Page 88 - Modern physical chemistry

4.13 Coefficients of Thermal Expansion and Isothermal Compressibility 77

dE~(:l dT+(:!)j(:~)p dT+(:;t dP J

Comparing this result with (4.78) yields

(:~l =(:~l +(:~)J:~l

and

4. 12 The Difference Cp - Cv
At constant volume, any energy supplied to a system goes to increase its internal
energy E. But at constant pressure, some of the energy goes to do work against the exter-
nal pressure P and some to do work against the molecular interaction. The latter exhibits
itself as a cohesion in liquids and solids.
Consider a unifonn thennodynamic system at temperature T, pressure P, and volume
V. Let some energy be added as described in section 4.7. With fonnulas (4.41) and (4.35),
we have
c p -Cv =(aH) _(aE) . [4.79]

aT p aT v
Introducing expression (4.25) for the enthalpy H gives

c p -Cv =[~(E+pv)l _(aE) =(aE) +p(av) _(aE) [4.80]
aT
aT p aT v'
p aT v aT p
Combining this with the fonnula for (dEldT)p found in example 4.7 leads to

Cp-Cv =(:!wa +p(:a ~[(:!l +pn~L [4.BI[

The internal property (dEldV)r acts like pressure P in this expression. Consequently, it
is often called the internal pressure in the system. On adding energy, work is done against
this as well as against the external pressure P.

4.13 Coefficients of Thermal Expansion and
Isothermal Compressibility

How the volume of a given system varies with temperature and pressure can be
described by empirically measured coefficients.
Consider a system with the volume V at temperature T and pressure P. The funda-
mental fonn for an exact differential, (4.75), then becomes

dV =(av) dT+(av) dP. [4.82]
aT p ap T