Half-Life of Reactions

Half-Life of a Reaction

The half-life ($t_{1/2}$) of a reaction is the time taken for the concentration of a reactant to fall to half its initial value. It is one of the most useful and most tested quantities in kinetics.

The form of $t_{1/2}$ depends on the order:

First order: from $k = \dfrac{2.303}{t}\log\dfrac{[R]_0}{[R]}$, put $[R] = [R]_0/2$:

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

Strikingly, this is independent of the initial concentration — a defining feature of first-order reactions.

Zero order: $t_{1/2} = \dfrac{[R]_0}{2k}$ — directly proportional to $[R]_0$.

Key contrast: First-order half-life does not depend on how much you start with; zero-order half-life does. This single difference is a favourite exam discriminator.

Half-Life and Radioactive Decay

All radioactive decays are first order, so they have a constant half-life independent of the amount present. This is the basis of radiocarbon dating and nuclear-fallout calculations.

After $n$ half-lives, the fraction remaining is $\left(\dfrac{1}{2}\right)^n$:

Half-lives elapsed	Fraction remaining
1	1/2 = 50%
2	1/4 = 25%
3	1/8 = 12.5%
4	1/16 = 6.25%
$n$	$(1/2)^n$

[NEET Important] $t_{1/2} = 0.693/k$ is one of the most-used formulas in the chapter. Note $0.693 = \ln 2$. The rate constant and half-life of a first-order reaction are inversely related.

Solved Examples

Example 1: Half-life from rate constant

A first-order reaction has $k = 1.386\times10^{-3}$ s$^{-1}$. Find its half-life.

Solution: $t_{1/2} = \dfrac{0.693}{k} = \dfrac{0.693}{1.386\times10^{-3}} = 500$ s.

Example 2: Rate constant from half-life

A first-order reaction has a half-life of 69.3 s. Find $k$.

Solution: $k = \dfrac{0.693}{t_{1/2}} = \dfrac{0.693}{69.3} = 0.01$ s$^{-1}$.

Example 3: Carbon-14 dating

The half-life of $^{14}$C is 5730 years. An archaeological sample has 25% of the original $^{14}$C. How old is it?

Solution: 25% remaining $= (1/2)^2$, so 2 half-lives have elapsed. Age $= 2\times5730 = 11460$ years.

Example 4: Fraction left after time

For a first-order reaction with $t_{1/2} = 10$ min, what fraction remains after 30 min?

Solution: Number of half-lives $= 30/10 = 3$. Fraction $= (1/2)^3 = 1/8 = 0.125$ (12.5%).

Example 5: Zero-order half-life

A zero-order reaction has $[R]_0 = 0.50$ M and $k = 0.05$ mol L$^{-1}$ s$^{-1}$. Find its half-life.

Solution: $t_{1/2} = \dfrac{[R]_0}{2k} = \dfrac{0.50}{2\times0.05} = \dfrac{0.50}{0.10} = 5$ s.

Example 6: Independence of concentration (first order)

For a first-order reaction, if the initial concentration is tripled, how does the half-life change?

Solution: $t_{1/2} = 0.693/k$ does not depend on initial concentration, so the half-life is unchanged.

Example 7: Sr-90 fallout

$^{90}$Sr has a half-life of about 28.1 years. What fraction of a sample remains after 84.3 years?

Solution: Number of half-lives $= 84.3/28.1 = 3$. Fraction remaining $= (1/2)^3 = 1/8 = 0.125$ (12.5%).

Example 8: Time for 75% completion (first order)

For a first-order reaction, how many half-lives are needed for 75% completion?

Solution: 75% complete means 25% remains $= (1/2)^2$, so 2 half-lives.

Example 9: k and t-half together

A first-order reaction is 75% complete in 60 minutes. Find its half-life.

Solution: 75% complete = 25% remaining = 2 half-lives in 60 min, so $t_{1/2} = 60/2 = 30$ min. (Check: $k = 0.693/30 = 0.0231$ min$^{-1}$.)

Example 10: Comparing zero and first order

Two reactions have the same $k$ numerically. One is zero order ($[R]_0 = 0.2$ M), the other first order. Which has the longer half-life, given $k = 0.1$ (in appropriate units)?

Solution: Zero order: $t_{1/2} = [R]_0/2k = 0.2/(2\times0.1) = 1$ s. First order: $t_{1/2} = 0.693/0.1 = 6.93$ s. The first-order reaction has the longer half-life here.