IB DP Physics: HL复习笔记12.2.6 The Law of Radioactive Decay

The Law of Radioactive Decay

  • Since radioactive decay is spontaneous and random, it is useful to consider the average number of nuclei which are expected to decay per unit of 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:

Screenshot-2021-12-09-at-9.38.04-am

  • 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
  • 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:

23.2-Exponential-Decay-Graph

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 law of radioactive decay states:

The rate of decay of a nuclide is proportional to the amount of radioactive material remaining

  • 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)

Exam Tip

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)

Make sure you are confident using the exponential and natural logarithmic functions, they are a major component of the mathematics in this topic!

Problems Involving the Radioactive Decay Law

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–λt

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

6.-The-Exponential-Nature-of-Radioactive-Decay-Worked-Example-equation-1

Step 4: Calculate the ratio A/A0

6.-The-Exponential-Nature-of-Radioactive-Decay-Worked-Example-equation-2

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

Worked Example

Americium-241 is an artificially produced radioactive element that emits α-particles.

In a smoke detector, a sample of americium-241 of mass 5.1 µg is found to have an activity of 5.9 × 105 Bq. The supplier’s website says the americium-241 in their smoke detectors initially has an activity level of 6.1 × 105 Bq.

(a)Determine the number of nuclei in the sample of americium-241.
(b)Determine the decay constant of americium-241.
(c)Determine the age of the smoke detector in years.

Part (a)

Step 1: Write down the known quantities

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

Step 2: Write down the equation relating to the number of nuclei, mass and molecular massTherefore, the smoke detector is 22.7 years old

 

 

 

 

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