## Table of Contents

- What is Order of Reaction?
- Zeroth Order Reaction
- First Order Reaction
- Second-Order Reaction
- Lesson Summary

Explore reaction order graphs. Understand how to use data from the graphs and integrated rate law to determine the order of reaction and find the rate constant.
Updated: 03/07/2022

- What is Order of Reaction?
- Zeroth Order Reaction
- First Order Reaction
- Second-Order Reaction
- Lesson Summary

Chemical reactions never happen instantaneously, it takes time for molecules to react with one another to form the products of a reaction. Each reaction has its own reaction kinetics, which are defined by its rate constant and order of the reaction. **Order of reaction** is defined as how the reactants' concentrations affect the rate of the reaction. The three most common are zero-, first-, and second-order reactions. The order is the sum of the exponents on each of the reactants' concentration terms in the rate law. If a reactant has a bigger exponent, its concentration affects the reaction rate more. For a general reaction:

{eq}aA + bB \rightarrow cC + dD \\\ Rate= k[A]^x[B]^y \hspace{0.5cm} Order=x+y {/eq}

Where k is the rate constant, and x and y are independent of a and b, respectively. The **rate constant** is a proportionality constant that generally determines how slow or fast a reaction occurs.

The simplest order of the reaction is a zero-order reaction. In zero-order reactions, the concentration of the reactants does not affect the rate of the reaction. The zero-order rate law is

{eq}Rate=k {/eq}

k is a constant value, so the rate of the reaction does not change unless other factors such as temperature and pressure change.

Rate law graphs, usually called kinetics graphs, show how reactants of a reaction change over time. This graph shows a zero-order reaction.

In rate order graphs, the X-axis depicts time, usually in seconds, where time=0 is the start of the reaction. The Y-axis of zero-order reaction order graphs depicts the concentration of a reactant/reactants, in mol per Liter. This graph depicts that a reactant has a concentration of 12 {eq}\frac{mol}{L} {/eq} before the reaction starts. Then the reaction starts at time=0, and 15 seconds after the reaction started, there are no reactants left. This means the reaction has run to completion. More time means less concentration for reactants because they are being transformed into products.

The order of a reaction can be determined from its kinetics graph. The rate of a reaction is defined as how fast reactants are being converted into products, measured in {eq}\frac{mol}{L*s} {/eq}, and happens to be the slope of the line of the graph shown.

The line in the zero-order reaction graph is linear, which means the slope of the line does not change as the concentration of the reactant decreases, which is true for zero-order reactions.

If a reaction is zero-order, the graph of Concentration versus Time is linear.

Once the order of the reaction is found, the rate constant can be found using that information. As far as how to find the rate constant from the graph, the integrated rate law of a zero-order reaction is used. The integrated rate-law is found by integrating Rate=k for a zero-order reaction and rearranging, which gives

{eq}[A]=-kt+[A]_0 {/eq}

Where A is the concentration at time=t, and {eq}A_0 {/eq} is the initial concentration. The formula fits nicely into the y=mx+b formula for a line, so the rate constant is the negative slope of the line plotted.

To find this, the initial and final values are taken, and the slope is found using m=change in Y / change in X. So, to do so for the zero-order reaction graph,

{eq}Rate \hspace{.02cm} Constant=-\frac{0\frac{mol}{L}-12\frac{mol}{L} }{15s-0s}=0.8 \frac{mol}{L*s} {/eq}

A **first-order reaction** means that the rate of the reaction is directly related to the concentration of a reactant. If a reactant's concentration is doubled, the rate of reaction at that specific time is doubled.

A first-order reaction has a rate law of:

{eq}Rate=k[A] {/eq}

The first-order reaction graph of the differential rate law is not as simple as the zero-order reaction graph but is still useful to compare.

The graph uses the same axes as the zero-order reaction and is read in the same way. Since the rate of the reaction changes with the concentration of the reactant, there is not an easy way to glean much information from the graph.

However, if the integrated rate law is plotted with a specific axis, the reaction order and rate constant can be determined.

A straight line is shown in a graph for a first-order reaction where the X-axis is time and the Y-axis is the natural log of concentration.

To determine the order of reaction for first-order reactions using rate order graphs, the graph of ln(concentration) vs time is plotted. If the graph is linear, the reaction follows first-order kinetics.

Like the zero-order example, the equation for the integrated rate law graphs is used. The integrated rate law for a first-order reaction is

{eq}ln[A]=-kt+ln[A]_0 {/eq}

If it is graphed on a time versus ln(concentration) graph, the equation follows the general equation y=mx+b, so once again the negative slope of the line equals the rate constant and is solved the same way. The constant's units in a first-order reaction are {eq}s^{-1} {/eq}.

The rate constant for the graph is

{eq}Rate \hspace{0.2cm} Constant=-\frac{0-1.7}{1.35s-0s}=1.26 s^{-1} {/eq}

A second-order reaction is when the rate of reaction is directly related to the square of a reactant's concentration. If the reactant's concentration doubles, the rate of reaction at that specific time would quadruple.

The rate law for a second-order reaction is

{eq}Rate=k[A]^2 {/eq}

The kinetics graph for a **second-order reaction** graphed on time versus concentration follows a general y=1/x equation.

The initial slope of the graph is much steeper than zero-and first-order reactions because the concentration is squared, leading to a larger rate of reaction at high concentrations. This graph is read like the prior time vs. concentration graphs. Much like the first-order graph of the rate law, finding the reaction order and rate constant is a challenge, but if the line is graphed on a different Y-axis, these values are easily found. Second-order reactions are graphed with a 1/concentration X-axis and time Y-axis.

If the integrated rate law for a chemical reaction is plotted as a straight-line using a {eq}\frac{1}{concentration} {/eq} vs. time graph, the reaction follows second order kinetics.

The rate constant can be solved for with the integrated rate law, which is

{eq}\frac{1}{ [A] }=-kt+\frac{1}{[A]_0} {/eq}

for second-order reactions. So, the rate constant is the positive slope of the line plotted on the{eq}\frac{1}{concentration} {/eq} vs. time graph. The units for a second-order rate constant are {eq}\frac{L}{mol*s} {/eq}.

Since the slope of the integrated second-order rate law is positive, the Y-and X- intercepts cannot be used to find the slope, so an arbitrary point is chosen.

{eq}Rate \hspace{0.2cm} Constant=-\frac{4.08\frac{L}{mol}-0.9\frac{L}{mol} }{7s-0s}=0.57 \frac{L}{mol*s} {/eq}

All chemical reactions have a rate constant and order of reaction that define their kinetics. A **rate constant** is a proportionality constant that determines the general speed of a reaction. The **order of reaction** is how the concentration of the reactants affects the rate of the reaction. The concentration of the reactants will not affect the rate in a **zero-order reaction**. In a **first-order** reaction, the reactant's concentration is directly related to the rate of the reaction. In **second-order reaction**, the square of the reactant's concentration is related to the rate of the reaction.

A zero-order reaction plots a straight line onto a Concentration vs. Time graph so no transformation is necessary. Both first- and second-order reactions must transform the y and b terms of a y = mx + b to plot a straight line. First-order reactions plot a straight line onto a natural log of Concentration vs. Time graph, and second-order reactions plot a straight line on the inverse of Concentration vs. Time graph. Using these graphs, the rate constant is the negative slope (m in a y = mx + b equation) for zero- and first-order reactions, and the positive slope for second-order reactions.

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Frequently Asked Questions

The reaction plotted against different Y-axes is used to determine the order of a reaction. If the line is straight when the Y-axis is concentration, the reaction is zero-order. If the line is straight when the Y-axis is ln(concentration), the reaction is first-order. If the line is straight when the Y-axis is 1/concentration, the reaction is second-order.

Once the graph shows a straight line, the rate constant is found from the slope of the line. For zero- and first-order reactions, the rate constant is the negative slope, and for second-order reactions, the rate constant is the slope.

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