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

(2) |

(3) |

(4) |

(5) |

(6) |

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:*

- If the density changes slowly, so that heat conduction keeps the
temperature
*T*constant, eq. (5.4) reduces to , where is the*isothermal compressibility*(5.3). As we have already noted, is positive (because a compression corresponds to a positive*dp*and a negative*dV*), and does not depend on the size of*V*(because for a larger*V*we get a correspondingly larger*dV*for the same*dp*). However, can depend on*p*and on the temperature*T*. For example, in an ideal gas, . - If the density changes rapidly, so that no heat can flow, we have
instead , where
(7) *adiabatic compressibility,*which we now discuss. The subscript*S*denotes a process at constant entropy, in accordance with the fact that an adiabatic volume change (*dQ*=0, no heat transfer) is also isentropic (*dS*=0). In such a process all the work done in changing the volume goes into the internal energy*E*of the fluid and eq. (5.1) reduces to This condition allows us to compute . For example, in an ideal gas*dE*=*C*_{V}*dT*, hence for an adiabatic process*C*_{V}*dT*= -*pdV*. Using this result in eq. (5.5), we find*(1 + Nk*and by the definition (5.8): Recalling that_{B})pdV + Vdp = 0*C*_{p}=*C*_{V}+*Nk*_{B}is the specific heat at constant pressure^{}and comparing with equation (5.5), we see that(8) *C*_{p}>*C*_{V}, so that : it is harder to compress a fluid if heat does not flow out of it.

In a typical liquid, there is little difference between and , but in a monatomic ideal gas the difference is large: *C*_{V}=(3/2)*Nk*_{B} and *C*_{p}=(5/2)*Nk*_{B}. The ratio *C*_{p}/*C*_{V} 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 *C*_{V}, 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

(9) |

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/m^{2}),
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.