Uranium and radium are found in many rocky soils throughout the world. Both undergo radioactive decay, and one of the products is radon-222, the heaviest noble gas \(\left(t_{1 / 2}=\right.\) 3.82 days). Inhalation of this gas contributes to many lung cancers. According to the Environmental Protection Agency, the level of radioactivity from radon in homes should not exceed \(4.0 \mathrm{pCi} / \mathrm{L}\) of air. (a) What is the safe level of radon in Bq/L of air? (b) A home has a radon measurement of \(41.5 \mathrm{pCi} / \mathrm{L}\). The owner vents the basement in such a way that no more radon enters the living area. What is the activity of the radon remaining in the room air (in Bq/L) after 9.5 days? (c) How many more days does it take to reach the EPA recommended level?

Step by step solution

01

Conversion Factor for Activity Units

Know that 1 picocurie (pCi) is equivalent to 0.037 becquerels (Bq). This is the conversion factor we'll use to go from pCi to Bq.

02

Convert Safe Level from pCi/L to Bq/L

To convert the safe level of radon from 4.0 pCi/L to Bq/L, use the conversion factor: \(4.0 \text{ pCi/L} \times 0.037 \text{ Bq/pCi} = 0.148 \text{ Bq/L}\). So, the safe level is 0.148 Bq/L.

03

Initial Activity of Radon

Given the initial radon measurement is 41.5 pCi/L, convert this to Bq/L: \(41.5 \text{ pCi/L} \times 0.037 \text{ Bq/pCi} = 1.5355 \text{ Bq/L}\).

04

Determine Remaining Activity After 9.5 Days

Use the decay formula: \( A = A_0 \times \frac{1}{2}^{t/t_{1/2}} \). Here, \(A_0 = 1.5355 \text{ Bq/L}\), \(t = 9.5 \text{ days} \) and \(t_{1/2} = 3.82 \text{ days} \). Thus, \( A = 1.5355 \text{ Bq/L} \times \frac{1}{2}^{9.5/3.82} \). Calculate the decay factor: \(\frac{9.5}{3.82} \approx 2.487\). Then \( A = 1.5355 \text{ Bq/L} \times \frac{1}{2}^{2.487} \approx 1.5355 \text{ Bq/L} \times 0.174 \approx 0.2673 \text{ Bq/L} \).

05

Remaining Time to Reach Safe Level

To find the remaining time to decay to 0.148 Bq/L from 0.2673 Bq/L, use the decay formula: \(0.148 = 0.2673 \times \frac{1}{2}^{t/3.82} \). Isolate \(t\): \( \frac{0.148}{0.2673} = \frac{1}{2}^{t/3.82} \). Calculate the ratio: \( \frac{0.148}{0.2673} \approx 0.5538 \). Take the natural logarithm on both sides: \( \text{ln}(0.5538) = \text{ln}(\frac{1}{2}^{t/3.82}) \). \(\text{ln}(0.5538) = (t/3.82) \text{ln}(0.5) \approx -0 -0.5901 \). \( -0.5901 = \frac{-0.693 \times t}{3.82} \). \( t = (\text{ln}(0.5538) \times 3.82) / \text{ln}(0.5) \approx 3.21 \text{ days} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Uranium and Radium Decay

Uranium and radium are naturally occurring elements found in many rocks and soils. Over time, these elements undergo a process known as radioactive decay. This means they spontaneously transform into other elements by emitting radiation. One significant decay product of both uranium and radium is radon-222. This decay process is essential for understanding why and how radioactive elements transform over time. During decay, these elements lose particles and energy. This not only changes their structure but also their chemical properties. It's important to note that radon-222 is part of this decay chain, and being a gas, it can seep into homes, posing health risks.

Radon-222

Radon-222 is a radioactive noble gas that forms from the decay of radium-226, which in turn comes from the decay of uranium-238. Radon-222 has a half-life of 3.82 days. This relatively short half-life means it decays rather quickly compared to other radioactive elements. When inhaled, radon-222 can contribute to lung cancer because it emits alpha particles during decay. These particles can damage lung tissue, increasing the risk of cancer. The Environmental Protection Agency (EPA) recommends that indoor air levels of radon-222 should not exceed 4.0 picocuries per liter (pCi/L) to minimize health risks.

Half-Life Calculation

The concept of half-life is crucial in understanding radioactive decay. The half-life of a radioactive element is the time it takes for half of the substance to decay. For radon-222, the half-life is 3.82 days. You can calculate the remaining activity of a radioactive substance using the formula: \( A = A_0 \times \frac{1}{2}^{t/t_{1/2}} \).Here, \( A \) is the remaining activity, \( A_0 \) is the initial activity, \( t \) is the time elapsed, and \( t_{1/2} \) is the half-life. This formula allows you to find out how much of the radioactive substance remains after a certain period. For instance, if you start with 1.5355 Bq/L of radon-222 and wait 9.5 days, the remaining activity can be calculated to determine how much radon is left.

Activity Unit Conversion

In the context of radioactive materials, activity is often measured in two units: picocuries (pCi) and becquerels (Bq). One picocurie is equivalent to 0.037 becquerels. This conversion factor is essential when comparing different measurement standards. For example, if the safe level recommended by the EPA is 4.0 pCi/L, you can convert this to becquerels by multiplying it by 0.037:\( 4.0 \text{ pCi/L} \times 0.037 \text{ Bq/pCi} = 0.148 \text{ Bq/L} \). Similarly, if a home's radon level is 41.5 pCi/L, converting it provides a clearer understanding in Bq/L, which is the SI unit for activity. Conversion ensures consistency in measurement and helps in accurate calculation of decay over time.