Problem 174 - A Complete Resource Book in Chemistry for JEE Main

Question

A radioactive element, \(\mathrm{X}\), decays by the sequence and with half lives, given below: $$ \begin{aligned} &\mathrm{X}\left(\mathrm{t}_{1 / 2}=30 \mathrm{~min}\right) \\ &\mathrm{Y}\left(\mathrm{t}_{1 / 2}=2 \text { days }\right) \quad \stackrel{\lambda_{1}}{\longrightarrow_{2}} \longrightarrow \mathrm{Y}+\alpha \\\ &\cline { 1 } \longrightarrow \mathrm{Z}+2 \beta \end{aligned} $$ Which of the following statement(s) is/are incorrect? (a) Atomic numbers of \(\mathrm{X}\) and \(\mathrm{Z}\) are same (b) Disintegration constant \(\lambda_{2}>\lambda_{1}\) (c) The mass number of \(\mathrm{Y}\) is greater than that of \(\mathrm{X}\). (d) \(\mathrm{Y}\) and \(\mathrm{Z}\) are isotopes.

Step-by-Step Solution

Answer

All statements (a), (b), (c), and (d) are incorrect.

Step 1: Understanding the Problem

This problem deals with radioactive decay and compares the properties of the elements involved in the decay process: X, Y, and Z. The decay chain shows element X decays to element Y by another process that includes emission of an alpha particle, and element Y further decays to Z by emitting two beta particles.

Step 2: Analyzing Statements

Evaluate each statement to determine its correctness:(a) Atomic numbers of X and Z are the same. This is incorrect because the decay chain leads to a change in atomic numbers.(b) Disintegration constant \(\lambda_2 > \lambda_1\). This statement is incorrect because, usually, a shorter half-life (t) implies a greater decay constant (λ), but here \(t_{1/2}\) of X < Y, so \(\lambda_{1} > \lambda_{2}\).(c) The mass number of Y is greater than that of X. This is incorrect: during nuclear decay, alpha emissions will reduce the mass number.(d) Y and Z are isotopes. This is incorrect; isotopes have the same atomic number, but Y and Z are formed by different decay processes.

Step 3: Conclusion

Reviewing the statements, all are determined to be incorrect. Thus, all options (a), (b), (c), and (d) are incorrect statements based on the provided decay chain and standard nuclear reactions.

Key Concepts

Radioactive Half-life

Radioactive half-life is the duration for which half of a radioactive substance decays into another element. It is a pivotal concept in understanding radioactive decay, dictating how quickly or slowly a material breaks down.
The formula used to denote this is \[ t_{1/2} = \frac{0.693}{\lambda} \]
Here, \( t_{1/2} \) represents the half-life, and \( \lambda \) is the decay constant. This equation shows that a longer half-life means a slower rate of decay and vice versa.
In the context of the problem, element X has a half-life of 30 minutes, whereas element Y has a half-life of 2 days, indicating that X will transform more quickly than Y will transform into Z. Understanding this helps predict the time frames over which different elements will remain before entirely decaying.

Alpha Decay

Alpha decay is a type of radioactive decay wherein an unstable atom emits an alpha particle, which consists of 2 protons and 2 neutrons. This emission results in a reduction of both mass number and atomic number of the original element.
For the original element being studied, as X decays into Y, it emits an alpha particle. This process alters the mass number and atomic number of the resulting element Y.
An alpha decay reaction can be represented as follows: \[ \text{X} \rightarrow \text{Y} + \alpha \]
In this context, the atomic number of Y will be 2 less than that of X, and its mass number will also be reduced by 4. Such an understanding allows us to determine changes in the properties of elements involved following the emission.

Beta Decay

Beta decay entails the transformation of a neutron into a proton and the emission of a beta particle, which is a high-speed electron or positron. This process does not alter the mass number but results in a change in the atomic number.
In our exercise, once Y is formed, it undergoes beta decay to become element Z. During this decay, two beta particles are emitted, leading to an increase in the atomic number by 2 without changing the mass number.
The beta decay can be depicted as: \[ \text{Y} \rightarrow \text{Z} + 2\beta \]
Understanding beta decay is crucial since it affects the positioning of elements in the periodic table without altering their mass quantities, giving insights into the identification of isotopes.

Nuclear Reactions

Nuclear reactions are processes where the nucleus of an atom changes due to external factors or decay processes. These reactions are quite potent, often involving significant energy levels.
In the given sequence of radioactive decay, both alpha and beta decay are types of nuclear reactions.
The primary focus here is the transformation of elements with distinct impacts on their atomic and mass numbers. Typically, in such reactions, the conservation of mass and energy principles hold true, which allows us to balance and predict the outcome of nuclear reactions.
Understanding these reactions aids in comprehending nuclear stability, decay chains, and even applications in medicine, energy, and more.