From Equation 1.42, we have
Substituting Equation 1.43 for Equation 1.41 leads to Equation 1.44, the rate law for Reaction 1.40.
where kobs = k1k2/(k−1 + k2) is the observed rate constant.
There are several limiting situations for such a stepwise process [3]. If k2 ≫ k−1 (the intermediate Y is converted to the product Z much faster than going back to the reactant X), Equation 1.44 can be simplified to
In this case, the first step of Reaction 1.40 is the rate‐determining step and actually irreversible.
If k2 ≪ k−1 (the intermediate Y is converted to the product Z much more slowly than going back to the reactant X), there is a fast preequilibrium between the reactant X and the intermediate Y before the product Z is formed. In this case, the Equation 1.44 can be simplified to
where Keq = (k1/k−1) is the equilibrium constant for the fast preequilibrium between X and Y (Keq = [Y]/[X]). Therefore, r = k2[Y], and the second k2 step is the rate‐determining step. Since the fast preequilibrium between X and Y is established prior to the formation of the product, the steady‐state assumption is not necessary if k2 ≪ k−1.
If the values of k2 and k−1 are comparable, the full steady‐state assumption is needed to establish the rate equation as shown in Equation 1.44.
1.5 THERMODYNAMICS
1.5.1 Enthalpy, Entropy, and Free Energy
Enthalpy (H), entropy (S), and free energy (G) are all thermodynamic state functions. Enthalpy (H) is defined as the sum of internal energy (U) and the product of pressure (P) and volume (V), formulated as
From Equation 1.45, the change in enthalpy (ΔH) can be calculated as
At constant pressure (P), Equation 1.46 becomes
According to the first law of thermodynamics, qP = ΔU − w (heat).
Therefore,
The physical meaning of Equation 1.47 is that the enthalpy change in a process (including a chemical reaction) at constant pressure is equal to the heat evolved. Since most of the organic reactions are conducted at constant pressure, the reaction heat can be calculated on the basis of the enthalpy change for the reaction.
Entropy (S) is considered as the degree of disorder. In thermodynamics, the infinitesimal change in entropy (dS) is defined as the reversible heat (dqrev) divided by the absolute temperature (T), formulated as
For a finite change in state,
(1.48)
Free energy (G) is defined as
At constant temperature and pressure, the change in free energy (ΔG) can be calculated as
1.5.2 Reversible and Irreversible Reactions
In general, chemical reactions in thermodynamics can be classified as two types, reversible and irreversible reactions. An irreversible reaction is such a reaction that proceeds only in one direction. As a result, the reactant is converted to the product completely (100%) in the end of the reaction. In contrast, a reversible reaction is such a reaction that can proceed to both forward and backward directions. In other words, there is an interconversion between the reactants and the products in a reversible reaction. As a result, all the reactants and the products coexist in the end of the reaction, and the conversion is incomplete.
The reversibility of a chemical reaction can be judged by the second law of thermodynamics. Originally, the second law is stated based on the entropy criterion as follows: A process (including a chemical reaction) is reversible if the universal entropy change (ΔSUNIV) associated to the process is zero; and a process is irreversible if the universal entropy change (ΔSUNIV) associated to the process is positive (greater than zero). ΔSUNIV = ΔS + ΔSSURR, the sum of the entropy change in the system (ΔS) and the entropy change in surroundings
