# 12 Reaction Rate Law

Learning Objectives

By the end of this section, you should be able to:

Define reaction rate law and reaction rate constant (k)

Reaction rate law Definition: The relationship between the rate of reaction and the concentration of reactants.

The rate law is usually proportional to the concentrations of reactants raised to a certain power:

Take the reaction we used as an example before: $A + 2B → 3C + D$

The general form for reaction rate law is

 $r=k_{r}[A]^a[B]^b$

For gas cases, we can use partial pressure

$r=k_{r}p_{A}^a p_{B}^b$

The rate constant $k_{r}$ is independent of species concentration but generally dependent on temperature.

For example, let’s look at the rate of the gas-phase decomposition of dinitrogen pentoxide,

$2 N_{2}O_{5} ⇌ 4 NO_{2} + O_{2}$

Say the rate law is found to be directly proportional to the concentration of $N_{2}O_{5}$, we can express the rate law by$^{}$:

$r = k_{r} [N_{2}O_{5}]$

Reaction rate laws can be complicated and may tell us about the mechanism of the reactions. For example, consider the reaction between hydrogen and bromine:

Simple stoichiometry:

$H_{2(g)} + Br_{2(g)} → 2 HBr_{(g)}$

Complicated rate law:

$r = \frac{k_{a}[H_{2}][Br_{2}]^{3/2}}{[Br_{2}]+k_{b}[HBr]}$

Rate Law vs. Equilibrium Constant

Be careful not to confuse equilibrium constant expressions with rate law expressions. The expression for $K_{eq}$ can always be written by inspecting the balanced reaction equation, and often contains a term for each species of the reaction (raised to the power of its coefficient) whose concentration changes during the reaction. The equilibrium constant for the reaction $2 N_{2}O_{5} ⇌ 4 NO_{2} + O_{2}$ is given below:

$K_{eq}=\frac{[NO_{2}]^4[O_{2}]}{[N_{2}O_{5}]^2}$

In contrast, the expression for the rate law generally bears no relation to the reaction equation and must be determined experimentally. $^{}$

## Reaction Rate Law Units

Reaction rate (r) is generally expressed in units of concentration over time (e.g. $\frac{mol}{L·s}$, $\frac{kPa}{min}$, $\frac{mol}{m^3·h}$ ).

This means the rate constant $k_{r}$ needs to be such that r is expressed in units of concentration over time.

Exercise: Rate Constant Units

For the following example, what are the units for the reaction rate constant ($k_{r}$)?

$r=k_{r}*p_{A}*p_{B}^2$

with p in Pa and time in seconds

### Solution

Since r is expressed in concentration over time, the units of r are $\frac{Pa}{s}$.

\begin{align*}
\frac{Pa}{s}& = k_{r}*Pa*Pa^2 \\
k_{r}& =\frac{1}{Pa^2s}
\end{align*}

## References 