Page 71 - Physical chemistry eng

48 CHAPTER 3 The Importance of State Functions: Internal Energy and Enthalpy

Suppose this expression needs to be evaluated for a specific substance, such as N 2 gas.
What quantities must be measured in the laboratory in order to obtain numerical values
for (0P>0T) V and (0P>0V) T ? Using Equations (3.6) and (3.7),
0P 0V 0T
a b a b a b =-1
0V T 0T P 0P V
0V
a b
0P 0P 0V 0T P b
a b =- a b a b =- = and
0T V 0V T 0T P a 0V b k
0P T
0P 1
a b =- (3.8)
0V T kV
where b and k are the readily measured isobaric volumetric thermal expansion
coefficient and the isothermal compressibility, respectively, defined by

1 0V 1 0V
b = a b and k =- a b (3.9)
V 0T P V 0P T
Both (0V>0T) P and (0V>0P) T can be measured by determining the change in volume
of the system when the pressure or temperature is varied, while keeping the second
variable constant.
The minus sign in the equation for is chosen so that values of the isothermal
k
compressibility are positive. For small changes in T and P, Equations (3.9) can be
written in the more compact form: V(T ) = V(T )(1 + b[T - T ]) and
2
1
1
2
k
V(P ) = V(P )(1 - k[P - P ]) . Values for b and for selected solids and liq-
1
2
1
2
uids are shown in Tables 3.1 and 3.2, respectively.
Equation (3.8) is an example of how seemingly abstract partial derivatives can be
directly linked to experimentally determined quantities using the mathematical properties of
state functions. Using the definitions of and , Equation (3.3) can be written in the form
k
b
b 1
dP = dT - dV (3.10)
k kV
which can be integrated to give
T f V f
b 1 b 1 V f
¢P = dT - dV L (T - T ) - ln (3.11)
i
f
k kV k k V
3 3 i
T i V i
TABLE 3.1 Volumetric Thermal Expansion Coefficient for Solids and Liquids
at 298 K
-1
4
6
-1
Element 10 b>(K ) Element or Compound 10 b>(K )
Ag(s) 57.6 Hg(l) 1.81
Al(s) 69.3 CCl 4 (l) 11.4
Au(s) 42.6 CH 3 COCH 3 (l) 14.6
Cu(s) 49.5 CH 3 OH(l) 14.9
Fe(s) 36.9 C 2 H 5 OH(l) 11.2
Mg(s) 78.3 C 6 H 5 CH 3 (l) 10.5
Si(s) 7.5 C 6 H 6 (l) 11.4
W(s) 13.8 H 2 O(l) 2.04
Zn(s) 90.6 H 2 O(s) 1.66

Sources: Benenson, W., Harris, J. W., Stocker, H., and Lutz, H. Handbook of Physics. New York: Springer,
2002; Lide, D. R., ed. Handbook of Chemistry and Physics. 83rd ed. Boca Raton, FL: CRC Press, 2002;
Blachnik, R., ed. D’Ans Lax Taschenbuch für Chemiker und Physiker. 4th ed. Berlin: Springer, 1998.