KAR - Integrated Rate Law (Lesson)

Integrated Rate Laws

In the previous lesson differences in concentration were used to determine rate laws. An examination of a plot of concentration vs. time for a chemical reaction also reveals that as time passes, the rate at which the reaction proceeds decreases as well indicating that there is a dependence not only upon concentration but also time. The three different ways that rates can be affected (three different rate orders) were also discussed. In this lesson, these rate orders will be determined in a different way using different data.

Zero Order Rate Law

The first possibility with regard to rate laws is a zero order relationship. As mentioned previously, a zero order relationship is one in which a change in concentration has no effect on the rate of reaction. A generic zero order rate law would therefore be:

A  $\longrightarrow$  B

Rate = k[A]⁰ 

Rate = k

A plot of concentration of A versus time reveals a linear relationship. Importantly, the slope of this line is equal to the rate constant, k in the rate law shown above.

 

 

First Order Rate Law

A reaction that is first order would have the generic rate law:

A  $\longrightarrow$  B

Rate = k[A]¹

This form is referred to as the differential form of the rate law because it relates changes or differences in concentration vs. time ($\Delta\left\lbrack A\right\rbrack/\Delta t$).  However, an interesting thing occurs when we look at the integrated form of the rate law:

ln[A]_t = -kt + ln[A]₀

y = mx + b

It can now be seen that the integrated form can be written in such a way as to demonstrate that it is in the form of a linear equation. Therefore if a reaction exhibits first order kinetics, then a plot of ln[A] vs. time will yield a linear plot as shown below.

With the equation in this format, it is easy to see that the graph with the ln[A]_t plotted on the y-axis and time plotted on the x-axis will produce a line with a slope of -k.  

Below are graphs of the decomposition of N₂O₅ over time (at 45°C).

• Graph A shows the concentration versus time for the decomposition.
• Graph B shows a straight line is obtained from a natural log versus time plot. The slope is the negative rate constant.

Second Order Rate Law

The third and final possibility for reaction rate order is for there to be a second order or exponential dependence. An example of this type of rate law is shown below.

A  $\longrightarrow$  B

Rate = k[A]²

As before, the form written above is in the differential form, but can also be written in the integrated form:

$\frac{1}{\left\lbrack A\right\rbrack_t}=kt+\frac{1}{\left\lbrack A\right\rbrack_0}$

As before, this form is also in the form of y = mx + b, and a plot of 1/[A] vs. t will yield a straight line with a positive slope if the reaction in question exhibits second order kinetics.  Take for example the reaction shown below:

2 C₄H₆ (g)  $\longrightarrow$  C₈H₁₂ (g)

A plot of ln [C₄H₆] does not yield a straight line, indicating that it does not proceed via first order kinetics; however, a plot of 1/[C₄H₆] is linear revealing second order kinetics. Once again, the slope of this line can be interpreted as the rate constant for this reaction.

Half Life

While the rate constant, k, is the standard way to compare reaction rates among chemical reactions, there is a second way to directly compare reaction rates. This second method of describing rates involves describing the time it takes for the concentration of a reactant to decrease by half. This time, or half-life, is commonly used to describe radioactive decay, but can also be used in the context of chemical reactions. As can be imagined, the equations used to calculate kinetics half-lives are dependent on the order of reaction. Due to this dependence, there are different half-life equations for each rate order as described in the figure below.

 

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