Isothermal and isentropic compressibilities

Reference: Daniel V. Schroeder, An Introduction to Thermal Physics, (Addison-Wesley, 2000) – Problem 5.16.

An expression similar to that relating the heat capacities can be derived to relate the isothermal and isentropic compressibilities {\kappa_{T}} and {\kappa_{S}}, defined as

\displaystyle   \kappa_{T} \displaystyle  \equiv \displaystyle  -\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_{T}\ \ \ \ \ (1)
\displaystyle  \kappa_{S} \displaystyle  \equiv \displaystyle  -\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_{S} \ \ \ \ \ (2)

These quantities measure the fractional change in volume of a substance in response to a change in pressure. To obtain the relation between them, we use a method similar to that for heat capacities {C_{V}} and {C_{P}}.

If we write {S=S\left(P,T\right)} then

\displaystyle  dS=\left(\frac{\partial S}{\partial P}\right)_{T}dP+\left(\frac{\partial S}{\partial T}\right)_{P}dT \ \ \ \ \ (3)

Also, starting with {V=V\left(P,S\right)} we have

\displaystyle  dV=\left(\frac{\partial V}{\partial P}\right)_{S}dP+\left(\frac{\partial V}{\partial S}\right)_{P}dS \ \ \ \ \ (4)

Substituting 3 into 4 we get

\displaystyle  dV=\left[\left(\frac{\partial V}{\partial S}\right)_{P}\left(\frac{\partial S}{\partial P}\right)_{T}+\left(\frac{\partial V}{\partial P}\right)_{S}\right]dP+\left(\frac{\partial V}{\partial T}\right)_{P}dT \ \ \ \ \ (5)

At constant temperature {dT=0} and we get

\displaystyle   \left(\frac{\partial V}{\partial P}\right)_{T} \displaystyle  = \displaystyle  \left(\frac{\partial V}{\partial S}\right)_{P}\left(\frac{\partial S}{\partial P}\right)_{T}+\left(\frac{\partial V}{\partial P}\right)_{S}\ \ \ \ \ (6)
\displaystyle  -V\kappa_{T} \displaystyle  = \displaystyle  \left(\frac{\partial V}{\partial S}\right)_{P}\left(\frac{\partial S}{\partial P}\right)_{T}-V\kappa_{S} \ \ \ \ \ (7)

From the Maxwell relation from the Gibbs energy

\displaystyle  \left(\frac{\partial S}{\partial P}\right)_{T}=-\left(\frac{\partial V}{\partial T}\right)_{P} \ \ \ \ \ (8)

Also, from the definition of the thermal expansion coefficient {\beta}

\displaystyle  \beta\equiv\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_{P} \ \ \ \ \ (9)

Combining these last two equations gives

\displaystyle  -V\kappa_{T}=-\beta V\left(\frac{\partial V}{\partial S}\right)_{P}-V\kappa_{S} \ \ \ \ \ (10)

To get rid of the last partial derivative, we observe that the volume change {dV} due to a temperature change {dT} at constant pressure is

\displaystyle  dV=\beta V\;dV \ \ \ \ \ (11)

The entropy change due to an influx of heat {dQ} at constant pressure at temperature {T} is

\displaystyle   dS \displaystyle  = \displaystyle  \frac{dQ}{T}\ \ \ \ \ (12)
\displaystyle  \displaystyle  = \displaystyle  C_{P}\frac{dT}{T} \ \ \ \ \ (13)

Dividing these two relations gives

\displaystyle  \left(\frac{\partial V}{\partial S}\right)_{P}=\frac{TV\beta}{C_{P}} \ \ \ \ \ (14)

Inserting this into 10 and cancelling off a factor of {-V} gives the final result

\displaystyle  \kappa_{T}=\kappa_{S}+\frac{TV\beta^{2}}{C_{P}} \ \ \ \ \ (15)

For an ideal gas, we can use this equation to work out {\kappa_{S}}:

\displaystyle   \beta \displaystyle  = \displaystyle  \frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_{P}=\frac{Nk}{PV}=\frac{1}{T}\ \ \ \ \ (16)
\displaystyle  \kappa_{T} \displaystyle  = \displaystyle  -\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_{T}=\frac{NkT}{P^{2}V}=\frac{1}{P}\ \ \ \ \ (17)
\displaystyle  C_{P} \displaystyle  = \displaystyle  C_{V}+Nk\ \ \ \ \ (18)
\displaystyle  \displaystyle  = \displaystyle  Nk\left(1+\frac{f}{2}\right)\ \ \ \ \ (19)
\displaystyle  \kappa_{S} \displaystyle  = \displaystyle  \frac{1}{P}-\frac{V}{NkT\left(1+\frac{f}{2}\right)}\ \ \ \ \ (20)
\displaystyle  \displaystyle  = \displaystyle  \frac{1}{P}\frac{f}{\left(f+2\right)} \ \ \ \ \ (21)

where in the third line, we’ve used Schroeder’s equation 1.48, and {f} is the number of degrees of freedom of each gas molecule.

To check this, recall that for an isentropic (adiabatic) process in an ideal gas

\displaystyle   PV^{\gamma} \displaystyle  = \displaystyle  K\ \ \ \ \ (22)
\displaystyle  V \displaystyle  = \displaystyle  \left(\frac{K}{P}\right)^{1/\gamma} \ \ \ \ \ (23)

where {\gamma=\left(f+2\right)/f} and {K} is a constant. So

\displaystyle   \kappa_{S} \displaystyle  = \displaystyle  -\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_{S}\ \ \ \ \ (24)
\displaystyle  \displaystyle  = \displaystyle  -\left(\frac{P}{K}\right)^{1/\gamma}\left(-\frac{1}{\gamma}\right)\left(\frac{K}{P}\right)^{1/\gamma}\frac{1}{P}\ \ \ \ \ (25)
\displaystyle  \displaystyle  = \displaystyle  \frac{1}{P\gamma}=\frac{1}{P}\frac{f}{\left(f+2\right)} \ \ \ \ \ (26)

which is the same as 21, so equation 15 checks out for an ideal gas.

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