A first order reaction takes 40 min for 30% decomposition. Calculate t1/2.

Exercises · 3.19

t_{1/2}

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Concept understanding — First Order Kinetics

First Order Kinetics

Imagine you have a bucket of water with a small hole at the bottom. The water drains out. At the start, the bucket is full, so the pressure at the hole is high — water gushes out fast. As the water level drops, the pressure decreases, and the water trickles out more slowly. The rate at which water leaves is directly proportional to how much water is still in the bucket.

That's the core intuition behind first order kinetics: the rate of a process depends linearly on how much of the substance is left.

The Precise Statement

In chemistry, first order kinetics describes a reaction where the rate of the reaction is directly proportional to the concentration of one reactant.

If we have a reaction: $A \rightarrow \text{products}$ , then:

\text{Rate} = -\frac{d[A]}{dt} = k[A]

Here:

$[A]$ is the concentration of reactant A at any time $t$
$k$ is the rate constant (units: time $^{-1}$ , e.g., s $^{-1}$ )
The negative sign indicates that $[A]$ decreases over time

The key point: double the concentration, double the rate. Halve the concentration, halve the rate.

The Integrated Form — What Actually Happens Over Time

The differential equation above tells us the instantaneous rate. But what we usually want is: how does concentration change with time?

Integrating gives:

\ln[A]_t = \ln[A]_0 - kt

Where $[A]_0$ is the initial concentration and $[A]_t$ is the concentration at time $t$ .

This exponential decay is the hallmark of first order kinetics. The concentration drops rapidly at first, then more slowly, approaching zero asymptotically.

The Half-Life — A Beautiful Constant

For first order kinetics, the half-life ( $t_{1/2}$ ) — the time taken for half the reactant to be consumed — is independent of the starting concentration.

t_{1/2} = \frac{\ln 2}{k} \approx \frac{0.693}{k}

This is a powerful result. Whether you start with 100 g or 1 g, it always takes the same time to go from that amount to half of it. This is unique to first order kinetics — no other order has this property.

Note

For a first order process, after $n$ half-lives, the fraction remaining is $\left(\frac{1}{2}\right)^n$ . After 1 half-life: 50% remains. After 2: 25%. After 3: 12.5%. And so on.

How to Identify First Order Kinetics Experimentally

If you plot $\ln[A]$ versus time and get a straight line with slope $-k$ , the reaction is first order. This is the gold standard test.

Alternatively, if the half-life remains constant as you change the initial concentration, that's a strong indicator.

Real-World Examples

Radioactive decay: Every radioactive isotope decays by first order kinetics. Carbon-14 dating works because $t_{1/2} = 5730$ years, regardless of how much carbon-14 is present. …

Why this formula?

First Order Kinetics: Why the Formula Holds

Let's build this from the core idea — not just memorise the equation.

The Fundamental Assumption

In a first order reaction, the rate of reaction depends linearly on the concentration of only one reactant.

If we have:

A \rightarrow \text{products}

The rate law is:

\text{Rate} = -\frac{d[A]}{dt} = k[A]

Here:

$-\frac{d[A]}{dt}$ = rate of disappearance of A (negative because [A] decreases)
$k$ = rate constant (units: time $^{-1}$ , e.g., s $^{-1}$ )
$[A]$ = concentration of A at any time

Why linear? Because the probability of a single molecule reacting in a given time is constant — it doesn't depend on other molecules. This is the molecular logic behind first order.

Deriving the Integrated Rate Law

We start from the differential form:

-\frac{d[A]}{dt} = k[A]

Step 1: Separate variables

Bring all $[A]$ terms to one side, $dt$ to the other:

\frac{d[A]}{[A]} = -k \, dt

Step 2: Integrate both sides

Integrate from initial time $t=0$ (concentration $[A]_0$ ) to any time $t$ (concentration $[A]_t$ ):

\int_{[A]_0}^{[A]_t} \frac{d[A]}{[A]} = -k \int_0^t dt

The left side integrates to $\ln[A]$ :

\ln[A]_t - \ln[A]_0 = -kt

Step 3: Rearrange

\ln \frac{[A]_t}{[A]_0} = -kt

Or equivalently:

\boxed{\ln[A]_t = \ln[A]_0 - kt}

This is the integrated rate law for first order kinetics.

Why This Form Makes Sense

Exponential decay: Taking antilog:

[A]_t = [A]_0 e^{-kt}

The concentration decays exponentially — a hallmark of first order processes.

Constant half-life: The time for $[A]_t$ to become half of $[A]_0$ is:

\frac{[A]_0}{2} = [A]_0 e^{-k t_{1/2}}

\frac{1}{2} = e^{-k t_{1/2}}

\ln\left(\frac{1}{2}\right) = -k t_{1/2}

\boxed{t_{1/2} = \frac{\ln 2}{k}}

Key insight: $t_{1/2}$ is independent of initial concentration — unique to first order. This is why radioactive decay (a first order process) has a fixed half-life regardless of how much you start with.

Graphical Interpretation (Exam-Ready) …

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For a first-order reaction, the time for a given fraction to decompose is linked to the rate constant via the integrated rate law. Using the 30% decomposition data, we find $k$ , then compute the half-life. The half-life is approximately 78 minutes.

Why First-Order Kinetics?

In a first-order reaction, the rate depends linearly on the concentration of one reactant. The key property is that the time required for a fixed fraction to decompose is constant — it does not depend on the starting amount. That’s why we can use any initial concentration to find the rate constant.

The integrated rate law is:

\ln \frac{[A]_0}{[A]_t} = k t

where $[A]_0$ is the initial concentration and $[A]_t$ is the concentration after time $t$ .

The half-life $t_{1/2}$ is the time for half the reactant to decompose. For a first-order reaction, it is given by:

t_{1/2} = \frac{\ln 2}{k}

So once we find $k$ , the half-life follows directly.

Step-by-step solution

1. Interpret the given data.

30% decomposition means that 30% of the reactant has been consumed. So the remaining concentration is 70% of the initial.

If we take $[A]_0 = 100$ (arbitrary units), then $[A]_t = 70$ after $t = 40$ minutes.

2. Apply the integrated rate law.

\ln \frac{100}{70} = k \times 40

Simplify the fraction:

\frac{100}{70} = \frac{10}{7}

So:

\ln \left( \frac{10}{7} \right) = 40k

3. Compute the natural logarithm.

\ln \left( \frac{10}{7} \right) = \ln 10 - \ln 7 \approx 2.3026 - 1.9459 = 0.3567

Thus:

$0.3567 = 40k \quad \Rightarrow \quad k = \frac{0.3567}{40} = 0.0089175 \ \text{min}^{-1}$ …

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Common Mistakes in First Order Kinetics Problems (30% Decomposition)

Students often slip on this exact type of problem. Here are the most frequent errors and how to avoid each.

✗ Mistake 1: Confusing "30% decomposed" with "70% remaining"

The error:

Students plug $30\%$ as the remaining concentration.

They write:

k = \frac{2.303}{40} \log \frac{100}{30}

Why it's wrong:

If 30% has decomposed, then 70% remains. The fraction remaining is $\frac{70}{100} = 0.7$ .

✓ How to avoid:

Always ask: "What is left?"

Decomposed = gone
Remaining = initial − decomposed

For 30% decomposition:

\text{Remaining} = 100\% - 30\% = 70\%

Correct formula:

k = \frac{2.303}{t} \log \frac{[\text{A}]_0}{[\text{A}]_t} = \frac{2.303}{40} \log \frac{100}{70}

✗ Mistake 2: Using the wrong logarithm base

The error:

Using $\log_{10}$ when the formula demands natural log (or vice versa), or mixing them mid-calculation.

Why it's wrong:

The integrated rate law for first order is:

k = \frac{2.303}{t} \log_{10} \frac{[\text{A}]_0}{[\text{A}]_t}

The factor 2.303 converts natural log to base-10 log. If you use $\ln$ directly, omit 2.303.

✓ How to avoid:

Stick to one consistent form:
- With $\log_{10}$ : use $k = \frac{2.303}{t} \log \frac{[\text{A}]_0}{[\text{A}]_t}$
- With $\ln$ : use $k = \frac{1}{t} \ln \frac{[\text{A}]_0}{[\text{A}]_t}$
Never mix them (e.g., $2.303 \times \ln$ ).

✗ Mistake 3: Forgetting that $t_{1/2}$ is independent of initial concentration

The error:

Students calculate $k$ correctly, then try to find $t_{1/2}$ using the same 40 min data again — or assume $t_{1/2} = 20$ min (half of 40 min).

Why it's wrong:

For a first order reaction:

t_{1/2} = \frac{0.693}{k}

It does not depend on how much time was taken for 30% decomposition.

✓ How to avoid:

First find $k$ from the given data.
Then only use $k$ to find $t_{1/2}$ .
Never guess $t_{1/2}$ from the given time directly.

✗ Mistake 4: Arithmetic errors in the log calculation

The error: …

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Q.A first order reaction takes 40 min for 30% decomposition. Calculate $t_{1/2}$ .

Concept understanding — First Order Kinetics

First Order Kinetics

The Precise Statement

The Integrated Form — What Actually Happens Over Time

The Half-Life — A Beautiful Constant

How to Identify First Order Kinetics Experimentally

Real-World Examples

Why this formula?

First Order Kinetics: Why the Formula Holds

The Fundamental Assumption

Deriving the Integrated Rate Law

Step 1: Separate variables

Step 2: Integrate both sides

Step 3: Rearrange

Why This Form Makes Sense

Graphical Interpretation (Exam-Ready) …

Why First-Order Kinetics?

Step-by-step solution

Method: Integrated Rate Law for First-Order Kinetics

Step 1: Write the first-order integrated rate law

Step 2: Identify given data

Step 3: Calculate the rate constant $k$

Common Mistakes in First Order Kinetics Problems (30% Decomposition)

✗ Mistake 1: Confusing "30% decomposed" with "70% remaining"

✗ Mistake 2: Using the wrong logarithm base

✗ Mistake 3: Forgetting that $t_{1/2}$ is independent of initial concentration

✗ Mistake 4: Arithmetic errors in the log calculation

Q.A first order reaction takes 40 min for 30% decomposition. Calculate t1/2t_{1/2}t1/2​.

Why First-Order Kinetics?

Step-by-step solution

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Q.A first order reaction takes 40 min for 30% decomposition. Calculate $t_{1/2}$ .

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