Bulk Modulus

The bulk modulus of an ideal gas is a measure of its resistance to bulk compression, and is defined

$$\kappa =- V\left(\frac{\partial p}{\partial V}\right).$$ (5.146)

In fact, an ideal gas possesses a number of different bulk moduli depending on what is held constant as the pressure is varied. The two most important bulk moduli are the isothermal bulk modulus,

$$\kappa_T=-V\left(\frac{\partial p}{\partial V}\right)_T,$$ (5.147)

and the isentropic bulk modulus,

$$\kappa_S=-V\left(\frac{\partial p}{\partial V}\right)_S.$$ (5.148)

The former describes situations in which the gas undergoes isothermal compression, whereas the latter describes situations in which the gas undergoes adiabatic compression. (Note that $S$ actually denotes entropy. However, a gas that undergoes compression at constant entropy is such that no heat is added to the gas during the compression.)

According to the isothermal gas law, (5.114),

$$\ln p + \ln V = {\rm constant},$$ (5.149)

so

$$-\frac{\partial \ln p}{\partial\ln V} = -\frac{V}{p}\left(\frac{\partial p}{\partial V}\right)_T = 1,$$ (5.150)

which implies that

$$\kappa_T = p.$$ (5.151)

According to the adiabatic gas law, (5.124),

$$\ln p + \gamma\,\ln V = {\rm constant},$$ (5.152)

$$-\frac{\partial \ln p}{\partial\ln V} = -\frac{V}{p}\left(\frac{\partial p}{\partial V}\right)_S = \gamma,$$ (5.153)

which implies that

$$\kappa_S = \gamma\,p.$$ (5.154)

Note that the isentropic bulk modulus of an ideal gas is greater than its isothermal bulk modulus (because $\gamma>1$). In other words, an ideal gas resists adiabatic compression to a greater degree than it resists isothermal compression. This is the case because during adiabatic compression the work done on the gas causes its temperature to rise, leading to a greater increase in the pressure than would be obtained if the temperature were held constant.