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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).

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 K.
where p = increase in pressure; V = original volume; ΔV = change in volume.
The negative sign shows that with increase in pressure p, the volume decreases by ΔV, i.e., if p is positive, ΔV is negative. The reciprocal of bulk modulus is called compressibility.
C = compressibility = 48275.png
The SI unit of compressibility is N –1m2 and CGS unit is dyne–1 cm2.
Gases have two bulk moduli, namely, isothermal elasticity Eθ and adiabatic elasticity Eφ.
Isothermal elasticity (Eθ) Elasticity possessed by a gas in isothermal condition is defined as isothermal elasticity.
For isothermal process, PV = constant (Boyle’s law)
Differentiating both sides,
P dV V dP = 0  P dV = –V dP
48269.png = Eθ
∴ Eθ = P
i.e., isothermal elasticity is equal to pressure.
Adiabatic elasticity (Eφ) Elasticity possessed by a gas in adiabatic condition is defined as adiabatic elasticity.
For adiabatic process, PV γ = constant (Poisson’s law)
Differentiating both sides, Pγ Vγ–1 dV + Vγ dP = 0
γ P dV + V dP = 0
48262.png = 
Eφ = γP
i.e., adiabatic elasticity is equal to γ times pressure. [where γ = Cp Cv]

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