has a high value of bulk modulus which implies it is difficult to compress water. So bulk modulus is dissimilar to modulus of elasticity as water is not stiff but it is hard to compress.