18.4 Integrated Rate Laws

The rate laws we have seen thus far relate the rate and the concentrations of reactants. We can also determine a second form of each rate law that relates the concentrations of reactants and time. These are called integrated rate laws. We can use an integrated rate law to determine the amount of reactant or product present after a period of time or to estimate the time required for a reaction to proceed to a certain extent. For example, an integrated rate law is used to determine the length of time a radioactive material must be stored for its radioactivity to decay to a safe level.

Using calculus, the differential rate law for a chemical reaction can be integrated with respect to time to give an equation that relates the amount of reactant or product present in a reaction mixture to the elapsed time of the reaction. This process can either be very straightforward or very complex, depending on the complexity of the differential rate law. For purposes of discussion, we will focus on the resulting integrated rate laws for first-, second-, and zero-order reactions.

First-Order Reactions

Integration of the rate law for a simple first-order reaction (rate = $k[A]$) results in an equation describing how the reactant concentration varies with time:

$[A{]}_t=[A{]}_0{e}^{-kt}$

where $[A{]}_t$ is the concentration of $A$ at any time $t$, $[A{]}_0$ is the initial concentration of $A$, and $k$ is the first-order rate constant.

For mathematical convenience, this equation may be rearranged to other formats, including direct and indirect proportionalities:

$\text{ln}\left({\displaystyle \frac{{\left[A\right]}_t}{{\left[A\right]}_0}}\right)=-kt$

or

$\text{ln}\left({\displaystyle \frac{{\left[A\right]}_0}{{\left[A\right]}_t}}\right)=kt$

and a format showing a linear dependence of concentration in time:

$\text{ln}[A{]}_t=\text{ln}[A{]}_0-kt$

We can use integrated rate laws with experimental data that consist of time and concentration information to determine the order and rate constant of a reaction. The integrated rate law can be rearranged to a standard linear equation format:

$\begin{array}{ccc}\hfill \text{ln}{\left[A\right]}_t& =& (-k)\left(t\right)+\text{ln}{\left[A\right]}_0\hfill \\ \hfill y& =& mx+b\hfill \end{array}$

As shown in Figure 18.4.1, a plot of $\text{ln}[A{]}_t$ versus $t$ for a first-order reaction is a straight line with a slope of –k and a y-intercept of $\text{ln}[A{]}_0$. If a set of rate data are plotted in this fashion but do not result in a straight line, the reaction is not first order in $A$.

Example 18.4.2: Determination of Reaction Order by Graphing

Show that the data in Figure 18.4.2 can be represented by a first-order rate law by graphing ln$\mathrm{H}{\phantom{A}}_2\mathrm{O}{\phantom{A}}_2$ versus time. Determine the rate constant for the rate of decomposition of $\mathrm{H}{\phantom{A}}_2\mathrm{O}{\phantom{A}}_2$ from this data.

Show Solution

The data from Figure 18.4.2 with the addition of values of ln$\mathrm{H}{\phantom{A}}_2\mathrm{O}{\phantom{A}}_2$ are given in Figure 18.4.3.

Trial	Time (h)	$[\mathrm{H}{\phantom{A}}_2\mathrm{O}{\phantom{A}}_2]$ (M)	ln$[\mathrm{H}{\phantom{A}}_2\mathrm{O}{\phantom{A}}_2]$
1	0	1.000	0.0
2	6.00	0.500	–0.693
3	12.00	0.250	–1.386
4	18.00	0.125	–2.079
5	24.00	0.0625	–2.772

The plot of ln$[\mathrm{H}{\phantom{A}}_2\mathrm{O}{\phantom{A}}_2]$ versus time is linear, thus we have verified that the reaction may be described by a first-order rate law.

The rate constant for a first-order reaction is equal to the negative of the slope of the plot of ln$[\mathrm{H}{\phantom{A}}_2\mathrm{O}{\phantom{A}}_2]$ versus time where:

$\text{slope}={\displaystyle \frac{\text{change in }y}{\text{change in }x}}={\displaystyle \frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}}={\displaystyle \frac{\mathrm{\Delta }\text{ ln}\left[{\mathrm{H}}_2{\mathrm{O}}_2\right]}{\mathrm{\Delta }t}}$

In order to determine the slope of the line, we need two values of ln$[\mathrm{H}{\phantom{A}}_2\mathrm{O}{\phantom{A}}_2]$ at different values of t (one near each end of the line is preferable). For example, the value of ln$[\mathrm{H}{\phantom{A}}_2\mathrm{O}{\phantom{A}}_2]$ when t is 6.00 h is -0.693; the value when t = 12.00 h is −1.386:

$\begin{array}{ccc}\hfill \text{slope}& =& {\displaystyle \frac{-1.386-(-0.693)}{\text{12.00 h}-\text{6.00 h}}}\hfill \\ & =& {\displaystyle \frac{-0.693}{\text{6.00 h}}}\hfill \\ & =& -1.155\times {10}^{-2}{\text{h}}^{-1}\hfill \\ \hfill k& =& -\text{slope}=-(-1.155\times {10}^{-2}{\text{h}}^{-1})=1.155\times {10}^{-2}{\text{h}}^{-1}\hfill \end{array}$

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Second-Order Reactions

The equations that relate the concentrations of reactants and the rate constant of second-order reactions are fairly complicated. We will limit ourselves to the simplest second-order reactions, namely, those with rates that are dependent upon just one reactant’s concentration and described by the differential rate law:

$\text{rate}=k{\left[A\right]}^2$

For these second-order reactions, the integrated rate law is:

${\displaystyle \frac{1}{{\left[A\right]}_t}}=kt+{\displaystyle \frac{1}{{\left[A\right]}_0}}$

where the terms in the equation have their usual meanings as defined above.

The integrated rate law for our second-order reactions has the form of the equation of a straight line:

$\begin{array}{ccc}\hfill {\displaystyle \frac{1}{{\left[A\right]}_t}}& =& kt+{\displaystyle \frac{1}{{\left[A\right]}_0}}\hfill \\ \hfill y& =& mx+b\hfill \end{array}$

As shown in Figure 18.4.4, a plot of ${\displaystyle \frac{1}{{\left[A\right]}_t}}$ versus t for a second-order reaction is a straight line with a slope of k and an intercept of ${\displaystyle \frac{1}{{\left[A\right]}_0}}$. If the plot is not a straight line, then the reaction is not second order.

Example 18.4.4: Determination of Reaction Order by Graphing

Test the data given to show whether the dimerization of $\mathrm{C}{\phantom{A}}_4\mathrm{H}{\phantom{A}}_6$ is a first- or a second-order reaction.

Show Solution

Trial	Time (s)	$[\mathrm{C}{\phantom{A}}_4\mathrm{H}{\phantom{A}}_6]$ (M)
1	0	1.00 × 10–2
2	1600	5.04 × 10–3
3	3200	3.37 × 10–3
4	4800	2.53 × 10–3
5	6200	2.08 × 10–3

In order to distinguish a first-order reaction from a second-order reaction, we plot $\text{ln}[4\mathrm{H}{\phantom{A}}_6]$ versus t and compare it with a plot of ${\displaystyle \frac{\text{1}}{[{\mathrm{C}}_4{\mathrm{H}}_6]}}$ versus t. The values needed for these plots follow.

Time (s)	${\displaystyle \frac{1}{\left[{\text{C}}_4{\text{H}}_6\right]}}\left(M^{-1}\right)$	ln$[\mathrm{C}{\phantom{A}}_4\mathrm{H}{\phantom{A}}_6]$
0	100	–4.605
1600	198	–5.289
3200	296	–5.692
4800	395	–5.978
6200	481	–6.175

The plots are shown in Figure 18.4.5. As you can see, the plot of ln$[\mathrm{C}{\phantom{A}}_4\mathrm{H}{\phantom{A}}_6]$ versus t is not linear, therefore the reaction is not first order. The plot of ${\displaystyle \frac{1}{[\mathrm{C}{\phantom{A}}_4\mathrm{H}{\phantom{A}}_6]{\phantom{A}}_t}}$ versus t is linear, indicating that the reaction is second order.

According to the second-order integrated rate law, the rate constant is equal to the slope of the ${\displaystyle \frac{1}{{\left[A\right]}_t}}$ versus t plot. Using the data for t = 0s and t = 6200 s, the rate constant is estimated as follows:

$k=\text{slope}={\displaystyle \frac{(481{\text{ M}}^{-1}-100{\text{ M}}^{-1})}{(6200\text{s}-0\text{s})}}=0.0614{\text{ M}}^{-1}{\text{s}}^{-1}$

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Zero-Order Reactions

For zero-order reactions, the differential rate law is:

$\text{rate}=k{\left[A\right]}^0=k$

A zero-order reaction thus exhibits a constant reaction rate, regardless of the concentration of its reactant(s). This may seem counterintuitive, since the reaction rate certainly can’t be finite when the reactant concentration is zero. For purposes of this introductory text, it will suffice to note that zero-order kinetics are observed for some reactions only under certain specific conditions. These same reactions exhibit different kinetic behaviors when the specific conditions aren’t met, and for this reason the more prudent term pseudo-zero-order is sometimes used.

The integrated rate law for a zero-order reaction also has the form of the equation of a straight line:

$\begin{array}{ccc}\hfill {\left[A\right]}_t& =& -kt+{\left[A\right]}_0\hfill \\ \hfill y& =& mx+b\hfill \end{array}$

As shown in Figure 18.4.6, a plot of [A] versus t for a zero-order reaction is a straight line with a slope of −k and a y-intercept of [A]₀.

Figure 18.4.7 shows a plot of $[\mathrm{NH}{\phantom{A}}_3]$ versus t for the thermal decomposition of ammonia at the surface of two different heated solids. The decomposition reaction exhibits first-order behavior at a quartz $(\mathrm{SiO}{\phantom{A}}_2)$ surface, as suggested by the exponentially decaying plot of concentration versus time. On a tungsten surface, however, the plot is linear, indicating zero-order kinetics.

If we use the data from the plot in Figure 18.4.7, we can graphically estimate the zero-order rate constant for ammonia decomposition at a tungsten surface. The integrated rate law for zero-order kinetics describes a linear plot of reactant concentration, [A]_t, versus time, t, with a slope equal to the negative of the rate constant, −k. Following the mathematical approach of previous examples, the slope of the linear data plot (for decomposition on W) is estimated from the graph. Using the ammonia concentrations at t = 0 and t = 1000 s:

$k=-\text{slope}={\displaystyle \frac{(0.0015{\text{ mol L}}^{-1}-0.0028{\text{ mol L}}^{-1})}{(1000\text{s}-0\text{s})}}=1.3\times {10}^{-6}{\text{ mol L}}^{-1}{\text{s}}^{-1}$

The zero-order plot in Figure 18.4.7 shows an initial ammonia concentration of 0.0028 mol L⁻¹ decreasing linearly with time for 1000 s. Assuming no change in this zero-order behavior, you should be able to calculate the time (min) when the concentration will reach 0.0001 mol L⁻¹. The correct time is 35 minutes!

The Half-Life of a Reaction

The half-life of a reaction is the time required for one-half of a given amount of reactant to be consumed. In each succeeding half-life, half of the remaining concentration of the reactant is consumed. Using the decomposition of hydrogen peroxide in Figure 18.4.1 as an example, we find that during the first half-life (from 0.00 hours to 6.00 hours), the concentration of $\mathrm{H}{\phantom{A}}_2\mathrm{O}{\phantom{A}}_2$ decreases from 1.000 M to 0.500 M. During the second half-life (from 6.00 hours to 12.00 hours), it decreases from 0.500 M to 0.250 M; during the third half-life, it decreases from 0.250 M to 0.125 M. The concentration of $\mathrm{H}{\phantom{A}}_2\mathrm{O}{\phantom{A}}_2$ decreases by half during each successive period of 6.00 hours. The decomposition of hydrogen peroxide is a first-order reaction, and, as can be shown, the half-life of a first-order reaction is independent of the concentration of the reactant. However, half-lives of reactions with other orders depend on the concentrations of the reactants.

First-Order Reactions

We can derive an equation for determining the half-life of a first-order reaction from the alternate form of the integrated rate law as follows:

$\begin{array}{lll}\hfill \ln {\displaystyle \frac{{\left[A\right]}_0}{\left[A\right]}}& =& kt\\ \hfill t& =& \ln {\displaystyle \frac{{\left[A\right]}_0}{\left[A\right]}}\times {\displaystyle \frac{1}{k}}\end{array}$

If we set the time t equal to the half-life, $t_{1\text{/}2},$ the corresponding concentration of A at this time is equal to one-half of its initial concentration. Hence, when $t=t_{1\text{/}2},$ ${\left[A\right]}_t={\displaystyle \frac{1}{2}}{\left[A\right]}_0$.

Therefore:

$\begin{array}{cc}\hfill t_{1\text{/}2}& =\ln {\displaystyle \frac{{\left[A\right]}_0}{\frac{1}{2}{\left[A\right]}_0}}\times {\displaystyle \frac{1}{k}}\hfill \\ & =\ln 2\times {\displaystyle \frac{1}{k}}=0.693\times {\displaystyle \frac{1}{k}}\hfill \end{array}$

Thus:

$t_{1\text{/}2}={\displaystyle \frac{0.693}{k}}$

We can see that the half-life of a first-order reaction is inversely proportional to the rate constant k. A fast reaction (shorter half-life) will have a larger k; a slow reaction (longer half-life) will have a smaller k.

Example 18.4.5: Calculation of a First-order Rate Constant using Half-Life

Calculate the rate constant for the first-order decomposition of hydrogen peroxide in water at 40 °C, using the data given in Figure 18.4.8.

Show Solution

The half-life for the decomposition of $\mathrm{H}{\phantom{A}}_2\mathrm{O}{\phantom{A}}_2$ is 2.16 × 10⁴ s:

$\begin{array}{ccc}\hfill t_{1\text{/}2}& =& {\displaystyle \frac{0.693}{k}}\hfill \\ \hfill k& =& {\displaystyle \frac{0.693}{t_{1\text{/}2}}}={\displaystyle \frac{0.693}{2.16\times {10}^4\text{s}}}=3.21\times {10}^{-5}{\text{s}}^{-1}\hfill \end{array}$

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Second-Order Reactions

We can derive the equation for calculating the half-life of a second order as follows:

${\displaystyle \frac{1}{{\left[A\right]}_t}}=kt+{\displaystyle \frac{1}{{\left[A\right]}_0}}$

or

${\displaystyle \frac{1}{\left[A\right]}}-{\displaystyle \frac{1}{{\left[A\right]}_0}}=kt$

Restrict t

$t=t_{1\text{/}2}$

then we can define

${\left[A\right]}_t={\displaystyle \frac{1}{2}}{\left[A\right]}_0$

and substitute into the integrated rate law and simplify:

$\begin{array}{ccc}\hfill {\displaystyle \frac{1}{\frac{1}{2}{\left[A\right]}_0}}-{\displaystyle \frac{1}{{\left[A\right]}_0}}& =& k{t}_{1\text{/}2}\hfill \\ \hfill 2{\left[A\right]}_0-{\displaystyle \frac{1}{{\left[A\right]}_0}}& =& k{t}_{1\text{/}2}\hfill \\ \hfill {\displaystyle \frac{1}{{\left[A\right]}_0}}& =& k{t}_{1\text{/}2}\hfill \end{array}$

Thus:

$t_{1\text{/}2}={\displaystyle \frac{1}{k{\left[A\right]}_0}}$

For a second-order reaction, $t_{1\text{/}2}$ is inversely proportional to the concentration of the reactant, and the half-life increases as the reaction proceeds because the concentration of reactant decreases. Consequently, we find the use of the half-life concept to be more complex for second-order reactions than for first-order reactions. Unlike with first-order reactions, the rate constant of a second-order reaction cannot be calculated directly from the half-life unless the initial concentration is known.

Zero-Order Reactions

We can derive an equation for calculating the half-life of a zero order reaction as follows:

$\left[A\right]=-kt+{\left[A\right]}_0$

When half of the initial amount of reactant has been consumed $t=t_{1\text{/}2}$ and $\left[A\right]={\displaystyle \frac{{\left[A\right]}_0}{2}}.$ Thus:

$\begin{array}{ccc}\hfill {\displaystyle \frac{{\left[\text{A}\right]}_0}{2}}& =& -k{t}_{1\text{/}2}+{\left[\text{A}\right]}_0\hfill \\ \hfill k{t}_{1\text{/}2}& =& {\displaystyle \frac{{\left[\text{A}\right]}_0}{2}}\hfill \end{array}$

and

$t_{1\text{/}2}={\displaystyle \frac{{\left[A\right]}_0}{2k}}$

The half-life of a zero-order reaction increases as the initial concentration increases.

Equations for both differential and integrated rate laws and the corresponding half-lives for zero-, first-, and second-order reactions are summarized in Table 18.4.1.

Table 18.4.1. Summary of Rate Laws for Zero-, First-, and Second-Order Reactions
	Zero-Order	First-Order	Second-Order
rate law	rate = k	rate = k[A]	rate = k[A]2
units of rate constant	M s−1	s−1	M−1 s−1
integrated rate law	[A] = −kt + [A]0	ln [A] = −kt + ln[A]0	$\frac{1}{\left[A\right]}=kt+\left(\frac{1}{{\left[A\right]}_0}\right)$
plot needed for linear fit of rate data
relationship between slope of linear plot and rate constant	k = −slope	k = −slope	k = +slope
half-life	$t_{1\text{/}2}=\frac{{\left[A\right]}_0}{2k}$	$t_{1\text{/}2}=\frac{0.693}{k}$	$t_{1\text{/}2}=\frac{1}{{\left[A\right]}_{0}k}$

Glossary

half-life of a reaction (t_l/2): time required for half of a given amount of reactant to be consumed

integrated rate law: equation that relates the concentration of a reactant to elapsed time of reaction