Question

A \(60-\mathrm{kg}\) woman has an average of \(145 \mathrm{~g}\) of potassium in her body. Of this potassium, \(0.0117 \%\) is radioactive potassium-40, which has a half-life of \(1.2 \times 10^{9}\) yr. Calculate the radioactivity (Bq) emitted from the potassium- 40 .

Short answer

The radioactivity is about 468 Bq.

Step-by-step solution

Calculate the mass of radioactive potassium-40

The mass of potassium-40 is given by the fraction of potassium-40 in the total potassium content: \(0.0117\%\) of \(145\,\text{g}\). Calculate this as follows: \(\text{mass of } \text{K-40} = \frac{0.0117}{100} \times 145\,\text{g} = 0.016965\,\text{g}\).

Convert the mass from grams to moles

Now, convert the mass of potassium-40 to moles using the molar mass of potassium-40 (\(40\,\text{g/mol}\)): \(\text{moles of K-40} = \frac{0.016965\,\text{g}}{40\,\text{g/mol}} = 0.000424125\,\text{mol}\).

Determine the number of atoms of potassium-40

Use Avogadro's number \(6.022 \times 10^{23}\) atoms/mol to find the number of atoms: \(\text{number of atoms} = 0.000424125\,\text{mol} \times 6.022 \times 10^{23}\,\text{atoms/mol} \approx 2.55 \times 10^{20}\,\text{atoms}\).

Calculate the decay constant

The decay constant \(\lambda\) is calculated using the formula \(\lambda = \frac{\ln(2)}{T_{1/2}}\). For potassium-40, \(T_{1/2} = 1.2 \times 10^9\,\text{years}\). \[\lambda = \frac{0.693}{1.2 \times 10^9 \times 3.156 \times 10^7 \text{ seconds/year}}\approx 1.83 \times 10^{-18}\,\text{s}^{-1}\].

Calculate the activity (Bq)

Activity \(A\) can be calculated using the formula \(A = \lambda N\), where \(N\) is the number of atoms. \[A = 1.83 \times 10^{-18}\,\text{s}^{-1} \times 2.55 \times 10^{20}\,\text{atoms} \approx 467.6\text{ Bq}\].

Key concepts

Potassium-40 Decay

Potassium-40 is a naturally occurring radioactive isotope of potassium. Despite its long half-life, it can still have significant natural radioactivity owing to its abundance on Earth. As a beta emitter, it undergoes radioactive decay by losing a beta particle and transforming into calcium-40 or by positron emission turning into argon-40. The decay process is spontaneous, which means that it does not require an external input of energy.
In our bodies, potassium-40 represents a tiny fraction of the total potassium, but given the large number of potassium atoms, it adds up to a significant number of radioactive particles. As in our exercise, only 0.0117% of the potassium in a person is potassium-40, yet it is enough to measure its radioactivity.
The understanding of potassium-40 decay helps in various applications, including geological dating and medical diagnostics. By measuring the remaining amount of potassium-40, scientists can estimate the age of rocks or archaeological samples.

Half-life Calculation

The half-life of a radioactive isotope is a measure of the time it takes for half of the sample to decay. For potassium-40, this time span is remarkably long—about 1.2 billion years. Long half-lives indicate that the decay process is slow.
To calculate phenomena related to half-life, one essential piece of information is the decay constant, denoted by the symbol \( \lambda \). This is computed with the natural logarithm of 2 divided by the half-life: \[\lambda = \frac{\ln(2)}{T_{1/2}}\]
This decay constant helps scientists calculate the rate of radioactive decay and understand how quickly or slowly a substance is losing its radioactivity. In practical terms, using potassium-40's half-life allows us to explore its decay rate and create decay models for material dating.

Radioactivity Measurement

Radioactivity, commonly measured in Becquerels (Bq), quantifies how many atomic nuclei decay per second. It is a crucial parameter in understanding the level of natural or artificial radioactivity in an environment or object like the human body.
For potassium-40, measuring radioactivity involves calculating the number of decays using the formula:\[A = \lambda N\]
where \(A\) is the activity in Becquerels, \(\lambda\) is the decay constant, and \(N\) is the number of radioactive atoms. This measurement gives insights into how active a sample really is, in terms of emission of particles.
In everyday terms, when obtained values are low, such as hundreds of Becquerels, it indicates a relative stability in terms of radiation exposure. It offers a standard that allows us to compare different radioactive sources and understand their potential impact on health and the environment.