Page 62 - Introduction to chemical reaction engineering and kinetics

44 Chapter 3: Experimental Methods in Kinetics: Measurement of Rate of Reaction

3.1.3 Effect of Temperature: Arrhenius Equation; Activation Energy
A rate of reaction usually depends more strongly on temperature than on concentra-
tion. Thus, in a first-order (n = 1) reaction, the rate doubles if the concentration is
doubled. However, a rate may double if the temperature is raised by only 10 K, in the
range, say, from 290 to 300 K. This essentially exponential behavior is analogous to the
temperature-dependence of the vapor pressure of a liquid, p*, or the equilibrium con-
stant of a reaction, Keq. In the former case, this is represented approximately by the
Clausius-Clapeyron equation,
dlnp* AHVaP(T)
- = (3.1-4)
dT RT2
where AHvap is the enthalpy of vaporization. The behavior of K,, is represented (ex-
actly) by the van’t Hoff equation (Denbigh, 1981, p. 144)
d In K,, AH’(T)
~ = (3.1-5)
dT RT2
where AH” is the standard enthalpy of reaction.
Influenced by the form of the van7 Hoff equation, Arrhenius (1889) proposed a sim-
ilar expression for the rate constant k, in equation 3.1-2, to represent the dependence
of (-Y*) on T through the second factor on the right in equation 3.1-1:

(3.1-6)

where EA is a characteristic (molar) energy, called the energy of activation. Since ( -rA)
(hence k.J increases with increasing Tin almost every case, EA is a positive quantity
(the same as AHVaP in equation 3.1-4, but different from AH” in equation 3.1-5, which
may be positive or negative).
Integration of equation 3.1-6 on the assumption that EA is independent of T leads to

1 /
In kA = In A - E,IRT (3.1-7)
or
kA = A exp( -E,IRT) (3.1-8)

where A is a constant referred to as the pre-exponential factor. Together, EA and A are
called the Arrhenius parameters.
Equations 3.1-6 to -8 are all forms of the Arrhenius equation. The usefulness of this
equation to represent experimental results for the dependence of kA on T and the nu-
merical determination of the Arrhenius parameters are explored in Chapter 4. The in-
terpretations of A and EA are considered in Chapter 6 in connection with theories of
reaction rates.

It is sometimes stated as a rule of thumb that the rate of a chemical reaction doubles for
a 10 K increase in T. Is this in accordance with the Arrhenius equation? Determine the