8.1.5

# Half Life

Test yourself

## Half-Life

Although each nuclear decay is random, with a large collection of nuclei, we can statistically predict how many will decay in a certain time.

### Time taken to halve

• Half-life, T1/2, is the time taken:
• For the number of radioactive nuclei in a sample to halve.
• For the activity (the number of decays per second) to halve.

### Activity

• The activity of a sample is directly proportional to the number of nuclei remaining:
• Activity = decay constant, λ x number of nuclei remaining.
• Activity is related to half-life:
• $\lambda =\frac{ln (2)}{T_\frac{1}{2}}$ = 0.69 ÷ T1/2

### Activity 2

• By knowing the activity, we can infer how many nuclei are remaining.
• Knowing the atomic mass of an isotope and the mass of a sample of isotope enables the number of nuclei and so the activity to be found.

### Example

• Find the half-life of a sample of plutonium-239, which has a mass of 1200 g and an activity of 2.8 × 1012 Bq:
• Number of moles of Pu-239 = 1200 ÷ 239 = 5.021.
• Number of nuclei = number of moles × Avogadro’s constant = 3.02 × 1024.
• λ = activity ÷ number of nuclei = 9.26 × 10-13.
• So, half-life T1/2 = ln(2) ÷ λ = 7.5 × 1011 s.

## Half-Life

The half-life of a sample is the time taken for the number of radioactive counts to fall to half the initial value.

### Half-life

• Half-life is the time taken for half of the sample to decay.
• This is the same as saying the time taken for the count rate to halve.
• Knowing the half-life of a substance and the number of nuclei it has means we can find its age.

### Uses of half-life

• We can find the age of a material by considering the count rate of the material.
• By knowing the original count rate, the current count rate, and the half-life of the material - we can use the following equation to calculate the time passed:
• $A={A_0}e^{-{\lambda}t}$

## Graphical Method

Half life can be found in many ways, the easiest is to use a graph.

### Graphical method

• Firstly plot a graph of count rate or number of atoms against time.
• The plot should show exponential decay.
• Look for the initial count rate / number of atoms (the y-intercept).

### Graphical method 2

• Divide the initial count rate or number of atoms by two, this is the half count rate, or half the initial number of atoms.
• Draw a line across from the half count rate to the decay curve.
• Draw a line down to the time axis (x axis) and read off the time. This is the half-life.

### Graphical method 3

• Each half-life will be the same length.
• So it will take two half-lives for the sample to decay to a quarter of the initial count, or a quarter of the initial atoms.