Thermodynamics and Chemistry

CHAPTER 7 PURE SUBSTANCES IN SINGLE PHASES

7.9 STANDARDMOLARQUANTITIES OF AGAS 187

The expression forSmSm(g) in the middle column of Table7.5comes from this equation.
The equation, together with a value ofSmfor a real gas obtained by the calorimetric method
described in Sec.6.2.1, can be used to evaluateSm(g).
Now we can use the expressions forGmandSmto find expressions for molar quantities
such asHmandCp;mrelative to the respective standard molar quantities. The general
procedure for a molar quantityXmis to write an expression forXmas a function ofGmand
Smand an analogous expression forXm(g) as a function ofGm(g) andSm(g). Substitutions
forGmandSmfrom Eqs.7.9.3and7.9.6are then made in the expression forXm, and the
differenceXmXm(g) taken.
For example, the expression forUmUm(g) in the middle column Table7.5was derived
as follows. The equation defining the Gibbs energy,GDUTSCpV, was divided by
the amountnand rearranged to

UmDGmCTSmpVm (7.9.7)

The standard-state version of this relation is

Um(g)DGm(g)CTSm(g)pVm(g) (7.9.8)

where from the ideal gas lawpVm(g) can be replaced byRT. Substitutions from Eqs.7.9.3
and7.9.6were made in Eq.7.9.7and the expression forUm(g) in Eq.7.9.8was subtracted,
resulting in the expression in the table.
For a real gas at low to moderate pressures, we can approximateVmby.RT=p/CB
whereBis the second virial coefficient (Eq.7.8.17). Equation7.9.2then becomes

(g)CRTln

p

p

CBp (7.9.9)

The expressions in the last column of Table7.5use this equation of state. We can see what
the expressions look like if the gas is ideal simply by settingBequal to zero. They show that
when the pressure of an ideal gas increases at constant temperature,GmandAmincrease,
Smdecreases, andUm,Hm, andCp;mare unaffected.