## 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

- SO
_{2}Cl_{2}→ Cl_{2}+ SO_{2} - 2N
_{2}O_{5}→ O_{2}+ 4NO_{2} - 2H
_{2}O_{2}→ 2H_{2}O + O_{2}

## 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 e^{ln(x)} = x), the equation is transformed as follows:

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.

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.

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 ‘t_{1/2}’) is the time taken for the initial concentration of the reactant(s) to reach half of its original value. Therefore,

At t = t_{1/2} , [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 = t_{1/2} in the equation [A] = [A]_{0} e^{-kt}:

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]_{0}e

^{-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:

t_{1/2} = 0.693/k

Where ‘t_{1/2}’ 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 s^{1} (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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