Activity & The Decay Constant
- Since radioactive decay is spontaneous and random, it is useful to consider the average number of nuclei which 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:
- 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 – however, for calculations it can be omitted
Part (a)
Step 1: Write down the known quantities
Mass = 5.1 μg = 5.1 × 10-6 g
Molecular mass of americium = 241
Step 2: Write down the equation relating number of nuclei, mass and molecular mass
where NA is the Avogadro constant
The Exponential Nature of Radioactive Decay
- In radioactive decay, the number of nuclei falls very rapidly, without ever reaching zero
- Such a model is known as exponential decay
- The graph of number of undecayed nuclei and time has a very distinctive shape
Equations for Radioactive Decay
- The number of undecayed nuclei N can be represented in exponential form by the equation:
N = N0e–λt
- Where:
- N0 = the initial number of undecayed nuclei (when t = 0)
- 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 be represented in exponential form by the equation:
A = A0e–λt
- 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 = C0e–λt
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)
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 = A0e–λt