When the density changes, both the pressure p and the temperature T will change, in general. The usual way to describe these changes in thermodynamics is to change the volume V occupied by a fixed number N of particles, so that
which are positive quantities characteristic of the material (fluid or solid). To convert from dV/V to a change of density, we note that a fixed N implies that is a constant (it is the mass contained in V). Then and we get
As we said already, there are two limiting cases where heat transfer and fluid motions are effectively decoupled, and then the dynamics of the fluid is fully determined by the appropriate compressibility coefficient:
This condition allows us to compute . For example, in an ideal gas dE=CVdT, hence for an adiabatic process CVdT = -pdV. Using this result in eq. (5.5), we find (1 + NkB)pdV + Vdp = 0 and by the definition (5.8):
Recalling that Cp=CV+NkB is the specific heat at constant pressure and comparing with equation (5.5), we see that
In a typical liquid, there is little difference between and , but in a monatomic ideal gas the difference is large: CV=(3/2)NkB and Cp=(5/2)NkB. The ratio Cp/CV appears often in the physics of gases and is called . For a monatomic ideal gas, ; for air near room temperature is very close to 1.4, which is the value for a diatomic ideal gas when the molecules can be regarded as rigid (at higher temperatures, molecular vibrations increase CV, so that decreases).
The inverse of the compressibility coefficient is called the bulk modulus B. We can define and , but we can use simply B and let the context distinguish what we mean (keeping in mind that for liquids, and solids, the distinction is unimportant). We also want to relate pressure changes to density changes, rather than volume changes. Using rdV + Vdr = 0 as in the derivation of eq. (5.4), we see that
Values of 1/B are given in Table 7.16 of PQRG. We see that for common organic liquids B is about 1 GigaPascal (10N/m2), increasing slowly with pressure. Water is a little less compressible: GPa at C and MPa. Mercury is a lot less compressible: with GPa it is getting close to the range of solid metals.