Page 175 - Physical chemistry understanding our chemical world

142 REACTION SPONTANEITY AND THE DIRECTION OF THERMODYNAMIC CHANGE

provided that we know the way C p varies with temperature, expressed as a mathematical
power series in T . For example, C p for liquid chloroform CHCl 3 is

−2
C p /JK −1 mol −1 = 91.47 + 7.5 × 10 T
Alternatively, because Equation (4.9) has the form of an integral, we could plot a
graph of C p ÷ T (as y) against T (as x) and determine the area beneath the curve. We
would need to follow this approach if C p ÷ T was so complicated a function of T that
we could not describe it mathematically.

Justification Box 4.1

Entropy is the ratio of a body’s energy to its temperature according to the Clausius
equality (as defined in the next section). For a reversible process, the change in entropy
is defined by
dq
dS = (4.11)
T
where q is the change in heat and T is the thermodynamic temperature. Multiplying
the right-hand side of Equation (4.11) by dT/dT (which clearly equals one), yields
dq dT
dS = × (4.12)
T dT
If no expansion work is done, we can safely assume that q = H. Substituting H for q,
and rearranging slightly yields

dH 1
dS = × dT (4.13)
dT T
where the term in brackets is simply C p . We write

C p
dS = dT (4.14)
T
Solution of Equation (4.14) takes two forms: (a) the case where C p is considered not
to depend on temperature (i.e. determining the value of S over a limited range of
temperatures) and (b) the more realistic case where C p is recognized as having a finite
temperature dependence.

(a) C p is independent of temperature (over small temperature ranges).

S 2 T 2 1
dS = C p dT (4.15)
T
S 1 T 1
So
S = S 2 − S 1 = C p [ln T ] T 2 (4.16)
T 1