# 5.18 Equations for Nuclear 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

#### 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.

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 Step 3: Calculate the number of nuclei Part (b)

Step 1: Write the equation for activity

Activity, A = λN

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

• 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 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)

#### 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 = 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

• 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, AN 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 t½, 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 t½ can be calculated using the equation: • This equation shows that half-life t½ 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.

Step 1: Convert the half-life into seconds

• t½ = 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 Tip

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. ### Get unlimited access

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