# Radioactive Decay Equations(OCR A Level Physics)

## Revision Note

Author

Katie M

Expertise

Physics

• In radioactive decay, the number of undecayed nuclei falls very rapidly, without ever reaching zero
• Such a model is known as exponential decay

• The graph of number of undecayed nuclei against time has a very distinctive shape:

Radioactive decay follows an exponential pattern. The graph shows three different isotopes each with a different rate of decay

• The key features of this graph are:
• The steeper the slope, the larger the decay constant λ (and vice versa)
• The decay curves always start on the y-axis at the initial number of undecayed nuclei (N0)

• The number of undecayed nuclei N can be represented in exponential form by the equation:

N = N0 e–λt

• Where:
• N0 = the initial number of undecayed nuclei (when t = 0)
• N = number of undecayed nuclei at a certain time t
• λ = decay constant (s-1)
• t = time interval (s)

• The number of nuclei can be substituted for other quantities.
• For example, the activity A is directly proportional to N, so it can also be represented in exponential form by the equation:

A = A0 e–λt

• Where:
• A = activity at a certain time t (Bq)
• A0 = initial activity (Bq)

• The received count rate C is related to the activity of the sample, hence it can also be represented in exponential form by the equation:

C = C0 e–λt

• Where:
• C = count rate at a certain time t (counts per minute or cpm)
• C0 = initial count rate (counts per minute or cpm)

#### The exponential function e

• The symbol e represents the exponential constant
• It is approximately equal to e = 2.718

• On a calculator, it is shown by the button ex
• The inverse function of ex is ln(y), known as the natural logarithmic function
• This is because, if ex = y, then x = ln(y)

#### Worked example

Strontium-90 decays with the emission of a β-particle to form Yttrium-90.The decay constant of Strontium-90 is 0.025 year -1. Determine the activity A of the sample after 5.0 years, expressing the answer as a fraction of the initial activity A0.

Step 1: Write out the known quantities

• Decay constant, λ = 0.025 year -1
• Time interval, t = 5.0 years
• Both quantities have the same unit, so there is no need for conversion

Step 2: Write the equation for activity in exponential form

A = A0 e–λ

Step 3: Rearrange the equation for the ratio between A and A0

Step 4: Calculate the ratio A/A0

Therefore, the activity of Strontium-90 decreases by a factor of 0.88, or 12%, after 5 years

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