What units are used for the rate constant (k) in a rate law?

The rate constant, often denoted as $k$, in a rate law has units that depend on the overall order of the reaction. The rate law for a chemical reaction relates the rate of the reaction to the concentrations of the reactants, each raised to some power. The general form of a rate law is:

$$\text{Rate}=k[A{]}^m[B{]}^n$$

where:
- $\text{Rate}$ is the rate of the reaction.
- $k$ is the rate constant.
- $[A]$ and $[B]$ are the concentrations of reactants A and B, respectively.
- $m$ and $n$ are the orders of the reaction with respect to reactants A and B, respectively.

The overall order of the reaction is the sum of the exponents $m+n$. The units of the rate constant $k$ are determined so that the rate of the reaction has units of concentration per unit time (e.g., M/s, where M is molarity and s is seconds).

Let's consider the overall order of the reaction and the corresponding units for the rate constant:

1. Zero-Order Reactions:
For a zero-order reaction, where the overall order is 0, the rate law is:

$$\text{Rate}=k[A{]}^0$$

Since any concentration raised to the power of zero is 1, the rate law simplifies to:

$$\text{Rate}=k$$

The units of the rate constant $k$ for a zero-order reaction are the same as the rate, which is concentration/time. For example:

$$[k]=\frac{\text{M}}{\text{s}}$$

2. First-Order Reactions:
For a first-order reaction, where the overall order is 1, the rate law is:

$$\text{Rate}=k[A]$$

The units of the rate constant $k$ must cancel out the units of concentration, leaving units of 1/time. For example:

$$[k]=\frac{1}{\text{s}}$$

3. Second-Order Reactions:
For a second-order reaction, where the overall order is 2, the rate law is:

$$\text{Rate}=k[A{]}^2\text{ or }k[A][B]$$

The units of the rate constant $k$ must cancel out the units of concentration squared, leaving units of time/concentration. For example:

$$[k]=\frac{\text{L}}{\text{mol}\cdot \text{s}}$$

4. Third-Order Reactions:
For a third-order reaction, where the overall order is 3, the rate law is:

$$\text{Rate}=k[A{]}^2[B]\text{ or }k[A][B][C]$$

The units of the rate constant $k$ must cancel out the units of concentration cubed, leaving units of time/concentration squared. For example:

$$[k]=\frac{{\text{L}}^2}{{\text{mol}}^2\cdot \text{s}}$$

For reactions of higher order, the pattern continues similarly, with the units of $k$ being determined by the requirement that the rate of the reaction has units of concentration per unit time.

In summary, the units of the rate constant $k$ in a rate law are determined by the overall order of the reaction and are derived to ensure that the rate of the reaction is expressed in units of concentration per unit time.