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### Overview

The differential rate law relates the change in concentration of a reactant over a specified time. In other words, it tells us about the overall rate of a reaction from the beginning to the end of the reaction and the rate law also tells us how the speed of a reaction is impacted by the change in concentration. The differential rate law can be integrated with time to describe the change in concentration of reactants with respect to time. Using the integrated rate law expressions, we can find the concentration of a reactant or product present at a particular moment in time after the reaction has started. In this section, we will look at the integration of 1st, 2nd and 0th order reactions and some interesting graphs that the integration produces.

## First order reactions

For a first order reaction, we know that the rate of reaction is dependent on one 1st order reactant.

 Order of Reactions Rate Integrated Rate Law 1st rate = k•[A]

### Click to show integration

See how the integrated rate law is derived using calculus.

First order differential rate law:

Integrating both sides by time between t=0 and t = t:

## 2nd order reactions

For a 2nd order reaction, we know that the rate of reaction is dependent on the square of the reactant’s concentration.

 Order of reactions Rate Integrated Rate Law 2nd rate = k•[A]²

### Click to show integration

See how the integrated rate law is derived using calculus.

First order differential rate law:

Integrating both sides by time between t=0 and t = t:

## 0th order reactions

A 0th order reaction rate is defined by the decrease in concentration of reactants over time. A zeroth order reaction rate does not depend on the concentration of the reactant, it means that the rate of reaction is constant over time even though the reactants are decreasing as the reaction progresses.

The differential rate law:

### Click to show integration

See how the integrated rate law is derived using calculus.

0th order differential rate law:

Integrating both sides by time between t=0 and t = t:

Rearranging the equation:

## Graphs

The most useful aspect of the integration is to arrive at an equation y = mx + c, as this can be easily graphed. And by distinguishing the patterns of the graphs, we can quickly determine if the reaction is 0th, 1st, or 2nd order.

For a first order reaction, the graph ln[A] vs time is a straight line with a slope -k.

For a second order reaction, a plot of the inverse of a concentration vs time produces a straight line graph with a slope = k.

For a zeroth order reaction, the plot of the concentration of the reactant vs time is a straight line with a negative slope.

## Example

Given the experimental data (see table below), determine the rate law for the reaction with respect to reactant A.

You can determine the order from integrated rate law. Calculate the values of ln[A] and 1/[A] .

Plot the graphs.

[A] vs Time
ln[A] vs time
1/[A] vs time

Since graph ln[A] vs time gives a straight line and the others do not, we know that the reaction is a first order reaction with respect to A.

## Determine time

How long will it take for the reaction to reach 0.0200 M which has a rate constant of k = 3.5 x 10⁻³ s⁻¹. The initial concentration is 0.100 M/

To determine t, we need to know:

[A]₀ – the initial concentration

[A]t – the final concentration

The order of the reacton

The rate constant k is given 3.5 x 10⁻³ s⁻¹