The isothermal compressibility of a material is defined as the fractional decrease of volume per unit increase of pressure, at constant temperature. This is the compressibility usually employed in thermodynamic calculations.

Sometimes, as when considering the propagation of sound waves, the adiabatic compressibility is required. This quantity is defined as the fractional decrease of volume per unit increase of pressure, when no heat flows in or out of the system. Because there is no heat flow, the entropy remains unchanged according to the second law of thermodynamics (in a reversible process), so the adiabatic process is also isentropic, i.e., a process that takes place at constant entropy. In adiabatic compression, the temperature rises, so the pressure increases more sharply than in isothermal compression. Therefore, the compressibility at constant entropy is always smaller than that at constant temperature.

It follows from thermodynamic relationships that the ratio of the...