Problem 93 - Chemistry: An Atoms-Focused Approach

Question

In \(1999,\) the U.S. Environmental Protection Agency set a maximum radon level for drinking water at \(4.0 \mathrm{pCi}\) per milliliter. a. How many decay events occur per second in a milliliter of water for this level of radon radioactivity? b. If the above radioactivity were due to the decay of \(^{222} \operatorname{Rn}\left(t_{1 / 2}=3.8 \text { days }\right),\) how many \(^{222} \mathrm{Rn}\) atoms would there be in \(1.0 \mathrm{mL}\) of water?

Step-by-Step Solution

Answer

Question: Calculate the number of decay events occurring per second in a milliliter of water for a radon level of 4.0 pCi/mL and find the number of 222Rn atoms in 1.0 mL of water, given a half-life of 3.8 days. Answer: There are approximately 0.148 decay events per second in a milliliter of water at the given radon level, and there are approximately \(6.985 \times 10^{7}\) atoms of \(^{222}\mathrm{Rn}\) in 1.0 mL of water.

Step 1: a. Decay events per second

We are given the maximum radon level of 4.0 pCi/mL. We need to convert this to events per second. 1 Ci (curie) = \(3.7 \times 10^{10}\) decays per second. Therefore, we have: \(4.0 \,\text{pCi/mL} \times \frac{1 \,\text{Ci}}{10^{12} \,\text{pCi}} \times \frac{3.7 \times 10^{10}}{1 \,\text{Ci}}\) Now, multiplying the conversion factors: \(4.0 \,\text{pCi/mL} \times \frac{3.7 \times 10^{10}}{10^{12}} = \) \((4.0 \times 3.7) \times 10^{-2}\, \text{events/s/mL} = 14.8 \times 10^{-2}\, \text{events/s/mL}\) So there are 0.148 decay events per second in a milliliter of water at this radon level.

Step 2: b. Calculating the number of \(^{222}\mathrm{Rn}\) atoms

The half-life of \(^{222}\mathrm{Rn}\) is given as 3.8 days. We first need to convert this to seconds. \(3.8\, \text{days} \times \frac{24 \,\text{hours}}{1\, \text{day}} \times \frac{60 \,\text{minutes}}{1 \, \text{hour}}\times \frac{60\, \text{s}}{1 \, \text{minute}}= 328,320\, \text{s}\) Now we can use the decay equation, which is: \(R = \frac{0.693}{t_{\frac{1}{2}}} \times N\) where R is the decay rate, \(t_{\frac{1}{2}}\) is the half-life, and N is the number of atoms. We know the decay rate R from part (a) and the half-life from the given data. \(0.148 = \frac{0.693}{328,320} \times N\) Now, solving for N: \(N = \frac{0.148 \times 328,320}{0.693} \) \(N = 6.985 \times 10^{7}\) So there are approximately \(6.985 \times 10^{7}\) atoms of \(^{222}\mathrm{Rn}\) in 1.0 mL of water.

Key Concepts

Radon

Radon is a naturally occurring radioactive gas that results from the decay of uranium in the earth’s crust. It is colorless, odorless, and tasteless, making its detection more challenging. Radon can dissolve in water, and when it accumulates in enclosed areas like basements, it poses a health risk because of its radioactivity.
At high levels, radon exposure is a known cause of lung cancer, which is why it’s monitored in environments like homes and drinking water sources. The U.S. Environmental Protection Agency considers a maximum safe level of radon in drinking water to be 4.0 pCi/mL to minimize health risks.

Decay Events

Decay events refer to the process of radioactive atoms transforming into different elements or isotopes. This happens as unstable nuclei release radiation while seeking a stable state. The rate of decay helps to classify the intensity of radioactivity in a substance.
For radon, decay events can be quantified by counting the number of disintegrations per second. This count is crucial for determining how radioactive a substance is and predicting how it might affect its surroundings. The calculation in the exercise shows that 0.148 decay events per second occur in a milliliter of water at the defined radon level.

Half-Life

Half-life is the time required for half of the radioactive atoms in a sample to decay. For radon-222, the isotope discussed here, the half-life is 3.8 days. Understanding half-life allows scientists to predict how long a radioactive sample will remain active.
Half-life is used to calculate the decay rate and to estimate how much of a radioactive element is still present after a given time. It is a critical concept in nuclear chemistry and helps in managing the safety levels of radioactive materials. In general, a shorter half-life means a faster decay rate, and vice versa.

Curie

A Curie (Ci) is a unit of radioactivity that expresses the decay rate of a radioactive material. Named after Marie and Pierre Curie, one Curie is equivalent to approximately 3.7 x 10^10 decay events per second.
The Curie is a large unit, so often smaller units like the picocurie (pCi) are used in practice. In this exercise, the radon level is measured in picocuries per milliliter of water, where 1 pCi is equal to 10^-12 Curies. This unit of measurement helps determine the level of radioactivity and ensure it remains within safe limits for public health.