# What is the Difference Between Differential Rate Law and Integrated Rate Law

The main difference between differential rate law and integrated rate law is that the differential rate law focuses on instantaneous rates at a given moment, while the integrated rate law describes concentration changes over a specific time period.

Differential rate law and integrated rate law are fundamental concepts in chemical kinetics. Both provide insights into how the rate of a chemical reaction changes over time and its dependence on reactant concentrations.

### Key Areas Covered

1. What is Differential Rate Law
– Definition, Features, Role
2. What is Integrated Rate Law
– Definition, Features, Role
3. Similarities Between Differential Rate Law and Integrated Rate Law
– Outline of Common Features
4. Difference Between Differential Rate Law and Integrated Rate Law
– Comparison of Key Differences
5. FAQ: Differential Rate Law and Integrated Rate Law

### Key Terms

Differential Rate Law, Integrated Rate Law, Rate Law

## What is Differential Rate Law

The differential rate law is a crucial concept in chemical kinetics, describing the relationship between the rate of a chemical reaction and the concentrations of reactants. Unlike the integrated rate law, which expresses concentrations as a function of time, the differential rate law directly relates the rate of reaction to the concentrations of reactants at a specific moment.

The general form of a differential rate law for a reaction:

Rate = k[A]m[B]n

where:

• Rate is the reaction rate,
• k is the rate constant,
• A] and][B] are the concentrations of reactants A and B, respectively,
• m and n are the reaction orders with respect to A and B, respectively.

The exponents m and n represent the reaction orders and can be determined experimentally. The overall reaction order is the sum of the individual reaction orders (m+n). The rate constant k is a proportionality constant that depends on factors such as temperature, pressure, and the specific reaction mechanism.

The reaction order provides insights into the dependence of the reaction rate on each reactant’s concentration. A zero-order indicates no dependence, a first-order signifies a linear dependence, and a second-order suggests a quadratic dependence. The overall reaction order is the sum of individual orders. Determining the rate law experimentally involves measuring reaction rates under varying initial concentrations of reactants. Furthermore the method of initial rates is often employed, where initial rate data is collected and used to deduce the reaction order with respect to each reactant.

Understanding the differential rate law aids in predicting how changes in concentration impact reaction rates, facilitating the optimization of reaction conditions. Moreover, the rate constant (k) quantifies the reaction’s intrinsic speed and is influenced by temperature.

## What is Integrated Rate Law

The integrated rate law is a mathematical expression that describes the concentration of reactants or products over time in a chemical reaction. It provides a comprehensive view of how the concentration changes with time and is derived from the differential rate law, which relates the rate of reaction to the concentrations of reactants.

There are different forms of integrated rate laws, and the choice of which one to use depends on the reaction’s order. For a zero-order reaction, where the rate is independent of the concentration, the integrated rate law is expressed as:

[A]t = [A] – kt,

where [A]t is the concentration at time t, [A] is the initial concentration, k is the rate constant, and t is time.

However, in the case of a first-order reaction, where the rate is directly proportional to the concentration of the reactant, the integrated rate law takes the form:

ln[A]t = -kt + ln[A].

The natural logarithm is employed to linearize the data in this scenario.

For a second-order reaction, where the rate is proportional to the square of the concentration, the integrated rate law is represented as:

1/[A]t = kt + 1/[A].

This form is especially useful in determining reaction order and rate constants.

These integrated rate laws are powerful tools in the field of chemical kinetics. They help scientists to deduce reaction mechanisms, predict reaction outcomes, and optimize reaction conditions. By analyzing concentration changes over time, researchers gain valuable insights into the kinetics of chemical reactions, contributing to advancements in fields such as pharmaceuticals, materials science, and environmental chemistry.

## Similarities Between Differential Rate Law and Integrated Rate Law

• Both describe the rate of a chemical reaction in terms of the concentrations of reactants or products.
• Moreover, these laws involve the concept of reaction order.
• In addition, both laws incorporate the rate constant (k), which represents the speed of the reaction.

## Difference Between Differential Rate Law and Integrated Rate Law

### Definition

Differential rate law describes how the rate of a reaction changes with time and provides the relationship between the concentration of reactants and the rate at a specific point in time. Integrated rate law expresses the concentration of reactants as a function of time and provides a relationship between reactant concentrations at different points in time.

### Expressed as

Furthermore, differential rate law is typically expressed as an equation involving the instantaneous rate of the reaction and the concentrations of reactants, whereas integrated rate law is typically expressed as an equation relating initial concentrations, concentrations at a given time, and reaction time.

### Plots

Differential rate law often leads to nonlinear plots, as it focuses on the rate at a specific moment, while integrated rate law can result in linear plots, such as first-order, second-order, or zero-order reactions, depending on the integrated form used.

## FAQ: Differential Rate Law and Integrated Rate Law

### What are the two types of rate laws?

The two types of rate laws are differential rate law and integrated rate law.

### What is a rate law?

Rate law is a mathematical expressions that describe the relationship between the rate of a chemical reaction and the concentration of its reactants.

### Why is rate law used?

Rate law is used to predict the relationship between the rate of a reaction and the concentrations of reactants.

## Conclusion

The main difference between differential rate law and integrated rate law is that the differential rate law focuses on instantaneous rates at a given moment, while the integrated rate law describes concentration changes over a specific time period.

##### Reference:

1. “Rate Laws of Motion.” Byju’s.
2. “Integrated Rate Laws.” LibreTexts Chemistry.

##### Image Courtesy:

1. “Rateloglogplot” By Fabiuccio~enwikibooks at English Wikibooks – Transferred from en.wikibooks to Commons. (Public Domain) via Commons Wikimedia

### About the Author: Hasini A

Hasini is a graduate of Applied Science with a strong background in forestry, environmental science, chemistry, and management science. She is an amateur photographer with a keen interest in exploring the wonders of nature and science.