Page 71 - Introduction to chemical reaction engineering and kinetics

3.4 Experimental Strategies for Determining Rate Parameters 53

SOLUTION

From the rate law and the material-balance equation 2.2-10, the equation to be integrated
is

- dc, = k,dt
CACB
As in Example 3-3, cn iS not independent of CA, but is related to it through equation 3.4-5,
to which we add the extent of reaction to emphasize that there is only one composition
variable:

CA - cAo = cB - cBo = _ 5 (3.4-5a)
VA VB V
where 5 is the extent of reaction introduced in equation 2.2-5, and equation 2.2-7 has
been used to eliminate the mole numbers from 2.2-5. Equation 3.4-5a may then be used
to eliminate both cA and cn from equation 3.4-12, which becomes:

d5 = -adt (3.4-12a)
kA
(CA0 + d?(cB, + @)

where a = VA/V and b = v,lV. Integration by the method of partial fractions followed
by reversion from 5 to CA and cn results in

In(?) = In(z)+ 2(vBcAo - vAcBo)t (3.4-13)

I I

Thus, ln(cA/cn) is a linear function oft, with the intercept and slope as indicated, for this
form of rate law. The slope of this line gives the value of kA, if the other quantities are
known.

Equations 3.4-9, -10 or -11, and -13 are only three examples of integrated forms of
the rate law for a constant-volume BR. These and other forms are used numerically in
Chapter 4.

Fractional lifetime method. The half-life, t1,2, of a reactant is the time required for its
concentration to decrease to one-half its initial value. Measurement of t1,2 can be used
to determine kinetics parameters, although, in general, any fractional life, tfA, can be
similarly used.
In Example 2-1, it is shown that tfA is independent of cAO for a first-order reaction
carried out in a constant-volume BR. This can also be seen from equation 3.4-10 or -11.
Thus, for example, for the half-life,

t1,2 = (lll2)/kA (TZ = 1) (3.4-14)

and is independent of cAO. A series of experiments carried out with different values of
CA0 would thus all give the same value of tl,*, if the reaction were first-order.