# 28.2.2 Activity & The Decay Constant

### 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

Part (b)

Step 1:            Write the equation for activity

Activity, A = λN

Step 2:            Rearrange for decay constant λ and calculate the answer ### 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

• 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 ### Author: Katie

Katie has always been passionate about the sciences, and completed a degree in Astrophysics at Sheffield University. She decided that she wanted to inspire other young people, so moved to Bristol to complete a PGCE in Secondary Science. She particularly loves creating fun and absorbing materials to help students achieve their exam potential.
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