Problem 30 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

\({ }^{64} \mathrm{Cu}\) (half-life \(=12.8 \mathrm{~h}\) ) decays by \(\beta\) - emission \((38 \%), \beta^{+}\)emission \((19 \%)\) and electron capture (43\%). Write the decay products and calculate partial half-lives for each of the decay processes.

Step-by-Step Solution

Answer

Decay products: \( {}^{64}\text{Zn}, {}^{64}\text{Ni} \). Partial half-lives: 33.68 h (\(\beta^-\)), 67.37 h (\(\beta^+\)), 29.77 h (electron capture).

Step 1: Understanding Decay Processes

Copper-64 \( ({}^{64}\text{Cu}) \) undergoes decay through three processes: \( \beta^- \) emission, \( \beta^+ \) emission, and electron capture. To determine the decay products, observe that: \( \beta^- \) emission converts a neutron to a proton, resulting in \( {}^{64}\text{Zn} \); \( \beta^+ \) emission converts a proton to a neutron, leading to \( {}^{64}\text{Ni} \); electron capture causes \( {}^{64}\text{Ni} \) as well.

Step 2: Calculating Probability for Each Decay Mode

The percentages for each decay mode represent their probability: \( P_1(\beta^-) = 38\% \), \( P_2(\beta^+) = 19\% \), \( P_3(\text{electron capture}) = 43\% \). Since the sum of probabilities needs to equate to 100%, these are already normalized values for the probability of each decay occurring.

Step 3: Understanding Partial Half-Lives

The concept of partial half-life involves calculating the time it takes for half the atoms to decay by each specific mode. This is done by relating each decay mode probability to the overall half-life.

Step 4: Calculating the Partial Half-Life for \( \beta^- \) Emission

The partial half-life \( T_{1/2}^{\beta^-} \) is calculated using: \[ T_{1/2}^{\beta^-} = \frac{T_{1/2}}{P_1} \]. Substitute \( T_{1/2} = 12.8 \text{ h} \) and \( P_1 = 0.38 \): \[ T_{1/2}^{\beta^-} = \frac{12.8}{0.38} = 33.68 \text{ h} \].

Step 5: Calculating the Partial Half-Life for \( \beta^+ \) Emission

Similarly, for \( \beta^+ \) emission: \[ T_{1/2}^{\beta^+} = \frac{T_{1/2}}{P_2} = \frac{12.8}{0.19} = 67.37 \text{ h} \].

Step 6: Calculating the Partial Half-Life for Electron Capture

For electron capture: \[ T_{1/2}^{\text{EC}} = \frac{T_{1/2}}{P_3} = \frac{12.8}{0.43} = 29.77 \text{ h} \].

Key Concepts

Half-life Calculation

Nuclear decay is a fascinating process where unstable nuclei lose energy by emitting radiation. One important measure in this process is the half-life. The half-life is the time required for half the atoms of a radioactive substance to decay. In calculations involving different decay processes, we must correctly account for each decay mode individually.

Let's delve deeper into partial half-lives. In cases where multiple decay processes occur, like with Copper-64, each process has its own specific half-life, called a partial half-life. This is the time it takes for half of a given number of the atoms to decay solely through that particular mode.

To find a partial half-life, you use the overall half-life divided by the probability of the decay process occurring. Generally, partial half-lives for each decay mode can be calculated using the formula:

• For a decay process with probability \( P \):

\[ T_{1/2}^{ ext{partial}} = \frac{T_{1/2}}{P} \]
Where \( T_{1/2} \) is the actual half-life of the nucleus and \( P \) is the probability that a particular decay occurs.

Beta Decay

Beta decay is a radioactive decay process where energy is released and a beta particle is emitted from an unstable nucleus.

There are two types of beta decay: \( \beta^- \) and \( \beta^+ \) decay. In \( \beta^- \) decay, a neutron converts into a proton and an electron, along with an antineutrino, is emitted. For Copper-64, this means it transforms into Zinc-64 (\( {}^{64} ext{Zn} \)). On the other hand, \( \beta^+ \) decay involves a proton turning into a neutron and a positron, alongside a neutrino, being released. In the decay of Copper-64, this results in Nickel-64 (\( {}^{64} ext{Ni} \)).

• \( \beta^- \) decay: Neutron \( \rightarrow \) Proton + Electron + Antineutrino
• \( \beta^+ \) decay: Proton \( \rightarrow \) Neutron + Positron + Neutrino

These processes are fundamental in changing the identity of an element by altering the nuclear composition, making beta decay a crucial component of nuclear physics studies.

Electron Capture

Electron capture is a less commonly known nuclear decay process but plays an essential role in the transformation of atomic nuclei.

In electron capture, an inner orbital electron is captured by the nucleus, specifically a proton. This causes the proton to convert into a neutron, and a neutrino is emitted. This process also results in the formation of a new element with one fewer atomic number. For Copper-64, electron capture results in Nickel-64 (\( {}^{64} ext{Ni} \)), similar to \( \beta^+ \) decay.

The distinctive factor here is that while in \( \beta^+ \) decay a positron is emitted, in electron capture, the electron itself is absorbed. This distinction leads to different energy releases and can affect the stability of the resulting nucleus.

• Electron Capture: Proton + Electron \( \rightarrow \) Neutron + Neutrino

Understanding electron capture provides more completeness to nuclear decay mechanisms, ensuring a fuller grasp of how elements can transform within different contexts.