# 8.2.2 Exponential Decay

### Exponential Decay

• 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: 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)

#### Equations for Radioactive Decay

• 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)

#### 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 = A0 e–λt

### Using Molar Mass & The Avogadro Constant

#### Molar Mass

• The molar mass, or molecular mass, of a substance is the mass of a substance, in grams, in one mole
• Its unit is g mol-1
• The number of moles from this can be calculated using the equation: #### Avogadro’s Constant

• Avogadro’s constant (NA) is defined as:

The number of atoms in one mole of a substance; equal to 6.02 × 1023 mol-1

• For example, 1 mole of sodium (Na) contains 6.02 × 1023 atoms of sodium
• The number of atoms, N, can be determined using the equation: #### 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

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 down the known quantities

• Activity, A = 5.9 × 105 Bq
• Number of nuclei, N = 1.27 × 1016

Step 2: Write the equation for activity

Activity, A = λN

Step 3: Rearrange for decay constant λ and calculate the answer Part (c)

Step 1: Write down the known quantities

• Activity, A = 5.9 × 105 Bq
• Initial activity, A0 = 6.1 × 105 Bq
• Decay constant, λ = 4.65 × 10–11 s–1

Step 2: Write the equation for activity in exponential form

A = A0 e–λt

Step 3: Rearrange for time t Step 4: Calculate the age of the smoke detector and convert to years • Therefore, the smoke detector is 22.7 years old
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