First-Order Reactions

What is a First-Order Reaction?

A first-order reaction can be defined as a chemical reaction in which the reaction rate is linearly dependent on the concentration of only one reactant. In other words, a first-order reaction is a chemical reaction in which the rate varies based on the changes in the concentration of only one of the reactants. Thus, the order of these reactions is equal to 1.

Examples of First-Order Reactions

  • SO2Cl2 → Cl2 + SO2
  • 2N2O5 → O2 + 4NO2
  • 2H2O2 → 2H2O + O2

Differential Rate Law for a First-Order Reaction

A differential rate law can be employed to describe a chemical reaction at a molecular level. The differential rate expression for a first-order reaction can be written as:

Rate = -d[A]/dt = k[A]1 = k[A]

Where,

  • ‘k’ is the rate constant of the first-order reaction, whose units are s-1.
  • ‘[A]’ denotes the concentration of the first-order reactant ‘A’.
  • d[A]/dt denotes the change in the concentration of the first-order reactant ‘A’ in the time interval ‘dt’.

Integrated Rate Law for a First-Order Reaction

Integrated rate expressions can be used to experimentally calculate the value of the rate constant of a reaction. In order to obtain the integral form of the rate expression for a first-order reaction, the differential rate law for the first-order reaction must be rearranged as follows.

\(\frac{-d[A]}{dt} = k[A]\) \(\Rightarrow \frac{d[A]}{[A]} = -kdt\)

Integrating both sides of the equation, the following expression is obtained.

\(\int_{[A]_0}^{[A]}\frac{d[A]}{[A]} = -\int_{t_0}^{t}kdt\)

Which can be rewritten as:

\(\int_{[A]_0}^{[A]}\frac{1}{[A]}d[A] = -\int_{t_0}^{t}kdt\)

Since \(\int\frac{1}{x} = ln(x) \), the equation can be rewritten as follows:

ln[A] – ln[A]0 = -kt

ln[A] = -kt + ln[A]0 (or) ln[A] = ln[A]0 – kt

Raising each side of the equation to the exponent ‘e’ (since eln(x) = x), the equation is transformed as follows:

\(e^{ln[A]} = e^{ln[A]_0 – kt}\)

Therefore,

\([A] = [A]_0 e^{-kt}\)

This expression is the integrated form of the first-order rate law.

Graphical Representation of a First-Order Reaction

The concentration v/s time graph for a first-order reaction is provided below.

First-Order Reaction Graph

For first-order reactions, the equation ln[A] = -kt + ln[A]0 is similar to that of a straight line (y = mx + c) with slope -k. This line can be graphically plotted as follows.

ln[A] vs Time for a First-Order Reaction

Thus, the graph for ln[A] v/s t for a first-order reaction is a straight line with slope -k.

Half-Life of a First-Order Reaction

The half-life of a chemical reaction (denoted by ‘t1/2’) is the time taken for the initial concentration of the reactant(s) to reach half of its original value. Therefore,

At t = t1/2 , [A] = [A]0/2

Where [A] denotes the concentration of the reactant and [A]0 denotes the initial concentration of the reactant.

Substituting the value of A = [A]0/2 and t = t1/2 in the equation [A] = [A]0 e-kt:

\(\frac{[A]_0}{2} = [A]_0 e^{-kt_{1/2}}\) \(\Rightarrow \frac{1}{2} = e^{-kt_{1/2}}\)

Taking the natural logarithm of both sides of the equation in order to eliminate ‘e’, the following equation is obtained.

\(ln(\frac{1}{2}) = -kt_{1/2}\) \(\Rightarrow t_{1/2} = \frac{0.693}{k}\)

Thus, the half-life of a first-order reaction is equal to 0.693/k (where ‘k’ denotes the rate constant, whose units are s-1).

Frequently Asked Questions – FAQs

What is the definition of a first-order reaction?

A first-order reaction can be defined as a chemical reaction for which the reaction rate is entirely dependent on the concentration of only one reactant. In such reactions, if the concentration of the first-order reactant is doubled, then the reaction rate is also doubled. Similarly, if the first-order reactant concentration is increased five-fold, it will be accompanied by a 500% increase in the reaction rate.

What is the differential rate law and the integrated rate law for a first-order reaction?

The differential rate law for a first-order reaction can be expressed as follows:

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

The integrated rate equation for a first-order reaction is:

[A] = [A]0e-kt

Where,

  • [A] is the current concentration of the first-order reactant
  • [A]0 is the initial concentration of the first-order reactant
  • t is the time elapsed since the reaction began
  • k is the rate constant of the first-order reaction
  • e is Euler’s number (which is the base of the natural logarithm)

What is the relationship between the half-life and the rate constant for a first-order reaction?

The half-life of a chemical reaction is the time required for the concentration of the reactants to reach half of their initial value. For first-order reactions, the relationship between the reaction half-life and the reaction rate constant is given by the expression:

t1/2 = 0.693/k

Where ‘t1/2’ denotes the half-life of the reaction and ‘k’ denotes the rate constant.

What are the units of the rate constant for a first-order reaction?

For first-order reactions, the rate constant is expressed in s1 (reciprocal seconds). The units of the rate constant can be determined using the following expression:

Units of k = M(1-n).s-1 (where ‘n’ is the order of the reaction)

Since the reaction order of a first-order reaction is equal to 1, the equation is transformed as follows:

Units of k = M(1-1).s-1 = s-1

For a first-order reaction, if a graph is plotted with ln[A] on the Y-axis and time on the X-axis, what will it look like?

The graph will be a straight line with a slope of -k.

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