# Chemical Kinetics: Integrated Rate Laws

## Concepts

The differential rate law describes how the rate of reaction varies with the concentrations of the reactants. The rate of reaction is proportional to the rates of change in concentrations of the reactants and products; that is, the rate is proportional to a derivative of a concentration.

To illustrate this point, consider the reaction

A → B

The rate of reaction, *r*, is given by

*r*= –

*t*

Suppose this reaction obeys a first-order rate law:

*r* = *k* [A]

This rate law can then be written as

*r*= –

*t*

*k*[A]

This equation is a differential equation that relates the rate of change in the concentration of A to the concentration of A. Integration of this equation produces the corresponding **integrated rate law**, which relates the concentration of A to time. Plots of concentration vs time for a chemical reaction are depictions of the integrated rate law.

*k*d

*t*

At *t* = 0, the concentration of A is [A]_{0}. Integration of the above differential equation produces the integrated rate law.

[A] = [A]_{0} e^{ - k t}

Experimentally one almost always measures concentration (or a property proportional to concentration) as a reaction progresses. It is possible to calculate the reaction rate from the derivative of the concentration and observe how the rate varies with concentration in order to determine the differential rate law. While this approach might work for simple systems with numerically exact data, in practice this method does not work well. Random error in the concentrations leads to much larger error in the calculated rate of reaction, and one often only measures the concentration infrequently.

A much better practical approach is to make characteristic kinetics plots. For each integrated rate law, there is a characteristic plot that can be created that will produce a straight line. These characteristic plots are presented in the table shown below. Species A is a reactant in the chemical reaction.

Reaction Order |
Differential Rate Law |
Integrated Rate Law |
Characteristic Kinetic Plot |
Slope of Kinetic Plot |
Units of Rate Constant |
---|---|---|---|---|---|

Zero | r = –
d [A] d = t k |
[A] = [A]_{0} - k t |
[A] vs t |
- k |
mole L^{-1} sec^{-1} |

First | r = –
d [A] d = t k [A] |
[A] = [A]_{0} e^{ - k t} |
ln [A] vs t |
- k |
sec^{-1} |

Second | r = –
d [A] d = t k [A]^{ 2} |
[A] =
[A] _{0} 1 + k t [A]_{0} |
1 [A] vs t |
k |
L mole^{-1} sec^{-1} |

The series of three graphs shown below illustrate the use of the characteristic kinetic plots. The graph on the left shows [A] vs *t* plots for a zero-order (red line), first-order (gold line), and second-order (blue line) reaction. The graph in the middle shows ln [A] vs *t* plots for each reaction order, and the graph on the right shows 1/[A] vs *t* plots for each reaction order.

Note: In the First-Order Plot, the y-axis is labeled as "ln [A]" . In practice, one does not take the logarithm of a quantity with units. So the graph is actually presenting "ln([A] L/mole)" on the y-axis. [A] has units of mole/L, so the quantity "[A] L/mole" has no units.

### Zero-Order Plot

### First-Order Plot

### Second-Order Plot

Notice that for each characteristic kinetic plot, a specific rate law shows a straight line.

In the [A] vs *t* plot, only the zero-order reaction (red line) produces a straight line. The other lines curve.

In the ln [A] vs *t* plot, only the first-order reaction (gold line) produces a straight line.

In the 1/[A] vs *t* plot, only the second-order reaction (blue line) produces a straight line.

In performing this type of analysis, one typically collects experimental data until the reaction is at least 90% complete. Over a short period of time, several plots may be approximately linear. Thus one needs an extensive set of data to make this approach work.

## Experiment

### Objectives

- Determine the rate law for a chemical reaction.
- Determine the rate constant for a chemical reaction.

Consider the following reaction between the persulfate ion and the iodide ion:

S_{2}O_{8}^{2-} *(aq)* + 3 I^{ -} *(aq)* → 2 SO_{4}^{2-} *(aq)* + I_{3}^{-} *(aq)*

In this chemical system, the only species that absorbs visible light is the triiodide ion (I_{3}^{-}). Spectrophotometry can be employed to determine the concentration of triiodide ion. With knowledge of the initial concentration of iodide ion, one can then determine the concentration of iodide ion at any point in time.

In this stopped-flow experiment, the top syringe contains a 0.150 M aqueous solution of KI and the bottom syringe contains a 0.150 M aqueous solution of K_{2}S_{2}O_{8}. The two solutions are mixed in a 3:1 ratio, so that there is a stoichiometric amount of iodide and persulfate ions in the reaction solution. As the reaction is occurring, the concentration of iodide is plotted versus time in the left graph.

This reaction is expected to follow a generic rate law of the form

*r* = *k* [S_{2}O_{8}^{2-}]^{ a} [I^{ -}]^{ b}

This expression indicates that the reaction is of order *a* with respect to persulfate ion and order *b* with respect to iodide ion. The overall order of the reaction is *a* + *b*. Because iodide and persulfate ion are present in a stoichiometric ratio throughout the reaction,

[I^{ -}] = 3 [S_{2}O_{8}^{2-}]

_{2}O

_{8}

^{2-}] =

^{ -}]

Substituting this expression into the general rate expression, produces the effective rate law

*r*= –

^{ -}] d

*t*

*k*[S

_{2}O

_{8}

^{2-}]

^{ a}[I

^{ -}]

^{ b}=

*k*(

^{ -}] )

^{ a}[I

^{ -}]

^{ b}

^{ -}] d

*t*

*k*3

^{1 - a}[I

^{ -}]

^{ a + b}=

*k*[I

_{o}^{ -}]

^{ n}

In this last equation, *n* = *a* + *b* is the overall order of the reaction and the observed rate constant is

*k _{o}* =

*k*3

^{1 - a}.

Run the stopped-flow experiment to acquire experimental data for how [I^{ -}] varies with time. After the reaction is complete, prepare kinetic plots for the zero-, first-, and second-order reactions. Use the kinetics plots to determine the rate law for this reaction. Because of the design of this experiment, the linear kinetics plot indicates the value of overall order of the reaction, *n*. The slope of the kinetic plot can be used to determine the observed rate constant *k _{o}* .

In this experiment, absorbance measurements, and thus measurements of [I^{ -}], are made at intervals and plotted with markers.

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*© 2000, 2014, 2023 David N. Blauch*