Thermodynamics and Chemistry

CHAPTER 7 PURE SUBSTANCES IN SINGLE PHASES

7.5 PARTIALDERIVATIVES WITHRESPECT TOT,p,ANDV 179

appropriate partial derivative ofT,p, orV. For instance, from the partial derivative

.@U=@V /TD.T=T/p, we obtain

@U

@p



T

D

@U

@V



T

@V

@p



T

D

T

T

p



.TV /D.TCTp/ V (7.5.5)

The remaining partial derivatives can be found by differentiatingUDHpV,HD

UCpV,ADUTS, andGDHTSand making appropriate substitutions.

Whenever a partial derivative appears in a derived expression, it is replaced with an

expression derived in an earlier step. The expressions derived by these steps constitute

the full set shown in Tables7.1,7.2, and7.3.

Bridgman^7 devised a simple method to obtain expressions for these and many

other partial derivatives from a relatively small set of formulas.

7.5.2 The Joule–Thomson coefficient

The Joule–Thomson coefficient of a gas was defined in Eq.6.3.3on page 157 byJTD
.@T=@p/H. It can be evaluated with measurements ofTandpduring adiabatic throttling
processes as described in Sec.6.3.1.
To relateJTto other properties of the gas, we write the total differential of the enthalpy
of a closed, single-phase system in the form

dHD



@H

@T



p

dTC



@H

@p



T

dp (7.5.6)

and divide both sides by dp:

dH

dp

D



@H

@T



p

dT

dp

C



@H

@p



T

(7.5.7)

Next we impose a condition of constantH; the ratio dT=dpbecomes a partial derivative:

0 D



@H

@T



p



@T

@p



H

C



@H

@p



T

(7.5.8)

Rearrangement gives 
@T
@p



H

D

.@H=@p/T

.@H=@T /p

(7.5.9)

The left side of this equation is the Joule–Thomson coefficient. An expression for the partial
derivative.@H=@p/Tis given in Table7.1, and the partial derivative.@H=@T /pis the heat
capacity at constant pressure (Eq.5.6.3). These substitutions give us the desired relation

JTD

.T1/V

Cp

D

.T1/Vm
Cp;m

(7.5.10)

(^7) Ref. [ 22 ]; Ref. [ 21 ], p. 199–241.