How to Compute Half-Life?

How to Compute Half-Life?

You can compute the half-life of a radioactive substance or other exponentially decaying quantity by determining how long it takes for the amount to decrease to 50% of its initial value.

What Is Half-Life?

Half-life is the time required for a quantity to reduce to half its original amount.

For radioactive materials, it describes the rate at which unstable nuclei decay. The same mathematical concept is also used in areas such as pharmacology, chemistry, and environmental science.

For example, if you start with 100 grams of a substance:

After 1 half-life → 50 g remains.
After 2 half-lives → 25 g remains.
After 3 half-lives → 12.5 g remains.
After 4 half-lives → 6.25 g remains.

Half-Life Formula

The basic relationship is:

[
N=N_0\left(\frac{1}{2}\right)^n
]

Where:

N = amount remaining.
N₀ = original amount.
n = number of half-lives elapsed.

If you know the elapsed time (t) and half-life (T_{1/2}):

[
n=\frac{t}{T_{1/2}}
]

Therefore:

[
N=N_0\left(\frac{1}{2}\right)^{t/T_{1/2}}
]

Example: Finding the Amount Remaining

Suppose a radioactive substance has an initial amount of 80 grams and a half-life of 5 years.

How much remains after 15 years?

Step 1: Calculate the Number of Half-Lives

[
n=\frac{15}{5}=3
]

Step 2: Calculate the Remaining Amount

[
N=80\left(\frac12\right)^3
]

[
N=80\times\frac18
]

[
N=10\text{ grams}
]

So, 10 grams remain after 15 years.

How to Calculate Half-Life From Two Measurements

If you know the initial amount (N_0), the remaining amount (N), and the elapsed time (t), you can calculate the half-life using:

[
T_{1/2}=\frac{t\ln(2)}{\ln(N_0/N)}
]

Example

Suppose:

Initial amount = 100 mg
Remaining amount = 25 mg
Time elapsed = 12 hours

Since 25 mg is one-quarter of 100 mg, two half-lives have passed.

Therefore:

[
T_{1/2}=\frac{12}{2}=6\text{ hours}
]

The half-life is 6 hours.

Half-Life Calculation Table
Half-Lives Passed Percentage Remaining Example From 100 Units
0 100% 100
1 50% 50
2 25% 25
3 12.5% 12.5
4 6.25% 6.25
5 3.125% 3.125
Finding the Number of Half-Lives

If the remaining amount is a simple fraction of the original amount, you can identify the number of half-lives directly.

For example:

[
\frac{N}{N_0}=\frac14
]

Since:

[
\frac14=\left(\frac12\right)^2
]

two half-lives have passed.

For more complicated values, use logarithms:

[
n=\frac{\ln(N/N_0)}{\ln(1/2)}
]

Calculating Half-Life From the Decay Constant

If the decay constant (λ) is known, the half-life is:

[
T_{1/2}=\frac{\ln2}{λ}
]

Because:

[
\ln2\approx0.693
]

the calculation can also be written as:

[
T_{1/2}\approx\frac{0.693}{λ}
]

Example

If:

[
λ=0.1\text{ per hour}
]

then:

[
T_{1/2}=\frac{0.693}{0.1}=6.93\text{ hours}
]

So the half-life is approximately 6.93 hours.

Half-Life vs Decay Constant
Quantity Symbol Meaning
Half-life (T_{1/2}) Time required for quantity to fall by half
Decay constant (λ) Rate constant describing exponential decay
Initial amount (N_0) Amount at the starting time
Remaining amount (N) Amount remaining after time (t)

A larger decay constant corresponds to a shorter half-life.

Simple Half-Life Calculation Process
Find Initial Amount (N₀)
↓
Find Remaining Amount (N)
↓
Determine Elapsed Time (t)
↓
Calculate Number of Half-Lives
↓
Apply the Half-Life Formula
↓
Find T₁/₂ or Remaining Amount
Important Point About Half-Life

Half-life does not mean that the entire substance disappears after two half-lives.

For example, after two half-lives, 25% remains, not zero.

The quantity continues decreasing exponentially:

100% → 50% → 25% → 12.5% → 6.25% → ...

In radioactive decay, this describes the expected behavior of a large population of unstable nuclei; individual nuclei decay probabilistically.

Final Answer

To compute half-life, use the information available. If the number of half-lives is known, divide the elapsed time by that number. If the initial and remaining amounts are known, use (T_{1/2}=t\ln(2)/\ln(N_0/N)). If the decay constant is given, use (T_{1/2}=\ln(2)/λ).