# 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: [latex]A + 2B → 3C + D[/latex]

The general form for reaction rate law is

[latex]r=k_{r}[A]^a[B]^b[/latex] |

For gas cases, we can use partial pressure

[latex]r=k_{r}p_{A}^a p_{B}^b[/latex]

The rate constant [latex]k_{r}[/latex] 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,

[latex]2 N_{2}O_{5} ⇌ 4 NO_{2} + O_{2}[/latex]

Say the rate law is found to be directly proportional to the concentration of [latex]N_{2}O_{5}[/latex], we can express the rate law by[latex]^{[1]}[/latex]:

[latex]r = k_{r} [N_{2}O_{5}][/latex]

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:

[latex]H_{2(g)} + Br_{2(g)} → 2 HBr_{(g)}[/latex]

Complicated rate law:

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

**Rate Law vs. Equilibrium Constant**

Be careful not to confuse equilibrium constant expressions with rate law expressions. The expression for [latex]K_{eq}[/latex] 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 [latex]2 N_{2}O_{5} ⇌ 4 NO_{2} + O_{2}[/latex] is given below:

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

In contrast, the expression for the rate law generally bears no relation to the reaction equation and must be determined experimentally. [latex]^{[1]}[/latex]

## Reaction Rate Law Units

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

This means the rate constant [latex]k_{r}[/latex] 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 ([latex]k_{r}[/latex])?

[latex]r=k_{r}*p_{A}*p_{B}^2[/latex]

with p in Pa and time in seconds

### Solution

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

\begin{align*}

\frac{Pa}{s}& = k_{r}*Pa*Pa^2 \\

k_{r}& =\frac{1}{Pa^2s}

\end{align*}

## References

[1] Chemistry LibreTexts. 2020. *The Rate Law.* [online] Available at: <https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Kinetics/Rate_Laws/The_Rate_Law> [Accessed 23 April 2020].

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