First Order Reaction: Definition, Derivation, & Graph

A first order reaction is a reaction that proceeds at a rate proportional to the concentration of one of the reactants. So, as the concentration doubles, the reaction rate also doubles. 

• The concentration of other reactants will not have any effect on the rate of reaction.
• Thus, the Order of Reaction for a first-order reaction is always one.
• A first-order reaction is a chemical reaction where the rate varies, depending on the concentration changes of only one reactant.
• The rate of a first-order reaction is affected by both temperature and reactant concentration, in return which is affected by time.

The differential rate expression for a first-order reaction is expressed as:

⇒ Rate = -d[A]/dt = k[A]¹ = k[A]

Here,

• ‘k’ = rate constant of the first-order reaction
• ‘[A]’ = concentration of the first-order reactant ‘A’.
• d[A]/dt = change in the concentration of the first-order reactant ‘A’ in the time interval ‘dt’.

Key Terms: Rate of Reaction, Chemical Reactions, Order of Reactions, Reactant, Rate Constant, Rate of a Chemical Reaction, Pseudo First-Order Reaction

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What is a First Order Reaction?

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First Order Reaction can be defined as:

“A chemical reaction in which reaction rate is linearly dependent on the concentration of one reactant. In a first-order reaction, the rate of reaction varies on the basis of changes in the concentration of only one of the reactants.”

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Pseudo First-Order Reaction

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Pseudo-first-order reactions are not first-order but seem to be first-order because of the higher concentrations of the reactant(s) in comparison to other reactants.

• The order of a chemical reaction can be expressed as the product of the powers of the reactant concentrations in the rate law expression.
• Based on the reactant concentration, reactions can be divided as first-order, second-order, pseudo-first-order, etc.
• Accurate concentrations of every reactant is required to calculate the reaction rate.
• It might result in a high cost for the experiment in case one or both the required reactants are expensive. 
• The Pseudo-1st-order reaction, wherein the 2nd order reaction is treated like a 1st order reaction, is done to avoid more difficult, expensive experiments and calculations.

Zero, First, and Second-order Reactions

The rate of a zero-order reaction is typically constant. The concentration of one of the reactants is seen to impact the rate of a first-order reaction. Thus, the square of a reactant’s concentration or the product of the concentrations of two reactants helps to determine the second-order reaction rate.

Zero, First, and Second-order Reactions

The  differences between the zero, first, and second-order reactions are tabulated below:

Type	Zero Order	First Order	Second Order
Meaning	In a Zero-order reaction, the reaction rate is constant over time and independent of the concentration of the reactant.	In a First-order reaction, the reaction rate is proportional to the first power of the reactant’s concentration.	In a Second-order reaction, the reaction rate is proportional to the second power of the reactant’s concentration or product of two reactants’ concentration.
Rate Equation	R = constant	R = k [A]	R = k [A]2 or R = k [A] [B]
Concentration	[A] = kt	[A] = [A]o exp (-kt)	1 / [A] = 1 / [A]o + kt
Half-life	[A]o / 2k	0.693 / k	1 / k[A]o
Example	Nitrous oxide Decomposition	Hydrogen peroxide Decomposition	Nitrogen dioxide Decomposition

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Examples of First Order Reaction

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First Order Reaction examples are listed below:

• Decomposition of Nitrogen Pentoxide (N₂O₅): 2N₂O₅ → 2NO₂ + ½ O₂
• Decomposition of ammonium nitrate in aqueous solution: NH₄NO₂ → N₂ + 2H₂O
• Decomposition of H₂O₂ in aqueous solution: H₂O₂ → H₂O + ½ O₂
• Hydrogenation of ethane: C₂H₄ + H₂ → C₂H₆
• Hydrolysis of diazo derivatives: C₅H₅N = NCl + H₂O → C₆H₅OH + N₂ + HCl

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Differential Rate Law for a First Order Reaction

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A differential rate law is used to describe a chemical reaction at a molecular level. The differential rate expression for a first order reaction can be expressed as:

Rate = -d[A]/dt = k[A]¹ = k[A]

Where,

• k is the rate constant denoted by 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 dt time interval.

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Integral Derivation Of First Order Reaction

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

Step 1: For any reaction, A → products 
Let,
[A]₀ - Initial Concentration of A
[A]_t  - Concentration of A after time t

Step 2: The Integrated rate equation for a First order reaction is: 

Rate = \(\begin{array}{l}\frac{-d[A]}{dt} = k[A]\end{array}\)

→ d[A]/dt = - k[A] …(i)

Step 3: Integrating this equation,

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

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

We get,

ln[A] – ln[A]₀ = -kt

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

Step 4: Raising both the side of the equation to the exponent ‘e’ (since e^ln(x) = x), the equation can be written as follows:

\(\begin{array}{l}e^{ln[A]} = e^{ln[A]_0 – kt}\end{array}\)

Therefore,

\(\begin{array}{l}[A] = [A]_0 e^{-kt}\end{array}\)

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

Step 5: log_e[A] = -kt + log_e[A]₀

→ k = 1/t . log_e([A]₀/[A])

→ log_e = 2.303 log10

→ k = 2.303/t log10 ([A]₀/[A]) …(ii)

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Graphical Representation of a First Order Reaction

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Comparing Eq (ii) with the equation of a straight line i.e y = mx+c and thereby on plotting Eq(ii), we get a graph where slope = k/2.303

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

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Rate Constant

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Rate of a reaction is the change in concentration with time. For any reaction whose concentration is expressed in terms of moles/litre and time in seconds.

Rate of reaction = mol L^-1 sec^-1

For any First order reaction, the rate is expressed as:

⇒ mol L^-1 sec^-1 = k (mol L^-1)
⇒ k = sec^-1

Thus the rate constant units (k) for the First-order reaction is sec^-1

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Half-Life Period of First Order Reaction

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Half-time or half-life of a first-order reaction is the time taken to reduce the concentration of the reactant to half of its initial value. It is denoted by t1⁄2

For first-order reaction, we know that, k = 1/t . log_e([A]₀/[A])

At half life period, t = t1⁄2 and [A] =[A]₀/2

On substituting, k = 1/ t1⁄2. log_e{[A]₀/([A]₀/2)}
→ t1⁄2 = 0.693/k

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Things to Remember

• In a first-order reaction, the concentration of only one reactant affects the rate of reaction.
• The order of reaction for any first-order reaction is one.
• The unit of rate constant (k) for First-order reactions is sec^-1.
• The half-life period of a first-order reaction is given by t1⁄2 = 0.693/k.
• In a first-order reaction, the rate constant does not depend upon the concentration of the reactant. The half-life period for a first-order reaction is constant.
• If a reaction is overall first order with respect to one of the reactants, the rate of reaction is proportional to the amount of that reactant. For example, Nuclear Decay.

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