Page 170 - Physical Chemistry

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So far, only elements have been considered. Suppose we want the conventional en- Section 5.5
thalpy of liquid water at T. The formation reaction is H 1 2 O → H O. Therefore Temperature Dependence
2
2
2
of Reaction Heats
H° (H O, l) H° (H O, l) H° (H , g) 1 2 H° (O , g). Knowing the conventional
2
2
m,T
m,T
2
f
m,T
2
T
enthalpies H° m,T for the elements H and O , we use the experimental H° of H O(l) (de-
2
2
2
T
f
termined as discussed earlier) to find the conventional H° of H O(l). Similarly, we can
m,T 2
find conventional enthalpies of other compounds.
5.5 TEMPERATURE DEPENDENCE OF REACTION HEATS
Suppose we have determined H° for a reaction at temperature T and we want H°
1
at T . Differentiation of H° n H° [Eq. (5.3)] with respect to T gives d H°/dT
2 i i m,i
n dH° /dT, since the derivative of a sum equals the sum of the derivatives. (The
i i m,i
derivatives are not partial derivatives. Since P is fixed at the standard-state value 1 bar,
H° and H° depend only on T.) The use of (
H /
T) C [Eq. (4.30)] gives
m,i m,i P P,m,i
d ¢H°
a n C° ¢C° (5.18)
dT i i P,m,i P
where C° is the molar heat capacity of substance i in its standard state at the tem-
P,m,i
perature of interest, and where we defined the standard heat-capacity change C°
P
for the reaction as equal to the sum in (5.18). More informally, if pr and re stand for
stoichiometric numbers of moles of products and reactants, respectively, then
d ¢H° d1H° H° 2 dH° pr dH°
pr
re
re
C° C° ¢C°
dT dT dT dT P,pr P,re P
Equation (5.18) is easy to remember since it resembles (
H/
T) C .
P P
Integration of (5.18) between the limits T and T gives
1 2
T 1
¢H° ¢H° T 2 ¢C° dT (5.19)*
T 2 P
T 1
which is the desired relation (Kirchhoff’s law).
An easy way to see the validity of (5.19) is from the following diagram:
1a2
Standard-state reactants at T S standard-state products at T 2
2
1d2
T 1b2 c
1c2
Standard-state reactants at T S standard-state products at T 1
1
We can go from reactants to products at T by a path consisting of step (a) or by a path
2
consisting of steps (b) (c) (d). Since enthalpy is a state function, H is indepen-
dent of path and H H H H . The use of H T 2 C dT [Eq. (2.79)]
a b c d T 1 P
to find H and H then gives Eq. (5.19).
d b
Over a short temperature range, the temperature dependence of C° in (5.19) can
P
often be neglected to give H° H° C° (T T ). This equation is useful if
T 2 T 1 P,T 1 2 1
we have C° data at T only, but can be seriously in error if T T is not small.
P,m 1 2 1
The standard-state molar heat capacity C° of a substance depends on T only and
P,m
is commonly expressed by a power series of the form
2
C° a bT cT dT 3 (5.20)
P,m
where the coefficients a, b, c, and d are found by a least-squares fit of the experimen-
tal C° data. Such power series are valid only in the temperature range of the data
P,m
used to find the coefficients. The temperature dependence of C was discussed in
P
Sec. 2.11 (see Fig. 2.15).