Decay Constant & Half-Life (HL) | HL IB Physics Revision Notes 2025

Decay Constant & Half-Life

Since radioactive decay is spontaneous and random, it is useful to consider the average number of nuclei that are expected to decay per unit time
- This is known as the average decay rate

As a result, each radioactive element can be assigned a decay constant
The decay constant λ is defined as:

The probability that an individual nucleus will decay per unit of time

When a sample is highly radioactive, this means the number of decays per unit time is very high
- This suggests it has a high level of activity

Activity, or the number of decays per unit time can be calculated using:

$A = \frac{∆ N}{∆ t} = - λ N$

Where:
- A = activity of the sample (Bq)
- ΔN = number of decayed nuclei
- Δt = time interval (s)
- λ = decay constant (s^-1)
- N = number of nuclei remaining in a sample

The activity of a sample is measured in becquerels (Bq)
- An activity of 1 Bq is equal to one decay per second, or 1 s^-1

This equation shows:
- The greater the decay constant, the greater the activity of the sample
- The activity depends on the number of undecayed nuclei remaining in the sample
- The minus sign indicates that the number of nuclei remaining decreases with time

Half-life and decay constant can be linked, using an equation called the exponential decay equation

$N = N_{0} e^{- λ t}$

This equation shows how the number of undecayed nuclei, N, changes over time, t, where N₀ is the initial number of nuclei in the sample
When time t is equal to the half-life t_½, the number of undecayed nuclei in the sample, N, will fall to half of its original value $(N = \frac{1}{2} N_{0})$

$\frac{1}{2} N_{0} = N_{0} e^{- λ t_{1 / 2}}$

The formula linking half-life and decay constant can then be derived as follows:

divide both sides by N₀: $\frac{1}{2} = e^{- λ t_{1 / 2}}$

take the natural log of both sides: $\ln (\frac{1}{2}) = - λ t_{1 / 2}$

apply properties of logarithms: $λ t_{1 / 2} = \ln 2$

Therefore, half-life t_½ can be calculated using the equation:

$t_{1 / 2} = \frac{\ln 2}{λ}$

This equation shows that half-life t_½ and the radioactive decay rate constant λ are inversely proportional
- Therefore, the shorter the half-life, the larger the decay constant and the faster the decay

The half-life of a radioactive substance can be determined from decay curves and log graphs
Since half-life is the time taken for the initial number of nuclei, or activity, to reduce by half, it can be found by
- Drawing a line to the curve at the point where the activity has dropped to half of its original value
- Drawing a line from the curve to the time axis, this is the half-life

A linear decay curve. This represents the relationship: $N = N_{0} e^{- λ t}$

Straight-line graphs tend to be more useful than curves for interpreting data
- Due to the exponential nature of radioactive decay, logarithms can be used to achieve a straight-line graph
Take the exponential decay equation for the number of nuclei

$N = N_{0} e^{- λ t}$

Taking the natural logs of both sides

$\ln N = \ln N_{0} - λ t$

$\ln N = - λ t + \ln N_{0}$

In this form, this equation can be compared to the equation of a straight line

$y = m x + c$

Where:
- ln N is plotted on the y-axis
- t is plotted on the x-axis
- gradient = −λ
- y-intercept = ln N₀

Half-lives can be found in a similar way to the decay curve but the intervals will be regular as shown below:

Half Life Decay Curves 2, downloadable AS & A Level Physics revision notes

A logarithmic graph. This represents the relationship: $\ln N = - λ t + \ln N_{0}$

Note: experimentally, the measurement generally taken is the count rate of the source
- Since count rate ∝ activity ∝ number of nuclei, the graphs will all take the same shapes when plotted against time (or number of half-lives) linearly or logarithmically

Worked example

Radium is a radioactive element first discovered by Marie and Pierre Curie.

They used the radiation emitted from radium-226 to define a unit called the Curie (Ci) which they defined as the activity of 1 gram of radium.

It was found that in a 1 g sample of radium, 2.22 × 10¹² atoms decayed in 1 minute.

Another sample containing 3.2 × 10²² radium-226 atoms had an activity of 12 Ci.

(a) Determine the value of 1 Curie

(b) Determine the decay constant for radium-226

Answer:

(a)

Step 1: Write down the known quantities

Number of atoms decayed, ΔN = 2.22 × 10¹²
Time, Δt = 1 minutes = 60 s

Step 2: Write down the activity equation

Step 3: Calculate the value of 1 Ci

(b)

Step 1: Write down the known quantities

Number of atoms, N = 3.2 × 10²²
Activity, A = 12 Ci = 12 × (3.7 × 10¹⁰) = 4.44 × 10¹¹ Bq

Step 2: Write down the activity equation

A = λN

Step 3: Calculate the decay constant of radium

Therefore, the decay constant of radium-226 is 1.4 × 10^–11 s^–1

Worked example

Strontium-90 is a radioactive isotope with a half-life of 28.0 years. A sample of Strontium-90 has an activity of 6.4 × 10⁹ Bq.

(a)

Calculate the decay constant λ, in year^–1, of Strontium-90.

(b)

Determine the fraction of the sample remaining after 50 years.

Answer:

(a)

Step 1: List the known quantities

Half-life, t_½ = 28 years

Step 2: Write the equation for half-life

$t_{\frac{1}{2}} = \frac{\ln 2}{λ}$

Step 3: Rearrange for λ and calculate

$λ = \frac{\ln 2}{t_{\frac{1}{2}}} = \frac{\ln 2}{28}$ = 0.025 year⁻¹

(b)

Step 1: List the known quantities

Decay constant, λ = 0.025 year⁻¹
Time passed, t = 50 years

Step 2: Write the equation for exponential decay

$N = N_{0} e^{- λ t}$

Step 3: Rearrange for $\frac{N}{N_{0}}$ and calculate

$\frac{N}{N_{0}} = e^{- λ t}$

$\frac{N}{N_{0}} = e^{- (0.025) \times 50}$ = 0.287

Therefore, 28.7% of the sample will remain after 50 years

Decay Constant & Half-Life (HL) (HL IB Physics)

Revision Note