Answer to Question #109332 in Physical Chemistry for GHOST

Question

Derive integrated rate law for the first order reaction.

Answer 1

First, write the differential form of the rate law.

Rearrange to give:

Second, integrate both sides of the equation.

Recall from calculus that:

Upon integration,

Rearrange to solve for [A] to obtain one form of the rate law:

This can be rearranged to:

This can further be arranged into y=mx +b form:

and y-intercept b:

Now, recall from the laws of logarithms that

where [A] is the concentration at time t and [A]o is the concentration at time 0, and k is the first-order rate constant.

Because the logarithms of numbers do not have any units, the product −kt also lacks units. This concludes that unit of k in a first order of reaction must be time-1. Examples of time-1 include s-1 or min-1. Thus, the equation of a straight line is applicable:

To test if it the reaction is a first-order reaction, plot the natural logarithm of a reactant concentration versus time and see whether the graph is linear. If the graph is linear and has a negative slope, the reaction must be a first-order reaction.

To create another form of the rate law, raise each side of the previous equation to the exponent, e:

Simplifying gives the second form of the rate law:

The integrated forms of the rate law!