Integrated Rate Equations: Rate Law, Zero & First Order Reaction

Chemistry is about change where chemicals with well-defined properties are converted into other substances. Scientists and students try to find out the feasibility, extent and rate of the reaction for understanding the chemical reaction. The branch of chemistry that deals with reaction rates and their mechanisms is called chemical kinetics. Chemical kinetics helps to predict the rate of reaction. 

The rate equation for any chemical reaction is the expression that provides a relationship between the rate of the reaction and reactants participating in the chemical reaction. If the equation is dependent on the concentration of the reacting species, it is called the differential rate equation. We integrate the differential rate equation to get the relation between concentration at different points and rate constant. Integrated rate equations were introduced due to the difficulty in determining the rate of the reaction from the concentration-time graph. In this article, we are focusing on integrated rate equations

Key Terms: Kinetics of reaction, rates of chemical reactions, elements, chemical reaction, Rate equation

---

Explanation

[Click Here for Sample Questions]

Let us start the explanation part from the basic things: Rate equation. As explained before, the rate equation provides a relationship between the rate of the reaction and the concentration of reactants participating in that particular chemical reaction.

Rate of Reaction

Now, let us consider a general equation.

aA + bB → cC + dD

The rate law for the above reaction will be 

R α [A]x [B]y

- \(\frac{dR}{dt}\)= k[A]x[B]y

Here x and y are the power whose sum will denote the order of the reaction. The order of the reaction can be 0, 1, 2, 3 etc. In a zero-order reaction, the rate of the reaction is independent of the concentration of reactants. 

Also Read:

---

Integrated Rate Equations

[Click Here for Sample Questions]

Integrated rate equations were introduced due to the difficulty in determining the rate of the reaction from the concentration-time graph. Due to which we integrate the differential rate equations to obtain the relation between the concentration and the rate constant. But one thing you have to keep in mind that the integrated rate equations depend on the order of the reaction. Hence, we shall derive the integrated rate equation based on the order of the chemical reaction.

1. Zero Order Reaction

Zero-order reactions are not affected by the concentration of the reactants. For deriving the integrated rate equation, let us look into a general equation.

A→ B

Rate = -d[A]/dt = k[A]⁰

Since any quantity is raised to the power of zero it becomes unity.

=> - d[A]dt = k

=>d[A] = -k dt

Integrating both sides;

⇒[A] = -kt + I →(1)

Here “I” is the constant of integration

When t = 0, [A] =[A]₀ , where [A]₀ is the initial concentration of the reactant. 

Put the limits in equation 1,

[A]₀ = I

Using the value of I in equation 1

=> [A] = -kt + [A]₀

The equation given above is the integrated rate equation for zero-order reactions. If you plot a graph with the concentration on y and time on the x-axis, you get a straight line. The rate constant can be calculated by finding the slope of the line. Also, we can determine the rate constant with a minor modification in the integrated rate equation.

A = -kt + [A]₀

kt = [A]₀-[A]

 k = {[A₀] - [At]} / t

Graph depicting the integrated rate equation of the zero-order reaction.

2. First Order Reaction

Unlike zero-order reactions, the rate of the reaction in 1st order reactions depends on the 1st power of concentration of reactants. Let us derive the integrated rate equation for a 1st order reaction with a rate constant, k.

A→ B

Rate = -d[A]/dt = k[A]

=> - d[A]d[A] = k dt

=> d[A]d[A] = -k dt

Integrating both sides;

⇒ln[A] = -kt + I →(2)

Here “I” is the constant of integration

When t = 0, [A] =[A]₀ , where [A]₀ is the initial concentration of the reactant.

Hence the equation 2 can be modified as,

ln[A]₀ = -k*0 + I

ln[A]₀ = I

Substituting the value of I in equation 2

⇒ln[A] = -kt +ln[A]₀

The equation given above is the integrated rate equation for first-order reactions. We can determine the rate constant with a minor modification in the integrated rate equation. 

ln[A] = -kt +ln[A]₀

ln[A]/[A]₀ = -kt

⇒ k =-(ln [A]/[A]₀₎t

Or

k=1/t ln[R]₀/[R]

Another way to determine rate constant k is given below,

At time t1, equation 2 will be

 ln[A]₁=-kt₁+I →3

At time t2, equation 2 will be

ln[A]₂=-kt₂+I →4

Here, [A]₁ and [A]₂ are the concentrations of the reactants at the time t1 and t2 respectively.

Now, subtracting equation 4 from 3

ln[R]₁ - ln[R]₂ = -kt₁ - -kt₂

ln [R]₁ / [R]₂ = kt₂ -t₁

k= 1/(t₂ - t₁)ln ([R]₁/[R]₂₎

Plotting graph

ln[A] = -kt +lnA₀

It can be rewritten as,

ln[A]/[A]0=kt

Taking antilog of both sides

[R] = [R]₀ e^-kt

Now, let us compare the integrated rate equation of the 1st order reaction with x. We can see the similarity between the equations. If we plot against y, we will get a straight line. The rate constant can be calculated by finding the negative slope of the line.

---

Things to Remember

Anyone can determine the concentration and rate of the reaction at any given moment. All you need are the equations and an understanding of how they were derived. Now, let us look at the most important equations that we learned.

• The rate law for the given reaction is (aA + bB → cC + dD) = -dR/dt= k[A]^x [B]^y, Where the order of the reaction = x +y
• The Integrated rate equation for the zero-order reaction (A→ B) is = [A] = -kt + [A]₀
• The rate constant for zero-order reaction is = k = {[A]₀ - [A]}/t
• The Integrated rate equation for first order reaction = ln[A] = -kt +lnA₀ or ln[A][A]₀=kt
• The rate constant for the first-order reaction is = k=1/t.ln[R]₀R or k= 1/(t₂ - t₁)ln (R1/R2)

Also Read: