What is the bulk modulus of elasticity? And Bulk modulus of Gases

Modulus of Elasticity

The modulus of elasticity is also known as the elastic modulus. It is used in the measurement of materials elasticity. The modulus of elasticity is the ratio between stress and strain. When the material is under stress, it deforms but returns to the original shape once the stress is removed in the elastic region. The materials enter the plastic region beyond the yield point, in this region, the materials exhibit permanent deformation even after the tensile stress is removed. The modulus of elasticity is of three types, Bulk modulus, Shear modulus and Young’s modulus.

Bulk Modulus

Bulk modulus is a measure of the resistance to the change in the volume of the solid or fluid when external pressure is applied on the surface. When force is applied to all surfaces of the solid or fluid there is a reduction in the volume of the material, the object returns to its original volume when the pressure is removed. So, What is the bulk modulus of elasticity? Bulk modulus is given by the ratio of pressure increase to the decrease in the volume.

Bulk Modulus of Gases

There is a change in the volume of the gas when pressure is varied. The bulk modulus is given by the ratio of the volumetric stress by the volumetric strain. The pressure change can be done through an isothermal process (constant temperature) or an adiabatic process (constant entropy). The bulk modulus of gases is different for the isothermal process and adiabatic process.

The bulk modulus of the gas, B = – (ΔP /(ΔV/V))

The negative sign shows that the volume decreases when the pressure increases. The SI unit of the bulk modulus is the pascal.

Isothermal Bulk Modulus

The isothermal bulk modulus is the ratio of the change in the pressure (ΔP) to the change in the volume (ΔV/V) at a constant gas temperature.  The ideal gas equation PV = nRT is differentiated to get PΔV + VΔP = 0 i.e., ΔV/V = – ΔP/P. Here, dT = 0 since the temperature is a constant for an isothermal process. 

Substituting ΔV/V = – ΔP/p in the definition of bulk modulus we get

B_isothermal = – (Δp /(ΔV/V))  = P

The isothermal bulk modulus is equal to the pressure.

Adiabatic Bulk Modulus

The adiabatic bulk modulus is the ratio of the change in the pressure (ΔP) to the change in the volume (ΔV/V) when there is no exchange of heat with the surrounding. For an adiabatic process, the equation of state of an ideal gas PV^γ = constant. Here, γ is the ratio of the specific heats. The equation can be differentiated to get PγV^γ−1ΔV+ΔpV^γ=0 i.e., ΔV/V = – ΔP/γP.

 Substituting ΔV/V = – ΔP/γp in the definition of bulk modulus we get

B_adiabaticl = – (Δp /(ΔV/V))  = γP

B_adiabaticl = γB_isothermal

Therefore, B_adiabaticl ＞ B_isothermal

In an isobaric process, the bulk modulus is zero.

The compressibility of the gas is the reciprocal of the bulk modulus i.e., K = 1/B