A radioactive element, X, decays by the sequence and with half lives, given below: X(t12=30

step 1 Hello everyone here it is given in the question that the radioactive element x decays by the seq

A radioactive element, X, decays by the sequence and with...

A radioactive element, $\mathrm{X}$, decays by the sequence and with half lives, given below: $$ \begin{array}{ll} \mathrm{X}\left(\mathrm{t}_{12}=30 \mathrm{~min}\right) & \stackrel{\lambda_{1}}{\longrightarrow} \mathrm{Y}+\alpha \\ \mathrm{Y}\left(\mathrm{t}_{12}=2 \text { days }\right) & \stackrel{\lambda_{2}}{\longrightarrow} \mathrm{Z}+2 \beta \end{array} $$ Which of the following statement(s) is/are incorrect? (a) Atomic numbers of $\mathrm{X}$ and $Z$ 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 $Z$ are isotopes.

A radioactive element, $\mathrm{X}$, decays by the sequence and with half lives, given below:
$$
\begin{array}{ll}
\mathrm{X}\left(\mathrm{t}_{12}=30 \mathrm{~min}\right) & \stackrel{\lambda_{1}}{\longrightarrow} \mathrm{Y}+\alpha \\
\mathrm{Y}\left(\mathrm{t}_{12}=2 \text { days }\right) & \stackrel{\lambda_{2}}{\longrightarrow} \mathrm{Z}+2 \beta
\end{array}
$$
Which of the following statement(s) is/are incorrect?
(a) Atomic numbers of $\mathrm{X}$ and $Z$ 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 $Z$ are isotopes.

Key Concepts

Conservation of Atomic and Mass Numbers in Nuclear Reactions

In nuclear decay processes, conservation laws apply to the overall reaction. While alpha decay and beta decay alter the atomic and mass numbers in predictable ways, the processes can be arranged such that the net change in the atomic number of the original nuclide is zero. Understanding these conservation principles is key to tracking nuclear transformations in decay chains.

Decay Constant

The decay constant is a parameter that quantifies the probability per unit time that a nucleus will decay. It is mathematically related to the half-life by the formula ? = ln2/half-life; a larger decay constant corresponds to a faster decay process.

Isotopes

Isotopes are nuclides that have the same atomic number (i.e., the same number of protons) but differ in their mass numbers due to variations in the number of neutrons. This concept is essential in nuclear chemistry to understand the differences in nuclear composition while retaining the elemental identity.

Beta Decay

Beta decay involves the emission of beta particles, which are high-energy, high-speed electrons or positrons. This process effectively converts a neutron into a proton (or vice versa), thereby changing the nucleus's atomic number by ±1 while leaving the mass number essentially unchanged.

Radioactive Decay Series

This concept involves a sequence of radioactive decays where an unstable parent nuclide decays into a daughter nuclide, often through multiple steps. Each step in the series has its own decay process and associated half?life, and the overall decay behavior is governed by the chain of these sequential transformations.

Half-life

Half-life is the time required for half of a given sample of a radioactive substance to decay. It is a fundamental measure of the rate of radioactive decay and is inversely related to the decay constant, providing a direct way to compare the decay speeds of different processes.

Alpha Decay

Alpha decay is a type of radioactive decay where a nucleus emits an alpha particle, which consists of two protons and two neutrons. This emission results in a decrease in the atomic number by 2 and the mass number by 4, fundamentally altering the identity of the element.

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