JEE Main & Advanced Physics Elasticity Bulk Modulus

When a solid or fluid (liquid or gas) is subjected to a uniform pressure all over the surface, such that the shape remains the same, then there is a change in volume.

Then the ratio of normal stress to the volumetric strain within the elastic limits is called as Bulk modulus. This is denoted by K.

\[K=\frac{\text{Normal stress}}{\text{volumetric strain}}\]           

\[K=\frac{F/A}{-\Delta V/V}=\frac{-pV}{\Delta V}\]

where p = increase in pressure; V = original volume; \[\Delta V=\] change in volume

The negative sign shows that with increase in pressure p, the volume decreases by \[\Delta V\] i.e. if p is positive, \[\Delta V\] is negative. The reciprocal of bulk modulus is called compressibility.

C = compressibility = \[\frac{1}{K}=\frac{\Delta V}{pV}\]

S.I. unit of compressibility is \[{{N}^{1}}{{m}^{2}}\] and C.G.S. unit is \[dyn{{e}^{1}}c{{m}^{2}}\].

Gases have two bulk moduli, namely isothermal elasticity \[{{E}_{\theta }}\] and adiabatic elasticity \[{{E}_{\phi }}\].

(1) Isothermal elasticity \[({{E}_{\theta }})\]: Elasticity possess by a gas in isothermal condition is defined as isothermal elasticity.

For isothermal process, PV = constant       (Boyle's law)

Differentiating both sides

\[PdV+VdP=0\Rightarrow

PdV=VdP\] \[P=\frac{dP}{(-dV/V)}\]\[=\frac{\text{stress}}{\text{strain}}\]\[={{E}_{\theta }}\]

\[\therefore \]\[{{E}_{\theta }}=P\]

i.e., Isothermal elasticity is equal to pressure.

(2) Adiabatic elasticity \[({{E}_{\phi }})\] : Elasticity possess by a gas in adiabatic condition is defined as adiabatic elasticity.

For adiabatic process,   \[P{{V}^{\gamma }}\]= constant           (Poisson's law)

Differentiating both sides,

\[P\,\gamma \,{{V}^{\gamma -1}}dV+{{V}^{\gamma }}dP=0\] \[\Rightarrow \]\[\gamma \,PdV+VdP=0\] \[\gamma \,P=\frac{dP}{\left( \frac{-dV}{V} \right)}=\frac{\text{stress}}{\text{strain}}\]\[={{E}_{\varphi }}\]

\[\therefore \] \[{{E}_{\phi }}=\gamma P\]

i.e., adiabatic elasticity is equal to \[\gamma \] times pressure.                       

[Where \[\gamma =\frac{{{C}_{p}}}{{{C}_{v}}}\]]  

Note :

• Ratio of adiabatic to isothermal elasticity

\[\frac{{{E}_{\varphi }}}{{{E}_{\theta }}}=\frac{\gamma \,P}{P}=\gamma >1\] \[\therefore \] \[{{E}_{\phi }}>{{E}_{\theta }}\]

i.e., adiabatic elasticity is always more than isothermal elasticity.