D32.4 Zeroth-Order Reaction

Jia Zhou; John Moore

D32.4 Zeroth-Order Reaction

The “A ⟶ products” reaction that is zeroth-order (m = 0) with respect to [A] exhibits a constant reaction rate regardless of the concentration of A:

$\text{Rate} = k[A]^0 = k$

The integrated rate law for such a zeroth-order reaction is:

$\displaystyle{\int^{[\text{A}]_t}_{[\text{A}]_0} \dfrac{d[\text{A}]}{[\text{A}]^m}=\int^{[\text{A}]_t}_{[\text{A}]_0} d[\text{A}] = [\text{A}]_t - [\text{A}]_0 = -kt}$

This integrated rate law also has a standard linear equation format:

$\begin{array}{rcl} [\text{A}]_t &=& -kt + [\text{A}]_0 \\[0.5em] y &=& \; mx + \; b \end{array}$

A plot of [A]_t vs. t for a zeroth-order reaction would yield a straight line with a slope of −k and a y-intercept of [A]₀.

In the figure below, there are two plots for the decomposition reaction of ammonia. One reaction occurred on a hot tungsten (W) surface, while the other reaction occurred on a hot quartz (SiO₂) surface.

Time ( s ),” on the x-axis and, “[ N H subscript 3 ] M,” on the y-axis. The x-axis shows a single value of 1000 marked near the right end of the axis. The vertical axis shows markings at 1.0 times 10 superscript negative 3, 2.0 times 10 superscript negative 3, and 3.0 times 10 superscript negative 3. A decreasing linear trend line is drawn through six points. This line is labeled “NH3 decomposition on W.” A decreasing slightly concave up curve is similarly drawn through eight points. This curve is labeled “NH3 decomposition on S i O subscript 2.” — **Figure: Decomposition of ammonia**. The decomposition of NH₃ on a tungsten (W) surface (blue plot) is zeroth-order with respect to [NH₃] because the data, when plotted as [NH₃] vs t, fits the equation y = mx + b. However, when this reaction occurs on a quartz (SiO₂) surface (red plot), there is curvature in the plot: the reaction is not zeroth-order with respect to [NH₃].

We can see from this set of data that the reaction on tungsten is zeroth-order with respect to [NH₃]: the plot of [NH₃] vs t fits a straight line. From the slope, we find that the rate constant for this reaction under the experimental conditions is:

$k = -\text{slope} = 1.25 \times 10^{-6}\;\frac{M}{s}$

The decomposition on hot quartz, on the other hand, is not zeroth-order with respect to [NH₃] (analysis of the data shows that it is first order).

Equations for zeroth-, first-, and second-order reactions are summarized below.

	Zeroth-Order	First-Order	Second-Order
rate law	rate = k	rate = k[A]	rate = k[A]²
units of rate constant	$\frac{M}{s}$	$\frac{1}{s}$	$\frac{1}{M{\cdot}s}$
integrated rate law	[A]_t = −kt + [A]₀	ln[A]_t = −kt + ln[A]₀	$\frac{1}{[\text{A}]_t} = kt + \frac{1}{[\text{A}]_0}$
linear plot	[A] vs. t	ln[A] vs. t	$\frac{1}{[\text{A}]}$ vs. t
relationship between slope of linear plot and rate constant	k = −slope	k = −slope	k = +slope
Table: Summary of Rate Laws. for Zeroth-, First-, and Second-Order Reactions

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