### Rate Laws

_{2}) and iodine (I

_{2}).

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

Measurement | [H_{2}] |
[I_{2}] |
Measured Rate |
---|---|---|---|

1 | [H_{2}]_{1} |
[I_{2}]_{1} |
R_{1} |

2 | 2[H_{2}]_{1} |
[I_{2}]_{1} |
2R_{1} |

3 | 3[H_{2}]_{1} |
[I_{2}]_{1} |
3R_{1} |

4 | [H_{2}]_{1} |
2[I_{2}]_{1} |
2R_{1} |

5 | [H_{2}]_{1} |
3[I_{2}]_{1} |
3R_{1} |

6 | 2[H_{2}]_{1} |
2[I_{2}]_{1} |
4R_{1} |

**rate constant (**.

*k*)**rate expression**, or rate law, for the reaction. Consider a generalized form of a chemical equation:

*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:

*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:

_{2}H

_{6}] does not double the rate but quadruples it. Similarly, tripling [C

_{2}H

_{6}] 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

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 | ${\rm{L}}^2/{\rm{mol}}^2{\cdot}{\rm{s}}={\rm{M}}^{-2}{\cdot}\,{\rm{s}}^{-1}$ |

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

*k*are

^{–1}·s

^{–1}at a specific temperature.