Equations for Nuclear Physics (Edexcel IAL Physics)

Activity & The Decay Constant

Spec point — 142

• 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, per second, that a given nucleus will decay

• 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:

$A=\frac{ΔN}{Δt}=-λN$

• 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

Worked Example Americium-241 is an artificially produced radioactive element that emits α-particles. A sample of americium-241 of mass 5.1 μg is found to have an activity of 5.9 × 10⁵ Bq.

(a) Determine the number of nuclei in the sample of americium-241.

(b) Determine the decay constant of americium-241.

Answer:

Part (a)

Step 1: Write down the known quantities

• Mass = 5.1 μg = 5.1 × 10^-6 g
• Molecular mass of americium = 241
• N_A = Avogadro constant

Step 2: Write down the equation relating number of nuclei, mass and molecular mass

$\mathrm{Number} \mathrm{of} \mathrm{nuclei}=\frac{\mathrm{mass}\times N_{A}}{\mathrm{molecular} \mathrm{mass}}$

Step 3: Calculate the number of nuclei

$number of nuclei=\frac{(5.1\times10^{-6})\times(6.02\times10^{23})}{241}=1.27\times10^{16}$

Part (b)

Step 1: Write the equation for activity

Activity, A = λN

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

$λ=\frac{A}{N}=\frac{5.9\times10^{5}}{1.27\times10^{16}}=4.65\times10^{-11}s^{-1}$

Exponential Decay

Spec point — 142

• 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

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

Radioactive Decay Equation

• 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 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 e^x

• The inverse function of e^x is ln(y), known as the natural logarithmic function

  • This is because, if e^x = 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

Answer:

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

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

Half-Life

Spec point — 142

Half Life

• Half-life is defined as: The time taken for half the number of nuclei in a sample to decay
• This means when a time equal to the half-life has passed, the activity of the sample will also half
• This is because the activity is proportional to the number of undecayed nuclei, A ∝ N

When a time equal to the half-life passes, the activity falls by half, when two half-lives pass, the activity falls by another half (which is a quarter of the initial value)

• To find an expression for half-life, start with the equation for exponential decay:

N = N₀ e–λ^t

• Where:

  • N = number of nuclei remaining in a sample
  • N₀ = the initial number of undecayed nuclei (when t = 0)
  • λ = decay constant (s^-1)
  • t = time interval (s)

• When time t is equal to the half-life t_½, the activity N of the sample will be half of its original value, so N = ½ N₀

• The formula can then be derived as follows:

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

• 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

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

Answer:

Step 1: Convert the half-life into seconds

• t_½ = 28 years = 28 × 365 × 24 × 60 × 60 = 8.83 × 10⁸ s

Step 2: Write the equation for half-life

Step 3: Rearrange for λ and calculate

Exam Hint Although you may not be expected to derive the half-life equation, make sure you're comfortable with how to use it in calculations such as that in the worked example.