All About Integrated Rate Equations

The rate laws that have been addressed thus far are those that link reaction rates to reactant concentrations. A second form of each rate law, which is a relationship between the concentrations of reactants and the passage of time, can also be found. These are officially referred to as integrated rate law. We can use an integrated rate law to assist us determine the amount of reactant or product present after a period of time, or to estimate the length of time required for a reaction to proceed to a certain extent.An integrated rate law is used to calculate the period of time a radioactive material must be held for its radioactivity to decay to a safe level.

A chemical reaction’s differential rate law can be integrated with respect to time using calculus, resulting in an equation that connects the amount of reactant or product present in a reaction mixture to the amount of time that has transpired from the beginning of the reaction. Depending on the intricacy of the differential rate law, this method might be either very basic or extremely complex. For the sake of this study, we will concentrate on the integrated rate laws that are obtained for first-, second-, and zero-order reactions, respectively.

Integrated Rate Equations

Zero-order reaction

Reactions of zero-order are those in which the rate of reaction is determined by the concentration of reactants multiplied by their zeroth power. Zero-order reactions are extremely rare and are only detected in very small numbers. Zero-order reactions include, for example, the thermal breakdown of HI on a gold surface, the decomposition of gaseous ammonia on a heated platinum surface, and many others. The following is a general equation for a zero-order reaction with a rate constant k, which may be expressed as follows:

                             A → B

Rate = –d[A]dt = k[A]o

    => d[A] = -k dt

Integrating both sides:

⇒ [A] = -kt + c………………………..(1)

Where, c= constant of integration,

At time, t=0, [A] = [A]o

Putting the limits in equation (1) we get the value of c,

 [A]o=c

First-order reaction

In a first-order reaction, the rate of reaction is determined by the first power of the concentration of the reactants in the reaction mixture. First-order reactions include the natural and controlled radioactive decay of unstable nuclei, which are both instances of first-order reactions. The following is a general equation for a first-order reaction with a rate constant k, which can be expressed as:

A → B

Rate = –d[A]dt= k[A]

=>d[A][A]  = -k dt

Integrating both sides:

=> ln [A] = -kt + c          ……………….(2)

Where, c= constant of integration,

At time, t=0, [A] =[A]o

Putting the limits in equation (2) we get the value of c

=> ln[A] = c

Using the value of c in the preceding equation, we get the following result:

=> ln [A] = -kt + ln

We may also calculate the value of the rate constant, k, using the following equation:

ln[A][A]O = -kt

=> k = –ln[A][A]O t

Concentration can be defined as follows at any point in time:

[A] =[A]Oe-Kt

As a result, we can use the integrated rate equation for zero and first-order reactions to determine the concentration and rate of reaction at any given time.

Differential Rate Law vs Integrated Rate Law

Integrated rate law and differential rate law are two types of rate laws. The differential rate law is the most common. The difference between differential rate law and integrated rate law is that differential rate law calculates the rate of a chemical reaction as a function of the change in concentration of one or more reactants over a specific time period, whereas integrated rate law calculates the rate of a chemical reaction as a function of the initial concentration of one or more reactants after a specific period of time. Differential rate law and integrated rate law are both used in chemical reactions.

Chemical reactions are measured by the rate at which the concentration of reactants or products changes as they go through the process. Different rate laws are employed to explain the progression of the reaction. These rate laws are mathematical correlations between different factors that are stated as rate laws.

Conclusion

It is possible to express the concentrations of reactants or products as a function of time using an integrated rate law, which is a mathematical equation. When determining the amount of reactant or product present after a period of time, or when estimating the length of time required for a reaction to proceed to a particular extent, we can make use of an integrated rate law to help us.