nth Order Kinetics

n^th Order Reactions

The nth-order reaction is one in which the rate of the reaction depends upon the concentration of one or more reactants raised to some power, which is called the order of the reaction. The order of a reaction may be an integer or even a fraction and is generally considered representative of how the rate of reaction depends on the concentration of reactants. Mathematically, this rate law may be defined for an nth-order reaction by the expression given below:

Rate=k[A]ⁿ

Here,

• k is the rate constant,
• [A] is the concentration of reactant A,
• n is the order of the reaction.

n^th Order Kinetics

The rates of the reaction are proportional to the n^th power of the reactant

$
\begin{aligned}
& \frac{\mathrm{d}[\mathrm{A}]}{\mathrm{dt}}=-\mathrm{k}[\mathrm{A}]^{\mathrm{n}} \\
\Rightarrow & \frac{\mathrm{d}[\mathrm{A}]}{[\mathrm{A}]^{\mathrm{n}}}=-\mathrm{kdt} \\
\Rightarrow & \int_{\mathrm{A}_0}^{[\mathrm{A}]_t} \frac{\mathrm{d}[\mathrm{A}]}{[\mathrm{A}]^{\mathrm{n}}}=-\mathrm{k} \int_0^{\mathrm{t}} \mathrm{dt} \\
\Rightarrow & {\left[\frac{[\mathrm{A}]^{1-\mathrm{n}}}{1-\mathrm{n}}\right]_{[\mathrm{A}]_0}^{[\mathrm{A}]_t}=-\mathrm{k}[\mathrm{t}]_0^{\mathrm{t}} } \\
\Rightarrow & \frac{1}{(\mathrm{n}-1)}\left[\frac{1}{[\mathrm{~A}]_t^{(\mathrm{n}-1)}}-\frac{1}{[\mathrm{~A}]_0^{(\mathrm{n}-1)}}\right]=\mathrm{k}(\mathrm{t})
\end{aligned}
$

Half life for any $\mathbf{n}^{\text {th }}$ order reaction
$
\mathrm{t}_{\frac{1}{2}}=\frac{1}{(\mathrm{k})(\mathrm{n}-1)\left([\mathrm{A}]_0^{\mathrm{n}-1}\right)}\left[2^{\mathrm{n}-1}-1\right]
$

Thus for any general nth-order reaction, it is evident that,

$\mathrm{t}_{\frac{1}{2}} \propto[\mathrm{A}]_0^{1-\mathrm{n}}$

It is to be noted that the above formula is applicable for any general n^th order reaction except n=1.

so,

• Zero order: $t_{1 / 2} \propto[A]_0$
• For first order: $t_{1 / 2}$ is independent of concentration
• For second order: $t_{1 / 2} \propto \frac{1}{[A]_0}$

Types of n^th Order Reactions

Nth-order reactions can be divided into a number of different types based on the value of n. For example, a zero-order reaction (n=0) is one with a constant rate, independent of the concentration of reactants. A first-order reaction is one whose rate is directly proportional to the concentration of one reactant. Higher-order reactions have more complex dependences of the rate on the concentration of reactants.

Zero-Order Reactions

In zero-order reactions, the rate of reaction remains constant and does not depend upon the concentration of the reactants. It can be represented by the equation given below:

Rate $=k[A]^0=k$

Characteristics:

• Constant reaction rate
• Concentration decreases linearly with time
• Half-life depends on initial concentration

Example:

• Photochemical reactions
• Reactions on catalyst surfaces

Zero-order reactions are often seen in processes in which a catalyst becomes saturated by the reactant, for example, the decomposition of hydrogen peroxide on a platinum surface.

First-Order Reactions

First-order reactions are chemical reactions whose rate is dependent upon the concentration of one reactant. Otherwise stated, the rate law for a first-order reaction is given by the equation:

Rate=k[A]

Characteristics:

• Exponential decrease in concentration
• Half-life is constant
• Most radioactive decays follow first-order kinetics

Example:

• Radioactive decay in which isotopes decay at a rate dependent on the number of radioactive atoms present.
• Decomposition of $\mathrm{N}_2 \mathrm{O}_5$

Second-Order Reactions

Second-order reactions may either involve the square of one reactant concentration or the product of two reactant concentrations. This is the rate law for a second-order reaction:

$\text { Rate }=k[A]^2$

or

$\text { Rate }=k[A][B]$

Characteristics:

• Half-life inversely proportional to initial concentration
• Integrated form contains reciprocal concentration terms

Example:

• Saponification reactions
• Dimerization reactions

One example of a second-order reaction is the reaction of nitric oxide and oxygen to yield nitrogen dioxide.

Mixed-Order Reactions

Mixed-order reactions do not follow simple integer orders but can have fractional orders. They are more complex to represent, as a mix of different rate laws is needed.

For example, a reaction can behave as first order at low concentration and zero order at high concentration. In such cases, the overall kinetics become complex and are called mixed-order kinetics.

A general representation is:

Rate $=k[A]^m[B]^n$

where:

• $k=$ rate constant
• $[A]$ and $[B]=$ concentrations of reactants
• $m$ and $n=$ different reaction orders

If the total order $(m+n)$ is not a simple whole number or changes during the reaction, it is often categorized under mixed-order reactions.

Pseudo-First Order Reaction

Pseudo first order reaction is actually higher order, but one reactant is in large excess so its concentration remains nearly constant.

Rate $=k[A][B]$

If $[B]$ is constant:

$\text { Rate }=k^{\prime}[A]$

Some Solved Examples

Question:1
Among the following, which one is the unit of rate constant for an nth order reaction?

1) $({L^{(n-1)}mol^{(1-n)}t^{-1}})$
2) $({L^{(n-1)}mol^{-1}t^{-1}})$
3) $({L^{(n-1)}mol^{(1-n)}t^{-2}})$
4) None of the above

Solution:
The correct answer is option (1), $({L^{(n-1)}mol^{(1-n)}t^{-1}})$. For an nth-order reaction, the rate constant ( k ) has units that depend on the order of the reaction. It is derived from the differential rate law and integrated rate laws specific to nth-order kinetics.

Question:2
Which of the following statements is true about the half-life $(( t_{1/2} ))$ of an nth-order reaction?

1)$ ( t_{1/2} \propto [A]_0^{1-n})$
2) $( t_{1/2} \propto [A]_0^{n-1})$
3)$( t_{1/2} \propto [A]_0^{-1} $
4)$ ( t_{1/2} \propto [A]_0^{n} )$

Solution:
The correct answer is an option (1), $( t_{1/2} \propto [A]_0^{1-n})$. The half-life of an nth-order reaction is inversely proportional to the initial concentration raised to the power of ( 1-n ), as derived from the integrated rate law for nth-order kinetics.

Question:3
A reaction is second order with respect to the concentration of carbon monoxide. If the concentration of carbon monoxide is doubled, what happens to the rate of reaction?

1) Remain unchanged
2) Tripled
3) Increased by a factor of 4
4) Doubled

Solution:
The correct answer is option (3), increased by a factor of 4. For a second-order reaction, the rate is proportional to ( [CO]² ). When the concentration of carbon monoxide (( [CO] )) is doubled, the rate of reaction increases by a factor of ( 2² = 4 ).

These examples illustrate the application of rate constants, half-life in nth-order reactions, and the effect of concentration changes on reaction rates in accordance with the order of reaction.

Question: 4

For a reaction:
$A \rightarrow$ Products
The half-life is inversely proportional to the initial concentration of reactant. The order of the reaction is:
(A) Zero order
(B) First order
(C) Second order
(D) Third order

Solution:

For an $n^{\text {th }}$-order reaction:

$t_{1 / 2} \propto[A]_0^{1-n}$
Given:

$t_{1 / 2} \propto \frac{1}{[A]_0}$
So,

$\begin{gathered}
1-n=-1 \\
n=2
\end{gathered}$

Hence, the correct answer is option (C)

Question: 5

A reaction follows second-order kinetics. If the initial concentration of reactant is doubled, the half-life becomes:

(A) Doubled
(B) Halved
(C) Four times
(D) Unchanged

Solution:

For second-order reactions:

$t_{1 / 2}=\frac{1}{k[A]_0}$
Thus,

$t_{1 / 2} \propto \frac{1}{[A]_0}$
If initial concentration is doubled:

$t_{1 / 2} \rightarrow \frac{t_{1 / 2}}{2}$

Hence, the correct answer is option (B)

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