Equations for Nuclear Physics (Edexcel International A Level Physics)

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, 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:

• 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

Exponential 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

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)

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^–λt

• 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