Due to high compressibility, gases possess volume elasticity. However, the magnitude of the volume elasticity depends on conditions under which it is compressed.
If the gas is compressed such that the temperature is not allowed to change, i.e. under isothermal conditions than the corresponding volume elasticity is known as isothermal elasticity represented as KT.
If the gas is compressed such that no heat is allowed to enter or leave the system, i.e. under adiabatic conditions than the corresponding elasticity is known as adiabatic elasticity and is represented as KΦ.
Isothermal elasticity: For a perfect gas at constant temperature,
pV = constant
Differentiating the above, we get
For a perfect gas under adiabatic elasticity,
pVγ = constant
Differentiating the above wrt volume, we get
Ratio of the two is