Rate Constant Units and Enzyme Kinetics

Rate Constant Units

Understanding Rate Constants in Chemical Kinetics

The rate constant (k) is a proportionality factor in the rate law of a chemical reaction. Its units depend on the overall order of the reaction, which is the sum of the exponents in the rate law expression.

• Zero Order Reaction: The rate is independent of reactant concentration. Rate law: $\text{Rate} = k$ Units of k: $\text{M} \cdot \text{s}^{-1}$

• First Order Reaction: The rate is directly proportional to the concentration of one reactant. Rate law: $\text{Rate} = k[A]$ Units of k: $\text{s}^{-1}$

• Second Order Reaction: The rate depends on the concentration of two reactants, or the square of one. Rate law: $\text{Rate} = k[A][B]$ or $k[A]^2$ Units of k: $\text{M}^{-1} \cdot \text{s}^{-1}$

Note: The units of k always ensure that the rate has units of concentration per time ($\text{M} \cdot \text{s}^{-1}$).

Overall Reaction Order (n)	Units of Rate Constant (k)
Zero	$\text{M} \cdot \text{s}^{-1}$
First	$\text{s}^{-1}$
Second	$\text{M}^{-1} \cdot \text{s}^{-1}$

Additional info: For an nth order reaction, the general unit for k is $\text{M}^{1-n} \cdot \text{s}^{-1}$.

Example Calculation

For the rate law $\text{Rate} = k[A][B]^2$, the overall order is 1 (from [A]) + 2 (from [B]) = 3. Units of k: $\text{M}^{1-3} \cdot \text{s}^{-1} = \text{M}^{-2} \cdot \text{s}^{-1}$

Enzyme-Catalyzed Reaction Rate Constants

Enzyme Kinetics and Rate Laws

Enzyme-catalyzed reactions often follow a two-step mechanism involving the formation and breakdown of the enzyme-substrate (ES) complex:

$\text{E} + \text{S} \underset{k_{-1}}{\overset{k_1}{\rightleftharpoons}} \text{ES} \overset{k_2}{\rightarrow} \text{E} + \text{P}$

• k1 (Formation of ES): Rate law: $\text{Rate} = k_1 [E][S]$ Units: $\text{M}^{-1} \cdot \text{s}^{-1}$ (second order)

• k-1 (Breakdown of ES to E + S): Rate law: $\text{Rate} = k_{-1} [ES]$ Units: $\text{s}^{-1}$ (first order)

• k2 (Breakdown of ES to E + P): Rate law: $\text{Rate} = k_2 [ES]$ Units: $\text{s}^{-1}$ (first order)

Summary Table:

Step	Rate Law	Units of k
ES Formation (k1)	$k_1 [E][S]$	$\text{M}^{-1} \cdot \text{s}^{-1}$
ES Breakdown (k-1)	$k_{-1} [ES]$	$\text{s}^{-1}$
Product Formation (k2)	$k_2 [ES]$	$\text{s}^{-1}$

Practice Problem

Given the enzyme-catalyzed reaction above, determine the units for each rate constant:

• k1: $\text{M}^{-1} \cdot \text{s}^{-1}$

• k-1: $\text{s}^{-1}$

• k2: $\text{s}^{-1}$

Additional info: These units are essential for understanding enzyme kinetics and for calculating kinetic parameters such as $K_m$ and $V_{max}$ in Michaelis-Menten kinetics.