Problem 20 - Master Resource Book in JEE Main Physics

Question

A nucleus \({ }_{Z} X^{A}\) emits an \(\alpha\)-particle. The resultant nucleus emits a \(\beta^{+}\)particle. The respective atomic and mass numbers of the final nucleus will be (a) \(Z-3, A-4\) (b) \(Z-1, A-4\) (c) \(Z-2, A-4\) (d) \(Z, A-2\)

Step-by-Step Solution

Answer

The final nucleus has atomic number \(Z-3\) and mass number \(A-4\), so the answer is (a).

Step 1: Understand Alpha Decay

An alpha particle consists of 2 protons and 2 neutrons, so when a nucleus emits an alpha particle, its mass number decreases by 4 and its atomic number decreases by 2. Thus, after the emission of an alpha particle, the new nucleus can be represented as \({ }_{Z-2} Y^{A-4}\).

Step 2: Understand Beta-plus Decay

Beta-plus decay involves the conversion of a proton into a neutron with the emission of a positron (\(\beta^{+}\) particle). When a \(\beta^{+}\) particle is emitted, the atomic number decreases by 1, but the mass number remains unchanged. Thus, after the \(\beta^{+}\) decay of \({ }_{Z-2} Y^{A-4}\), the nucleus becomes \({ }_{Z-3} Y'^{A-4}\).

Step 3: Evaluate the Final Nucleus

After emitting an \(\alpha\) particle followed by a \(\beta^{+}\) particle, the atomic number of the nucleus is \(Z-3\) and the mass number is \(A-4\).

Key Concepts

Alpha Decay

Alpha decay is one of the most common types of radioactive decay. During alpha decay, a nucleus emits an alpha particle, which is essentially a helium nucleus composed of 2 protons and 2 neutrons. This emission has a significant impact on the nucleus:

• The atomic number of the nucleus decreases by 2, since it loses 2 protons.
• The mass number decreases by 4, as 2 protons and 2 neutrons are removed.

This process results in a transformation of the original element into a new element positioned two places back on the periodic table. For example, if the original nucleus is represented by \( { }_{Z}X^{A} \), after alpha decay, it becomes \( { }_{Z-2}Y^{A-4} \). This new nucleus now belongs to a different element with a lower atomic and mass number.

Beta-Plus Decay

Beta-plus decay, also known as positron emission, is another common radioactive decay process. In this decay type, a proton inside the nucleus converts into a neutron, emitting a positron (positively charged electron) and a neutrino. Unlike alpha decay, beta-plus decay has distinct effects:

• The atomic number decreases by 1, as a neutron replaces a proton.
• The mass number remains unchanged because the total number of nucleons stays the same.

So, if a nucleus is initially \( { }_{Z}X^{A} \), and then undergoes beta-plus decay, it transforms into \( { }_{Z-1}Y'^{A} \). Here, the element changes to another element that is one place to the left on the periodic table due to a decrease in the atomic number, even though the mass number is retained.

Atomic Number

The atomic number, denoted as \( Z \), is a fundamental property of an atom. It represents the number of protons in a nucleus, which defines the chemical element of the atom. Each element on the periodic table has a unique atomic number:

• Determines the element type and its position in the periodic table.
• Changes when a nucleus undergoes processes like alpha or beta-plus decay.

For alpha decay, the atomic number decreases by 2, while in beta-plus decay, it decreases by 1. Therefore, understanding changes in the atomic number helps us predict the resultant element after radioactive decays. For instance, in the given step-by-step analysis, the initial nucleus transitions from \( { }_{Z}X^{A} \) to final nucleus \( { }_{Z-3}Y'^{A-4} \) through both decay processes.

Mass Number

The mass number, represented by \( A \), is the sum of protons and neutrons in the nucleus. It is essential in describing the isotopes of elements, as different isotopes of an element have different mass numbers while sharing the same atomic number. Here’s how mass number changes:

• In alpha decay, the mass number decreases by 4 due to the loss of 2 protons and 2 neutrons.
• In beta-plus decay, the mass number remains constant because the conversion of a proton to a neutron doesn't change the count of nucleons.

Therefore, after an initial alpha decay causing a 4-unit drop, any subsequent beta-plus decay will not affect the mass number further. In explaining our problem, the sequence of nucleon changes results in a final mass number of \( A-4 \) for the nucleus \( { }_{Z-3}Y' \).