**Calculus in Chemistry**

Rate Laws

A rate law is a mathematical equation that describes the progress of the reaction.

There are two types of rate law expressions: differential and integrated.

The one which relates the rate of the reaction to the concentration (molarity) are the differential rate laws.

For differential rate laws, the unit for the rate is the concentration of the solution/second.

Introduction

Calculus has been around for hundreds of years during which it has been used in various fields of science, business, engineering, and much more.

Calculus has also had a big impact on chemistry and that is the focus of today's presentation.

The presentation will focus on calculus implementation within chemical reactions and rates of reactions.

Average Rate of Reaction

In calculus the average rate of a function is described as:

In chemistry, the average rate of reaction is described as :

The square brackets represent molarity which is moles of solute per liter of solution.

Things to remember: In the reaction A B, A is the reactant.

Think of a graph of concentration vs time

Instantaneous Rate of Reaction

The average rate of reaction gives the average rate from the beginning to the end of the reaction.

The instantaneous rate of reaction give the rate at some specific instant of time

Average rate of change:

Instantaneous rate of change:

Differential Rate Laws

For a reaction with the general equation:

aA+bB cC+dD

where A,B,C, and D are the chemicals and a,b,c,d are the coefficients.

For the reaction above the rate law is:

rate=k[A]^m[B]^n

The reaction rate depends on the rate constant k for the given set of reaction conditions and the concentration of A and B raised to the powers m and n, respectively

The overall reaction order is the sum of all the exponents in the rate law: m + n

In general, rate laws are categorized by their order. The laws begin with the zeroth order, then continue up to the first, second, and up to the nth order

Zero - Order Reaction

For a zero-order reaction, the rate of reaction is a constant.

When the limiting reactant is completely consumed, the reaction stops abruptly.

The Differential Rate Law expression should look like rate = k and the rate constant, k, has units of M/sec.

First - Order Reaction

For a first-order reaction, the rate of reaction is directly proportional to the concentration of one of the reactants.

The Differential Rate Law expression should look like rate = k [A] and the rate constant, k, has units of sec-1.

Second - Order Reaction

For a second-order reaction, the rate of reaction is directly proportional to the square of the concentration of one of the reactants.

The Differential Rate Law expression should look like rate = k [A]^2 and the rate constant, k, has units of 1/(M*sec).

Integrated Rate Laws

The integrated rate laws are used to relate both time and concentration of species.

Each nth order rate law has its own integrated rate law.

To find the integrated rate law for the nth order rate law, we integrate both sides of the nth order differential rate law.

[Look at board for the proof of the first and second order rate law]

Practice Problem

(Don't worry, we'll do it together)

The following reaction is aspirin (acetylsalicylic acid) mixing with water to form salicylic acid and acetic acid:

Q: Use the data given to determine the instantaneous rate of change at Time = 200 h

Conclusion

In the previous question we studied the instantaneous rate of reaction of the salicylic acid and this is studied by scientists as well. Because salicylic acid is what actually reduces pain, scientists study what causes the reaction to go faster or slower in order to improve the reaction.

In general, the application of calculus through average and instantaneous rates of change of a reaction and through differential and integrated rate laws help in many areas of chemistry, creating a major impact on many industries, from medicine, to food/agriculture, to the industrial production of almost every man-made material or chemical (through catalysts, enzymes, etc), who use those concepts. Without calculus, many of these concepts would not exist, and the world would be a much different place