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When the density $\rho $ 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  
dV=\left( \frac{\partial V}{\partial p}\right) _{T}dp+\left( \frac{\partial V}{\partial T}\right) _{p}dT

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It is convenient to consider the fractional volume change dV/V and to define the isothermal compressibility  
\kappa _{T}=-\frac{1}{V}\left( \frac{\partial V}{\partial p}\right) _{T}

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and the thermal expansion coefficient

\beta =\frac{1}{V}\left( \frac{\partial V}{\partial T}\right) _{p} 

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 $\rho V$ is a constant (it is the mass contained in V). Then $\rho dV+Vd\rho =0$ and we get  
\frac{d\rho }{\rho }=-\frac{dV}{V}=\kappa _{T}dp-\beta dT

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As an example, consider an ideal gas. From pV=NkBT, get pdV+Vdp=NkBdT, hence  
\kappa _{T}=1/p\quad \quad \quad \quad {\rm (ideal\;gas)}

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\beta =1/T\quad \quad \quad \quad {\rm (ideal\;gas)}
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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:

In a typical liquid, there is little difference between $\kappa _{S}$ and $\kappa _{T}$, 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 $\gamma $. For a monatomic ideal gas, $\gamma =5/3=1.667$; for air near room temperature $\gamma $ 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 $\gamma $ decreases).

The inverse of the compressibility coefficient is called the bulk modulus B. We can define $B_{T}=1/\kappa _{T}$ and $B_{S}=1/\kappa _{S}$, 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  
dp=\frac{B}{\rho }\,d\rho

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Values of 1/B are given in Table 7.16 of PQRG. We see that for common organic liquids B is about 1 GigaPascal (10$^{9}\,$N/m2), increasing slowly with pressure. Water is a little less compressible: $B=2.17\,\,$GPa at $T=22{{}^\circ}$C and $p=0.1\,$MPa. Mercury is a lot less compressible: with $B=25\,\,$GPa it is getting close to the range of solid metals.

next up previous
Next: Sound waves Up: Compressible fluids Previous: Compressible fluids
Vittorio Celli