# Rate Laws and Reaction Order

### Rate Laws

The rate law relates the rate of a reaction with the concentrations of reactants.
The concentrations of the reactants can be used to calculate reaction rates using a rate law. This can be illustrated by the chemical reaction between hydrogen gas (H2) and iodine (I2).
${\rm H}_2(\mathit g)+{\rm I}_2(\mathit g)\rightarrow2\rm{HI}(\mathit g)$
The initial rate of this reaction depends on the concentrations of the reactants. It is possible to experimentally determine how these concentrations affect the rate by a series of trials with varying concentrations.

### Initial Rate and Concentrations of Reactants for $\bold{\rm{H}}_2+{\rm{I}}_2\rightarrow2{\rm{HI}}$

Measurement [H2] [I2] Measured Rate
1 [H2]1 [I2]1 R1
2 2[H2]1 [I2]1 2R1
3 3[H2]1 [I2]1 3R1
4 [H2]1 2[I2]1 2R1
5 [H2]1 3[I2]1 3R1
6 2[H2]1 2[I2]1 4R1

The reaction of hydrogen gas (H2) and iodine (I2) to produce hydrogen iodide can be repeated with various concentrations of the reactants. The measured initial reaction rates depend on the concentrations of the reactants.

Doubling the concentration of one reactant and keeping the other the same doubles the reaction rate. Tripling the concentration of one reactant and keeping the other the same triples the reaction rate. Doubling concentrations of both reactants quadruples the reaction rate. These measurements show that rate is directly proportional to ($\propto$) both concentrations.
$\rm{rate}\propto\lbrack{\rm H}_2\rbrack\lbrack{\rm I}_2\rbrack$
A constant relating reaction rate to concentrations of reactants is called the rate constant (k).
${\rm{rate}}=k\lbrack{\rm H}_2\rbrack\lbrack{\rm I}_2\rbrack$
The rate constant is not dependent on concentrations. However, it is dependent on temperature. Rate constants are determined for a chemical reaction at a specific temperature. The mathematical expression that relates rate with a rate constant and concentrations of the reactants is called the rate expression, or rate law, for the reaction. Consider a generalized form of a chemical equation:
$a{\rm{A}}+b{\rm{B}}+\mathellipsis\rightarrow c{\rm{C}}+d{\rm{D}}+\mathellipsis$
The general form of the rate expression is:
$r=k\lbrack{\rm A}\rbrack^m\lbrack{\rm B}\rbrack^n\mathellipsis$
The exponents m and n indicate how the rate expression depends on the reactant concentrations. For elementary (single-step) chemical reactions, these exponents are the coefficients in the balanced chemical reaction. For an elementary reaction $a{\rm{A}}+b{\rm{B}}\rightarrow c{\rm{C}}$, for example, the rate expression is:
$r=k\lbrack\lbrack{\rm{A}}\rbrack^{a}\lbrack{\rm{B}}\rbrack^{b}\rbrack$
For reactions that are not elementary, however, the exponents m and n cannot be determined from the coefficients in the balanced chemical equation. Instead, they must be determined experimentally. For example, if an experiment shows that the reaction rate doubles when [A] is doubled, then the exponent m is 1. If the rate quadruples when [B] is doubled, the exponent n is 2. Alternatively, an experiment may show that the rate is does not change when one of the reactants changes. In this case, the exponent would be 0, so the reactant concentration would not appear in the rate expression. The rate law for the equation ${\rm{H_2}}(g)+{\rm{I_2}}(g)\rightarrow2{\rm{HI}}(g)$ is:
${\rm{rate}}=k\lbrack{\rm H}_2\rbrack\lbrack{\rm I}_2\rbrack$
This equation is in the form ${\rm{rate}}=k\lbrack{\rm A}\rbrack\lbrack\rm B\rbrack$. Not all reactions have rate expressions of this form. For example, in a decomposition reaction, $\rm A\rightarrow\rm B+\rm C$, doubling the concentration of A doubles the reaction rate. The rate expression therefore has the form $r=k\lbrack\rm A\rbrack$. In the reaction in which ethane decomposes into methyl radicals, ${\rm{C_2H_6}}(g)\rightarrow2{\rm{CH_3\cdot}}(g)$, doubling [C2H6] does not double the rate but quadruples it. Similarly, tripling [C2H6] increases the rate ninefold. In this example, rate is proportional to the square of the concentration, $r=k[\rm{A}]^2$. The exponent of [A] is 2 in the rate equation because the coefficient of A in the balanced equation is 2.

### Reaction Order

Reaction order is the sum of the powers that the concentrations of the reactants are raised to.

Rate expressions can take different forms. Forms such as $r=k\lbrack{\rm A\rbrack^{2}\lbrack B\rbrack}$, $r=k\lbrack{\rm A}\rbrack\lbrack{\rm B}\rbrack^{2}$, and $r=k\lbrack\rm A\rbrack^2\lbrack\rm B\rbrack^2$ are common. A rate expression is described as having the general form $r=k\lbrack{\rm A}\rbrack^m\lbrack{\rm B}\rbrack^n$. The sum of the powers that the reactant concentrations are raised to in the rate expression is the reaction order. The reaction order is equal to $m+n$ in the general form.

A zero-order reaction is a reaction in which rate is not dependent on the reactant concentrations. The rate law for a zero-order reaction has the form $r=k$.

A first-order reaction is a reaction in which the rate is dependent on the concentration of one of the reactants, raised to the power of one. The rate law for a first-order reaction has the form $r=k\lbrack\rm A\rbrack$. A first-order reaction may have more than one reactant. For example, it may have the form $\rm A+\rm B\rightarrow\rm C$. The concentration of B does not affect rate in this case. For first-order reactions, $m+n=1$.

A second-order reaction is a reaction in which the sum of the powers that the reactant concentrations are raised to in the rate expression is equal to two. The rate depends on the concentrations of two reactants each raised to the power of one, or the rate depends on the concentration of one reactant raised to the power of two. The rate law for a second-order reaction can have the form $r=k[\rm{A}][\rm{B}]$, or it can have the form $r\;=\;k\lbrack\rm A\rbrack^2$. For second-order reactions, $m+n=2$.

In a third-order reaction the powers that the concentrations of the reactants are raised to in the rate expression is equal to three. Third-order reactions can have multiple different forms, for example, $r=k\lbrack\rm A\rbrack\lbrack\rm B\rbrack\lbrack\rm C\rbrack$, $r=k\lbrack\rm A\rbrack^2\lbrack\rm B\rbrack$, or $r=k\lbrack\rm A\rbrack^3$. For third-order reactions, the sum of the exponents in the rate equation equals 3.

The units of the rate constant are dependent on the reaction order.

### Rate Constant Units and Reaction Order

Reaction Order Unit
0 ${\rm{mol}}/{\rm{L}}{\cdot}{\rm{s}}={\rm{M}}/{\rm{s}}$
1 $\rm s^{-1}$
2 ${\rm{L}}/{\rm{mol}}{\cdot}{\rm{s}}={\rm{M}}^{-1}{\cdot}\,{\rm{s}}^{-1}$
3 ${\rm{L}}^2/{\rm{mol}}^2{\cdot}{\rm{s}}={\rm{M}}^{-2}{\cdot}\,{\rm{s}}^{-1}$

The units of a reaction rate constant depends on the reaction order.

Rate expression does not need to be in whole numbers. In some reactions, the rate expression is a fraction, such as $r=k\lbrack\rm A\rbrack^{3/2}$.
Step-By-Step Example
Calculate Initial Reaction Rate

In a chemical reaction of the form $\rm{A}+\rm{B}\rightarrow\rm{C}$, the following experimental data are measured.

Measurement [A] (M) [B] (M) Measured Rate (M/s)
1 0.100 0.100 $2.0\times10^{-4}$
2 0.200 0.100 $4.0\times10^{-4}$
3 0.100 0.200 $8.0\times10^{-4}$

For this reaction, determine the following.

1. Rate expression

2. Reaction order

3. Rate constant

Step 1
To determine the rate expression, study the table, and observe how changing the concentrations affects rate. Doubling [A] doubles the reaction rate. Doubling [B] quadruples the reaction rate. This rate expression for this reaction is $r=k\lbrack\rm A\rbrack\lbrack\rm B\rbrack^2$.
Step 2
The reaction order is the sum of the powers of the concentrations in the rate expression. [A] has the power 1, and [B] has the power 2. The reaction order is $1+2=3$.
Step 3
The rate expression can be used with any of the experimental measurements to calculate the rate constant. For example, use the measurements in row 1.
\begin{aligned} r&=k\lbrack{\rm{A}}\rbrack\lbrack{\rm{B}}\rbrack^2\\k&=\frac{r}{\lbrack{\rm{A}}\rbrack\lbrack{\rm{B}}\rbrack^2}\\&=\frac{2.0\times10^{-4}}{(0.100)(0.100)^2}\\&=0.20\end{aligned}
Solution
The units for k are
$\frac{\displaystyle\frac{{\rm{M}}}{{\rm{s}}}}{{\rm{M}}^3}=\frac1{{\rm{M}}^2{\cdot}{\rm{s}}}=\frac{{\rm{L}}^2}{{\rm{mol}}^2{\cdot}{\rm{s}}}={\rm{L}}^2/{\rm{mol}}^{2}{\cdot}{\rm{s}}$
In another example, calculate the initial reaction rate for the chemical reaction with a rate constant of 0.04 M–1·s–1 at a specific temperature.
$\rm A+\rm B\rightarrow\rm C+\rm D$
The rate law for the reaction is $r=k\lbrack\rm A\rbrack\lbrack\rm B\rbrack$. The initial concentration of A is 0.050 M, and the initial concentration of B is 0.070 M. The initial rate of the reaction can be calculated using this information.
\begin{aligned} r&=k\lbrack\rm A\rbrack\lbrack\rm B\rbrack\\&=(0.04\;\rm M^{-1}{\cdot}\,{\rm{s}}^{-1})(0.050\;\rm M)(0.070\;\rm M)\\&=1.4\times10^{-4}\;\rm M/\rm s\end{aligned}