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 O3 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 Cl2 and F2 and the formation of ClF3.
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 C4H6 gave the data shown in the table:
$$2C_4H_6⟶C_8H_{12}$$| Time (s) | 0 | 1600 | 3200 | 4800 | 6200 |
| [C4H6] (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 C4H6. (Rate of reaction is $ \frac{1}{2}$ that of the rate of disappearance of C4H6).
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 [C4H6]. 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 C4H6.
(c) Determine the average rate of formation of C8H12 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 C8H12 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 Br2(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 Br2 is $2.1 \times 10^{-4} M/s$.