Question

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.

Short answer

Statements (b), (c), and (d) are incorrect.

Step-by-step solution

Understanding Alpha Decay

In alpha decay, the element X emits an alpha particle. An alpha particle consists of 2 protons and 2 neutrons, thus reducing the atomic number by 2 and the mass number by 4. If the atomic number of X is Z, then the atomic number of Y is Z-2.

Understanding Beta Decay

In beta decay, a neutron changes into a proton, causing the atomic number to increase by 1 without changing the mass number. β-decay is the process that converts Y into Z. Thus, the atomic number of Z is (Z-2)+2 = Z.

Comparing Atomic Numbers

After understanding the decay process, observe that atomic number of Z remains the same as that of X as both alpha decay and double beta decay compensatively modify the atomic number.

Disintegration Constant Comparison

The disintegration constant ( extlambda) is inversely proportional to the half-life (t_{1/2}). Therefore, lambda_{1}>lambda_{2} since t_{1/2} for X (30 min) is much shorter than for Y (2 days).

Mass Number Analysis

During the alpha decay of X to Y, the mass number of Y equals the mass number of X minus 4. Thus, the mass number of Y is smaller than the mass number of X.

Isotopic Relationship Analysis

Isotopes of an element have the same atomic number but different mass numbers. Since Y and Z have different atomic numbers, they cannot be isotopes.

Key concepts

Alpha Decay

Alpha decay is a process that occurs when a radioactive element emits an alpha particle. An alpha particle is made up of 2 protons and 2 neutrons. Because of this, the element that undergoes alpha decay experiences a change in its atomic structure. Specifically, the atomic number of the element decreases by 2, and its mass number decreases by 4.
For example, in the solution provided in the original exercise, element X undergoes alpha decay to become element Y. If X has an atomic number Z, after alpha decay, Y’s atomic number would be Z-2. This reduction is because two protons have been emitted as part of the alpha particle.

• Alpha particle: 2 protons + 2 neutrons
• Decrease in atomic number: by 2
• Decrease in mass number: by 4

Understanding this concept is crucial for determining how the identity of a radioactive element changes during alpha decay.

Beta Decay

Beta decay is another type of radioactive decay that involves a neutron being transformed into a proton within an atom’s nucleus. When a neutron becomes a proton, an electron, known as a beta particle, is ejected from the atom. This results in an increase in the atomic number of the element by 1, since a neutron's conversion to a proton adds to the total count of protons, but the mass number remains unchanged.
In the scenario described in the original problem, element Y undergoes beta decay to become element Z. This process does not affect the mass number, but it does increase the atomic number, meaning if the atomic number of Y was Z-2, after two beta decays, Z's atomic number matches X's atomic number, Z.

• Neutron becomes proton
• Atomic number increases by 1
• Mass number remains the same

This process is essential for identifying how elements transmute into new ones during radioactive decay, particularly when examining the role of beta particles.

Half-Life

The concept of half-life is central to understanding how quickly a radioactive element undergoes decay. The half-life of a radioactive substance is the time it takes for half of the substance to decay. Each element's half-life is different and highly specific to its properties.
In the initial solution review, we learn that element X has a half-life of 30 minutes, while element Y has a half-life of 2 days. It's important to note that a longer half-life means a slower rate of decay. This makes Y decay slower than X, demonstrating that different elements have unique decay rates based on their half-lives.

• Definition: time for half the substance to decay
• Shorter half-life: faster decay
• Longer half-life: slower decay

Understanding half-life helps in making predictions about how much of an element remains after a given time, and the rate at which different elements transform into others during decay.