Problem 13 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

\({ }_{92}^{238} \mathrm{U}\) is known to undergo radioactive decay to form \({ }_{82}^{206} \mathrm{~Pb}\) by emitting alpha and beta particles. A rock initially contained \(68 \times 10^{-6}\) g of \({ }_{92}^{238} \mathrm{U}\). If the number of alpha particles that it would emit during its radioactive decay of \({ }_{92}^{238} \mathrm{U}\) to \({ }_{82}^{206} \mathrm{~Pb}\) in three half- lives is \(Z \times 10^{18}\), then what is the value of \(Z\) ?

Step-by-Step Solution

Answer

The value of \(Z\) is 0.086.

Step 1: Determine Number of Decay Steps

Each alpha decay reduces the atomic number by 2, while each beta decay increases it by 1. The change from \(_{92}^{238} \mathrm{U}\) to \(_{82}^{206} \mathrm{~Pb}\) involves a change in the atomic number from 92 to 82, a difference of 10. Given the decay process, this can be explained by 8 alpha decays (which decrease the atomic number by 16), along with 6 beta decays adding 6 back to reach a total reduction of 10.

Step 2: Calculate Total Atoms Before Decay

The number of \(\mathrm{U}^{238}\) atoms initially present in the rock can be computed by dividing the total mass by the molar mass and then multiplying by Avogadro's number. Using the molar mass of uranium-238 as 238 g/mol, the calculation is: \[\text{Initial atoms} = \frac{68 \times 10^{-6} \, \text{g}}{238 \, \text{g/mol}} \times 6.022 \times 10^{23} \, \text{atoms/mol} \approx 1.72 \times 10^{17} \text{ atoms}.\]

Step 3: Determine Remaining Quantity After Decay Period

In a duration of three half-lives, the amount of uranium-238 will decay to one eighth of the original. Therefore, the remaining uranium atoms are: \[\text{Remaining atoms} = \frac{1.72 \times 10^{17}}{8} = 2.15 \times 10^{16} \text{ atoms}.\]

Step 4: Calculate Total Decayed Atoms

The total number of atoms that have decayed is the initial number minus those remaining, calculated as: \[\text{Decayed atoms} = 1.72 \times 10^{17} - 2.15 \times 10^{16} = 1.505 \times 10^{17} \text{ atoms}.\]

Step 5: Find Total Alpha Particles Emitted

Since each decay step involves 8 alpha decay events (per earlier determination), the total number of alpha particles is: \[\text{Alpha particles} = 1.505 \times 10^{17} \times \frac{8}{14} \approx 8.60 \times 10^{16}.\]

Step 6: Calculate Value of Z

The total alpha emissions are given as \(Z \times 10^{18}\). So, equating \(8.60 \times 10^{16}\) to \(Z \times 10^{18}\), we solve for \(Z\): \[Z = \frac{8.60 \times 10^{16}}{10^{18}} = 0.086.\]

Key Concepts

Alpha Decay

Alpha decay is a type of radioactive decay where an unstable atom emits an alpha particle to become more stable. An alpha particle consists of 2 protons and 2 neutrons. This means that when a nucleus undergoes alpha decay, it loses 2 protons and 2 neutrons, reducing its atomic mass by 4 units and its atomic number by 2.
This is why, during the alpha decay of uranium-238 ({ }_{92}^{238} ext{U}) to lead-206 ({ }_{82}^{206} ext{Pb}), we see a decrease in the atomic number. Alpha decay results in a state change of the atom, leading to significant alterations in its structure and properties.

• An example of this is the transmutation of uranium-238 into thorium-234:

\[{ }_{92}^{238} ext{U} \rightarrow { }_{90}^{234} ext{Th} + { }_{2}^{4} ext{He}\]Through emitting an alpha particle ({ }_{2}^{4} ext{He}), uranium-238 transforms, continuing its radioactive decay series.

Beta Decay

Beta decay is another common form of radioactive decay, which includes the emission of a beta particle. A beta particle is essentially an electron (\(\beta^-\) decay) or a positron (\(\beta^+\) decay), released from the nucleus alongside an antineutrino or neutrino, respectively.
Unlike alpha decay, beta decay changes the atomic number by increasing or decreasing it by one, which in turn changes the element without altering the atomic mass.

• In the context of Uranium-238's decay, beta decay compensates for the atomic number reduction caused by multiple alpha decays, essentially retuning the atomic number:

For instance, in beta-minus decay:\[{ }_{n}^{1} ightarrow { }_{p}^{1} + e^- + \bar{u}_e\]This process transforms a neutron into a proton, increasing the atomic number of the atom by one while maintaining mass number.

Uranium-238 Decay

Uranium-238, abbreviated as { }_{92}^{238} ext{U} , is a well-known radioactive isotope and a prominent player in nuclear decay processes. Its journey to become { }_{82}^{206} ext{Pb} consists of a complex series of transformations incorporating both alpha and beta decays.
The half-life of uranium-238 is about 4.5 billion years, a time span that highlights the slow rate of its transformation. Over three half-lives, evidence of its decay can be drawn significantly, reducing the number of uranium atoms to an eighth of the initial amount.

• In this decay chain, uranium-238 undergoes a series of alpha and beta decays:
• 8 alpha decays – decrease atomic number by 16
• 6 beta decays – increase atomic number by 6

This intricate balance results in a cumulative atomic number reduction of 10, changing uranium-238 into lead-206 through various intermediary elements in a lengthy process.

Isotopic Transformation

Isotopic transformation refers to the change of one isotope into another, a process commonly seen in nuclear decay series. When an atom undergoes decay, it changes from one element to another by altering its proton and neutron count.
The uranium-238 to lead-206 transformation is a classic example of isotopic transformation. Through the decay chain, uranium changes its identity through a series of steps involving isotopes of thorium, radium, radon, and several others along its journey.

• This transformation is governed by:
• Alpha particles reducing proton and neutron count significantly.
• Beta particles altering the proton count by converting neutrons to protons or vice versa.

The journey from uranium-238 to lead-206 illustrates isotopic transformation in a sequence where the unstable isotopes shed particles to gradually reach a more stable state.