Problem 76 - University Physics with Modern Physics

Question

The most common isotope of uranium, \({ }_{92}^{238} \mathrm{U},\) produces radon \({ }_{86}^{222} \mathrm{Rn}\) through the following sequence of decays: $$\begin{array}{c}{\underline{\phantom{xx}}}^{238} \mathrm{U} \rightarrow{ }^{234} \mathrm{Th}+\alpha,{ }^{234} \mathrm{Th} \rightarrow{ }^{234} \mathrm{~Pa}+\beta^{-}+\bar{\nu}_{e}, \\\\{ }_{91}^{234} \mathrm{~Pa} \rightarrow{ }_{92}^{234} \mathrm{U}+\beta+\bar{\nu}_{e},{ }^{234} \mathrm{U} \rightarrow{ }^{230} \mathrm{Th}+\alpha ,\\\\{ }_{91}^{230} \mathrm{Th} \rightarrow{ }_{90}^{226} \mathrm{Ra}+\alpha,{ }_{88}^{226} \mathrm{Ra} \rightarrow{ }_{86}^{222} \mathrm{Rn}+\alpha,\end{array}$$. A sample of \({ }_{92}^{238} \mathrm{U}\) will build up equilibrium concentrations of its daughter nuclei down to \({ }_{88}^{226} \mathrm{Ra} ;\) the concentrations of each are such that each daughter is produced as fast as it decays. The \({ }_{88}^{226} \mathrm{Ra}\) decays to \({ }_{86}^{222} \mathrm{Rn},\) which escapes as a gas. (The \(\alpha\) particles also escape, as helium; this is a source of much of the helium found on Earth.) In high concentrations, the radon is a health hazard in buildings built on soil or foundations containing uranium ores, as it can be inhaled. a) Look up the necessary data, and calculate the rate at which \(1.00 \mathrm{~kg}\) of an equilibrium mixture of \({ }_{92}^{238} \mathrm{U}\) and its first five daughters produces \({ }_{86}^{222} \mathrm{Rn}\) (mass per unit time). b) What activity (in curies per unit time) of radon does this represent?

Step-by-Step Solution

Answer

Answer: The rate at which 1.00 kg of the equilibrium mixture of uranium-238 and its first five daughters produces radon-222 is 2.58 x 10^(-21) g/s. The activity of radon production from the mixture is 1.90 x 10^(-10) curies per unit time.

Step 1: Gathering necessary data

We can find the decay constants for all the daughter nuclei involved in the decay process from a reference, such as a book or a reliable online source. Here, we have listed the decay constants (in per second) for the considered elements: - Uranium-238: λ_238 = 4.916 x 10^(-18) s^(-1) - Thorium-234: λ_234Th = 2.826 x 10^(-12) s^(-1) - Protactinium-234: λ_234Pa = 2.199 x 10^(-6) s^(-1) - Uranium-234: λ_234 = 2.788 x 10^(-18) s^(-1) - Thorium-230: λ_230 = 9.158 x 10^(-23) s^(-1) - Radium-226: λ_226 = 1.373 x 10^(-11) s^(-1)

Step 2: The equilibrium condition

In an equilibrium mixture, each daughter nucleus is produced at the same rate as it decays. Thus, we can find the rate of radon production (n_Rn) as follows: n_Rn = λ_226 * N_Ra = λ_230 * N_Th Where λ_226 and λ_230 are the decay constants of radium-226 and thorium-230, and N_Ra and N_Th are the number of radium-226 and thorium-230 nuclei, respectively.

Step 3: Calculating the number of Uranium-238 nuclei

First, let us find the number of uranium-238 nuclei (N_U) in the mixture: N_U = (1.00 kg) * (6.022 x 10^23 atoms/mol) / (238 g/mol) = 2.529 x 10^24 atoms

Step 4: Using the decay constant to find the number of Thorium-230

Since the number of uranium-238 atoms in equilibrium with thorium-230, we can find the number of thorium-230 nuclei (N_Th) by using the decay constant of uranium-234: N_Th = N_U * (λ_234 / λ_230) = 2.529 x 10^24 * (2.788 x 10^(-18) / 9.158 x 10^(-23)) = 7.686 x 10^20 atoms

Step 5: Using the decay constant to find the number of Radium-226

Similarly, we can find the number of radium-226 nuclei (N_Ra) by using the decay constant of thorium-230: N_Ra = N_Th * (λ_230 / λ_226) = 7.686 x 10^20 * (9.158 x 10^(-23) / 1.373 x 10^(-11)) = 5.107 x 10^11 atoms

Step 6: Calculating the rate of radon production

Now we can calculate the rate of radon production (in atoms per second): n_Rn = λ_226 * N_Ra = (1.373 x 10^(-11) s^(-1)) * (5.107 x 10^11 atoms) = 7.023 x 10^0 atoms/s

Step 7: Converting the rate of radon production to mass per unit time

To find the rate of radon production in mass per unit time, we will use the molar mass of radon-222 (222 g/mol): Rate (in g/s) = (7.023 x 10^0 atoms/s) * (222 g/mol) / (6.022 x 10^23 atoms/mol) = 2.58 x 10^(-21) g/s (a) Therefore, the rate at which 1.00 kg of an equilibrium mixture of uranium-238 and its first five daughters produces radon-222 is 2.58 x 10^(-21) g/s.

Step 8: Calculating the activity in curies per unit time

Now, we need to find the activity (in curies) of radon production. One curie is defined as 3.7 x 10^10 decays per second. Hence: Activity (in curies per unit time) = (7.023 x 10^0 atoms/s) / (3.7 x 10^10 atoms/Ci) = 1.90 x 10^(-10) Ci/s (b) The activity of radon production from 1.00 kg of the equilibrium mixture of uranium-238 and its first five daughters is 1.90 x 10^(-10) curies per unit time.

Key Concepts

Uranium-238

Uranium-238, known for its atomic number 92 and mass number 238, is the most prevalent isotope of uranium found in nature. It is known for its stability compared to other uranium isotopes, but it is still radioactive and undergoes alpha decay to transform into Thorium-234. This process is part of a lengthy decay chain that eventually results in the formation of Radon Gas.
Uranium-238 is significant because of its role in nuclear reactions and the generation of heat in Earth's crust, contributing to geothermal energy. Its long half-life of about 4.5 billion years makes it a key factor in radiometric dating, helping scientists determine the age of Earth and various geological formations.
In nuclear reactors and weaponry, Uranium-238 acts as a fertile material. Though it is not fissile itself, it can absorb neutrons to become Plutonium-239, which is fissile and used as a nuclear fuel.

Radon Gas

Radon Gas, represented by the chemical symbol Rn, is a colorless, tasteless, and odorless gas at room temperature. It is formed as a part of the decay chain of Uranium-238, specifically from the decay of Radium-226.
A notable characteristic of radon is its radioactivity. Once it is produced, it can seep out of soil, rocks, and building materials, and accumulate in homes, especially basements and ground floors. When inhaled, the radioactive particles can damage lung tissue and increase the risk of lung cancer over long-term exposure.
To minimize radon exposure, it is essential to ventilate indoor spaces and use radon detection kits to measure radon levels. If high levels are detected, professional mitigation may be necessary to reduce radon concentrations in buildings.

Decay Chain

The decay chain, also known as a radioactive series, is a sequence of decays that radioactive materials undergo to reach a stable form. Uranium-238 follows such a chain, involving several steps of alpha and beta decays, before eventually becoming Lead-206, a stable isotope.
Key steps in the decay chain of Uranium-238 involve the formation of various daughter isotopes such as Thorium-234, Protactinium-234, Uranium-234, Thorium-230, Radium-226, and finally Radon-222. Each step in the chain has its own decay constant, which influences how quickly each decay occurs.
The decay chain is significant for understanding the presence of naturally occurring radioactive materials and how they contribute to radiation in our environment. Throughout the decay chain, different isotopes contribute to radioactivity until a stable isotope is formed, impacting both geological processes and human health.

Equilibrium Concentration

Equilibrium concentration in the context of radioactive decay refers to a state where the rate of production of a daughter isotope equals its rate of decay. For Uranium-238 and its decay chain, equilibrium is reached when each daughter product is produced as quickly as it decays, maintaining constant concentrations over time.
This concept is essential for understanding how long-term radioactive series remain in balance in geological formations. It explains why certain concentrations of radioactive elements can persist in soil and rocks without increasing over time, as the proportion of each isotope remains stable.
Reaching an equilibrium concentration is particularly important for isotopes like Radium-226 in the uranium-238 decay chain because it leads to the formation of Radon Gas, which can escape and accumulate in the atmosphere. Understanding these equilibria helps in mitigating health risks associated with radon exposure in indoor environments.