Q.Calculate the half-life of a first order reaction from their rate constants given below:

Concept: First Order Kinetics

For a first-order reaction, the half-life ($t_{1/2}$) is independent of initial concentration and given by:

Step 1 – Apply the formula

Substitute each rate constant $k$ into $t_{1/2} = 0.693/k$.

Step 2 – Compute each case

1. $k = 200\ \text{s}^{-1}$:

   $t_{1/2} = \frac{0.693}{200} = 0.003465\ \text{s}$

2. $k = 2\ \text{min}^{-1}$: $t_{1/2} = \frac{0.693}{2} = 0.3465\ \text{min}$ …

Why half-life is constant for first-order reactions

In a first-order reaction, the rate depends linearly on the concentration of one reactant:

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

The integrated rate law is $[A] = [A]_0 e^{-kt}$. Half-life is the time when $[A] = \frac{1}{2}[A]_0$. Substituting gives $\frac{1}{2}[A]_0 = [A]_0 e^{-k t_{1/2}}$, so $e^{-k t_{1/2}} = \frac{1}{2}$. Taking natural logs: $-k t_{1/2} = \ln\frac{1}{2} = -\ln 2$. Hence:

This is the central result. Notice that $\ln 2 \approx 0.693$. The half-life depends only on $k$, not on the starting amount — that’s the hallmark of first-order kinetics.

Step-by-step calculation

1. For $k = 200\ \text{s}^{-1}$

Plug into the formula:

In scientific notation: $3.47 \times 10^{-3}\ \text{s}$.

2. For $k = 2\ \text{min}^{-1}$

That’s about $0.347\ \text{min}$, or roughly $20.8$ seconds if you need it in seconds. …

Method: Half-Life Formula for First-Order Kinetics

For a first-order reaction, the half-life ($t_{1/2}$) is independent of initial concentration and is given by:

Where:

• $k$ = rate constant (must be in consistent time units)
• $\ln 2 \approx 0.693$

Steps to Solve

1. Identify the rate constant $k$ and its units.
2. Ensure time units are consistent — the half-life will have the same time unit as $k$.
3. Substitute into $t_{1/2} = \frac{0.693}{k}$.
4. Calculate and write the answer with correct units.

(i) $k = 200\ \text{s}^{-1}$

Answer: $\mathbf{3.47 \times 10^{-3}\ s}$ (or $3.47\ \text{ms}$)

(ii) $k = 2\ \text{min}^{-1}$

$t_{1/2} = \frac{0.693}{2} = 0.3465\ \text{min}$ …

Here are the common mistakes students make when calculating half-life from rate constants in first-order kinetics, along with how to avoid each.

Mistake 1: Using the Wrong Formula

The Error

Students often confuse the half-life formula for first-order reactions with those for zero-order or second-order reactions. For a first-order reaction, the correct formula is:

Using $t_{1/2} = \frac{1}{k}$ or $t_{1/2} = \frac{1}{k[A]_0}$ is incorrect.

How to Avoid

• Memorise the formula with reasoning: The half-life for a first-order reaction is independent of initial concentration. Only $k$ matters.
• Write the formula at the top of your solution before plugging in numbers.

Mistake 2: Ignoring Units of the Rate Constant

The Error

The rate constant $k$ is given in different units: $\text{s}^{-1}$, $\text{min}^{-1}$, $\text{years}^{-1}$. Students often forget to match the unit of $t_{1/2}$ with the unit of $k$.

Example of the mistake:

For (ii) $k = 2\ \text{min}^{-1}$, a student writes $t_{1/2} = \frac{0.693}{2} = 0.3465$ and leaves it unitless, or writes seconds instead of minutes.

How to Avoid

• Always write the unit of $t_{1/2}$ explicitly.
• If $k$ is in $\text{s}^{-1}$, $t_{1/2}$ is in seconds.
• If $k$ is in $\text{min}^{-1}$, $t_{1/2}$ is in minutes.
• If $k$ is in $\text{years}^{-1}$, $t_{1/2}$ is in years.

Correct answers:

• (i) $t_{1/2} = \frac{0.693}{200} = 3.465 \times 10^{-3}\ \text{s}$
• (ii) $t_{1/2} = \frac{0.693}{2} = 0.3465\ \text{min}$
• (iii) $t_{1/2} = \frac{0.693}{4} = 0.17325\ \text{years}$

Mistake 3: Rounding Off Too Early

The Error

Using $0.693$ is standard, but some students round it to $0.7$ or use $0.69$ inconsistently, leading to slightly off answers. In competitive exams, precision matters.

How to Avoid

• Use $0.693$ consistently (or $\ln 2$ if allowed).
• Do all calculations in one step on paper, then round only the final answer to 3–4 significant figures.

Mistake 4: Forgetting That Half-Life Is Independent of Initial Concentration

The Error

Some students try to find initial concentration $[A]_0$ from the given data, or assume it is needed. This wastes time and can lead to wrong formulas.

How to Avoid

• Remember the key property: For a first-order reaction, $t_{1/2}$ depends only on $k$. …