3.5 The rate constant based on
The observed reaction rate constant cannot explain the relationship between the reaction rate and sulfuric acid concentration very well. Therefore, it is necessary to study the rate constant based on . Extensive studies have shown that the rate-determining step for homogeneous aromatic nitration in mixed acids is the attack of nitrate ions on the aromatics, and the reaction rate can be expressed by the concentration of CB and nitronium ions as follows:3,32
where is the observed rate constant expressed for , and is the concentration.
Based on the Brønsted-Bjerrum rate law (transition-state theory),33,34 the nitration rate can be expressed as follows:
where k * is the rate constant based on and , and are the activity coefficients for CB, nitronium ions and the transition-state intermediate, respectively.
The value of the activity coefficient is not easy to obtain. To calculate the pK a of weak bases in a study of acid-base equilibrium, Marziano et al.26,34 introduced the activity coefficient function M c to describe the change in the activity coefficient with acid strength:
where n is a thermodynamic parameter related to the type of aromatic compounds. The M c for sulfuric acid represents a general scale describing its acidity, and its value depends only on the concentration of sulfuric acid at a given temperature. Therefore, the value corresponding to describes the variation in the activity coefficient ratio of Equation (12) as a function of the sulfuric acid concentration, and the effective concentration of the reactant is taken into consideration.
Combining Equations (7) and (13), the relationship betweenk * and k obs can be expressed as follows:
Taking the logarithm of Equation (14), we obtain:
Therefore, taking as the ordinate and M c as the abscissa, the values of n and k * can be obtained by the slope and intercept of the resulting straight line. However, before this can be done, the values ofM c and need to be determined.