The rates at which chemical reactions take place. Some are very slow, like iron rusting, where as others are very fast, like an explosion. Chemical kinetics is the study of reaction rates and the variables influencing them. Integrated rate equations, which show us how the concentration of reactants varies over time, are one of the main subjects in this field.
What is an Integrated Rate Equation?
The concentration of reactants or products as a function of time is expressed using an integrated rate equation. It helps forecast how much reactant will be left at any given time and is obtained by integrating the differential rate law. First-order, second-order, and zero-order reactions have different integrated rate equations.
1.Zero-Order Reaction
When the rate of a reaction is independent of the reactant concentration, it is said to be zero-order. This indicates that the pace of the reaction is constant.
Equation for a Zero-Order Reaction:
[A] = [A]0 − kt Where:
[A] = concentration of reactant at time t
[A]0 = initial concentration of reactant
k = rate constant
t = time
Graphical Representation:
If we plot [A] vs. t, we get a straight line with a negative slope. The slope of this line gives the value of k.
Example of a Zero-Order Reaction:
The decomposition of ammonia on a hot platinum surface:
2NH3 → N2 + 3H2
2. First-Order Reaction
A first-order reaction is one in which the concentration of one ingredient determines the rate of reaction.
Equation for a First-Order Reaction:
ln[A] = ln[A]0 − kt Or, in another form: [A] = [A]0e − kt
Graphical Representation:
If we plot ln[A] vs. t, we get a straight line with a negative slope. The slope gives the value of k.
Half-Life of a First-Order Reaction:
Half-life (t1/2) is the time required for half of the reactant to be used up. For a first-order reaction, t1/2 = 0.693 / k
This demonstrates that the half-life of first-order reactions is unaffected by the initial concentration.
3. Second-Order Reaction
A second-order reaction is one in which the concentration of either one reactant squared or two separate reactants determines the rate of reaction.
Equation for a Second-Order Reaction:
1 / [A] = 1 / [A]0 + kt
Graphical Representation:
If we plot 1 / [A] vs. t, we get a straight line with a positive slope, where the slope is equal to k.
Half-Life of a Second-Order Reaction:
For a second-order reaction, the half-life is given by: t1/2 = 1k / [A]0
The half-life is dependent on the initial concentration, in contrast to first-order processes.
Example of a Second-Order Reaction:
The reaction between hydrogen and bromine:
H2 + Br2 → 2HBr
Key Differences Between Reaction Orders
Order of Reaction | Rate Law | Integrated Equation | Graph | Half-Life (t1/2) |
Zero-Order | Rate = k | [A] = [A]0 − kt | [A] vs. t (straight line) | t1/2 = [A]0 / 2k |
First-Order | Rate = k[A] | ln[A] = ln[A]0 − kt | ln[A] vs. t (straight line) | t1/2 = 0.693 / k |
Second-Order | Rate = k[A]2 | 1 / [A] = 1 / [A]0 + kt | 1 / [A] vs. t (straight line) | t1/2 = 1 / k[A]0 |
Why Are Integrated Rate Equations Important?
They aid in figuring out the reactant concentration at any given moment.
They are employed in the computation of rate constants and half-lives.
They are vital for drug stability research in the pharmaceutical sector.
They aid in the comprehension of environmental processes and radioactive decay.
Summary
In chemical kinetics, integrated rate equations are essential for understanding how reactions develop over time. These equations are useful in daily life, whether it is for understanding nuclear decay or forecasting a medication’s shelf life.
A mathematical expression that links a reactant’s concentration to time is called an integrated rate equation. It helps ascertain how the concentration of reactants falls over time during a chemical reaction and is obtained by integrating the differential rate law.
Depending on the reaction’s order, there are three primary categories of integrated rate equations.
Zero-order reaction: [A] = [A]0 − kt
First-order reaction: ln[A] = ln[A]0 − kt
Second-order reaction: 1 / [A] = 1 / [A]0 + kt
Each equation is used depending on whether the reaction follows zero-order, first-order, or second-order kinetics.
To determine the order of a reaction:
Plot [A] vs. t → If it is a straight line, it’s zero-order.
Plot ln[A] vs. t → If it is a straight line, it’s first-order.
Plot 1 / [A] vs. t → If it is a straight line, it’s second-order.
The order is determined by checking which graph gives a linear relationship.
Half-life (t1/2) is the time required for half of the reactant to be consumed. It is useful in predicting how long a substance will last.
First-order reactions: t1/2 = 0.693 / k (constant, independent of concentration).
Zero-order & second-order reactions: Half-life depends on the initial concentration.
Half-life is widely used in radioactive decay, drug elimination, and chemical stability studies.
The rate constant k is determined by analysing the equation that fits the reaction:
In a zero-order reaction, k is the negative slope of [A] vs. t.
In a first-order reaction, k is the negative slope of ln[A] vs. t.
In a second-order reaction, k is the positive slope of 1 / [A] vs. t.
The slope of these plots gives the value of k.
Integrated rate equations are used in:
Pharmaceuticals – For calculating the shelf life of medicines.
Nuclear chemistry – For predicting radioactive decay (e.g., Carbon-14 dating).
Environmental science – For understanding pollutant degradation rates.
Food science – For estimating food spoilage and chemical preservation.
Temperature affects the rate constant (k) according to the Arrhenius equation: K = Ae−Ea/RT
Where:
A = frequency factor
Ea = activation energy
R = gas constant
T = temperature in Kelvin