# 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 (Fig. 13).

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

**Fig. 13**

*K*.where

*p*= increase in pressure;*V*= original volume; Î”*V*= change in volume.The negative sign shows that with increase in pressure

*p*, the volume decreases by Î”*V*, i.e., if*p*is positive, Î”*V*is negative. The reciprocal of bulk modulus is called compressibility.*C*= compressibility =

The SI unit of compressibility is N

^{â€“1}m^{2 }and CGS unit is dyne^{â€“1}cm^{2}.Gases have two bulk moduli, namely, isothermal elasticity

*E*_{Î¸}_{ }and adiabatic elasticity*E*._{Ï†}**Isothermal elasticity (**Elasticity possessed by a gas in isothermal condition is defined as isothermal elasticity.

*E*)_{Î¸}For isothermal process,

*PV*= constant (Boyleâ€™s law)Differentiating both sides,

*P dV*+

*V dP*= 0 â‡’

*P dV*= â€“

*V dP*

=

âˆ´ E

*E*_{Î¸}âˆ´ E

_{Î¸}= Pi.e., isothermal elasticity is equal to pressure.

**Adiabatic elasticity (**Elasticity possessed by a gas in adiabatic condition is defined as adiabatic elasticity.

*E*)_{Ï†}For adiabatic process,

*PV**= constant (Poissonâ€™s law)*^{Î³}Differentiating both sides,

*P**Î³**V*^{Î³}^{â€“1}*dV*+*V*^{Î³}*dP*= 0â‡’

*Î³**P dV*+*V dP*= 0 =

*EÎ¸*âˆ´

*E*=_{Ï†}*Î³P*i.e., adiabatic elasticity is equal to

*Î³*times pressure.*[where**Î³*=*C*/_{p }*C*]_{v}