### Key Concepts and Summary

The rate of a reaction can be expressed either in terms of the decrease in the amount of a reactant or the increase in the amount of a product per unit time. Relations between different rate expressions for a given reaction are derived directly from the stoichiometric coefficients of the equation representing the reaction.

### Key Equations

relative reaction rates for $aA⟶bB=-\frac{1}{a}\frac{ΔA}{Δt}$ are: $\frac{1}{b}\frac{ΔB}{Δt}$

### Practice Problems: Kinetics – Rates of Reaction

What is the difference between average rate, initial rate, and instantaneous rate?

## Solution

The instantaneous rate is the rate of a reaction at any particular point in time, a period of time that is so short that the concentrations of reactants and products change by a negligible amount. The initial rate is the instantaneous rate of reaction as it starts (as product just begins to form). Average rate is the average of the instantaneous rates over a time period.

Ozone decomposes to oxygen according to the equation $2O_3(g)⟶3O_2(g)$. Write the equation that relates the rate expressions for this reaction in terms of the disappearance of O_{3} and the formation of oxygen.

## Solution

$$- \frac{1}{2} \frac{\Delta [O_3]}{\Delta t} = + \frac{1}{3} \frac{\Delta [O_2]}{\Delta t} $$

In the nuclear industry, chlorine trifluoride is used to prepare uranium hexafluoride, a volatile compound of uranium used in the separation of uranium isotopes. Chlorine trifluoride is prepared by the reaction $Cl_2(g)+3F_2(g)⟶2ClF_3(g)$. Write the equation that relates the rate expressions for this reaction in terms of the disappearance of Cl_{2} and F_{2} and the formation of ClF_{3}.

## Solution

$$\text{rate}=+\frac{1}{2}\frac{Δ[ClF_3]}{Δt}=-\frac{Δ[Cl_2]}{Δt}=-\frac{1}{3}\frac{Δ[F_2]}{Δt}$$

A study of the rate of dimerization of C_{4}H_{6} gave the data shown in the table:

Time (s) | 0 | 1600 | 3200 | 4800 | 6200 |

[C_{4}H_{6}] (M) |
$1.00×10^{-2}$ | $5.04×10^{-3}$ | $3.37×10^{-3}$ | $2.53×10^{-3}$ | $2.08×10^{-3}$ |

(a) Determine the average rate of dimerization between 0 s and 1600 s, and between 1600 s and 3200 s.

## Solution

By determining the slope (rate of change) between each set of points, we can find:
0 – 1600 s: $ 1.55 \times 10^{-6} \frac{M}{s} $

1600 – 3200 s: $ 5.21 \times 10^{-7} \frac{M}{s} $

Remember that this rate is the rate of *reaction*, not rate of disappearance of C_{4}H_{6}. (Rate of reaction is $ \frac{1}{2}$ that of the rate of disappearance of C_{4}H_{6}).

Notice here and in the following questions that the rate of reaction is decreasing with time.

(b) Estimate the instantaneous rate of dimerization at 3200 s from a graph of time versus [C_{4}H_{6}]. What are the units of this rate?

## Solution

Your exact value will vary depending on how you drew your graph, but you should find something in the neighborhood of $3.9 \times 10^{-7} \frac{M}{s}$.

Again, the rate of dimerization is the overall rate of *reaction*, not rate of disappearance of C_{4}H_{6}.

(c) Determine the average rate of formation of C_{8}H_{12} at 1600 s and the instantaneous rate of formation at 3200 s from the rates found in parts (a) and (b).

## Solution

Since the average rate of formation of C_{8}H_{12} will be equal to the rate of reaction (its stoichiometric coefficient is 1) this rate is equal to the values found above.

A study of the rate of the reaction represented as $2A⟶B$ gave the following data:

Time (s) | 0.0 | 5.0 | 10.0 | 15.0 | 20.0 | 25.0 | 35.0 |

[A] (M) |
1.00 | 0.775 | 0.625 | 0.465 | 0.360 | 0.285 | 0.230 |

(a) Determine the average rate of disappearance of *A* between 0.0 s and 10.0 s, and between 10.0 s and 20.0 s.

## Solution

average rate, 0 − 10 s = 0.0375 mol L^{−1} s^{−1}

average rate, 10 − 20 s = 0.0265 mol L^{−1} s^{−1}

(b) Estimate the instantaneous rate of disappearance of *A* at 15.0 s from a graph of time versus [*A*]. What are the units of this rate?

## Solution

instantaneous rate, 15 s = 0.023 mol L^{−1} s^{−1}

(c) Use the rates found in parts (a) and (b) to determine the average rate of formation of *B* between 0.00 s and 10.0 s, and the instantaneous rate of formation of *B* at 15.0 s.

## Solution

average rate for B formation = 0.0188 mol L^{−1} s^{−1}

instantaneous rate for B formation = 0.012 mol L^{−1} s^{−1}

Consider the following reaction in aqueous solution:

If the rate of disappearance of Br^{–}(*aq*) at a particular moment during the reaction is $3.5×10^{-4}$ mol L^{−1} *s*^{−1}, what is the rate of appearance of Br_{2}(*aq*) at that moment?

## Solution

From the reaction stoichiometry, we can determine the relative rates:
$$ – \frac{1}{5} \frac{ \Delta [Br^{-}]}{\Delta t} = \frac{1}{3} \frac{ \Delta [Br_2]}{\Delta t} $$
Therefore: $ \frac{ \Delta [Br_2]}{\Delta t} = \frac{-3}{5} \frac{\Delta [Br^{-}]}{\Delta t} $.
The rate of appearance of Br_{2} is $2.1 \times 10^{-4} M/s$.