Page 74 - The engineering of chemical reactions

58 Reaction Rates, the Batch Reactor, and the Real World

fi PT = exp(-AGi/RT)
j=l .
We need to determine AG: at the temperature of the calculation. This is obtained from the
van’t Hoff equation
d(-AG;IRT) AH; d In K,,
Z---E
dT RT2 dT
which relates equilibrium composition to temperature. This equation can be integrated from
298 K to any temperature T to yield

T
AH,
In KT = In K298 + y&T
s
2 9 8
The enthalpy change of reaction varies with temperature as

AHR = C VjHj = C vj

where Hfj,298 is the heat of formation of species j at 298 K. The variation of Hfj with T is
determined by the variation of Cpj with T, as calculated by this integral.
It is a good approximation for estimations to ignore the Cpj term so that KT is given
approximately by the expression

In KT = In K298 - asp--)

While A HR does not vary strongly with T, AG: and K,, are strong functions of T. This is
illustrated in Figure 2-l 1, which shows that AG: varies by orders of magnitude from 298
to 1000 K. Note also that K,, either increases or decreases, depending on the sign of A HR.
Endothermic reactions have favorable equilibrium compositions at high temperatures, while
exothermic reactions have favorable equilibrium compositions at low temperatures. You
probably learned of these effects through Le Chatalier’s Principle, which states that, if a
reaction liberates heat, its equilibrium conversion is more favorable at low temperature, and
if a reaction causes an increase in the number of moles, its equilibrium is more favorable at
low pressure (and the converse situations). While these equations permit you to calculate
these effects, this simple principle permits you to check the sign of the change expected.
For multiple reactions we have simultaneous equilibrium equations

fi a? = exp(-AG&/RT) = Ki, i = 1,2,...,R
j=l
which is a set of R polynomials that must be solved simultaneously, as is considered in
thermodynamics courses.
Three commonly used sets of units for describing densities of fluids are partial pressure