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
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(2) |
It is convenient to consider the
fractional volume change dV/V and to define the isothermal
compressibility
![]() |
(3) |
![]() |
(4) |
As an example, consider an ideal gas. From pV=NkBT, get pdV+Vdp=NkBdT, hence
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(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:
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(7) |
![]() |
(8) |
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
,
or 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
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(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/m2),
increasing slowly with pressure. Water is a little less compressible:
GPa at
C and
p = 0.1 MPa (1 atm). Mercury is a lot less
compressible: with
GPa it is getting close to the range of solid
metals.
In general, the specific heat is dQ/dT, or TdS/dT. This can be rewritten using the basic equation of thermodynamics (just energy balance), dE = TdS - pdV. At constant volume, dE = TdS and