Question

What is an integrated rate law?

Answer 1

Hint: The rate law is an equation representing the dependence of the rate of reaction on the concentration of reacting species. It is a differential equation that describes the change in concentration of reactant(s) per unit time. It can be integrated to obtain an integrated rate law that will relate concentration terms with time directly.

Complete answer:
The rate of a reaction is the measure of the time required for products to be formed or reactants to be used up in a chemical reaction. It is dependent on the concentration of the reactants. The mathematical expression that is used to establish a relationship between the rate of chemical reaction and concentration of reactants is known as rate law or rate equation.
Sometimes, it is called differential rate law because it expresses the reaction rate in terms of change in concentration of reactants over a period of time.
For a general reaction:
\[\text{aA}+\text{bB}\to \text{cC}+\text{dD}\]
Differential rate law can be written as follows:
\[\text{rate}=-\dfrac{1}{\text{a}}\dfrac{\Delta \text{A}}{\Delta \text{t}}=-\dfrac{1}{\text{b}}\dfrac{\Delta \text{B}}{\Delta \text{t}}=\text{k}{{\left[ \text{A} \right]}^{\text{m}}}{{\left[ \text{B} \right]}^{\text{n}}}\]
Here, k is the rate constant.
We integrate the differential rate equation to obtain a relation between the concentration and reaction rate at an instantaneous time. The equation so obtained is known as the integrated rate equation. For reactions of a different order, we observe different integrated rate equations.
Below are given the integrated rate laws for first and second-order reactions:
\[\begin{align}
  & \text{1st-order: }\left[ \text{A} \right]={{\left[ \text{A} \right]}_{0}}{{\text{e}}^{-\text{kt}}} \\
 & \text{2nd-order: }\dfrac{1}{\left[ \text{A} \right]}=\text{kt}+\dfrac{1}{{{\left[ \text{A} \right]}_{0}}} \\
\end{align}\]

Additional information:
An integrated rate law is used to determine the time taken by a radioactive material to decay into a stable nucleus.

Note:
Rate laws are determined experimentally and cannot be predicted by reaction directly. The m and n values in rate law are the order of reaction with respect to reactants and describe how much a change in the amount of each substance affects the overall rate, and the overall order of a reaction is the sum of the orders for each substance present in the reaction.