Problem 142 - A Complete Resource Book in Chemistry for JEE Main

Question

Consider the following nuclear reactions \({ }_{92} \mathrm{M}^{238} \longrightarrow \mathrm{y}^{\mathrm{x}}+2{ }_{2} \mathrm{He}^{4}\) \({ }_{\mathrm{y}} \mathrm{N}^{\mathrm{x}} \longrightarrow{ }_{\mathrm{B}} \mathrm{L}^{\mathrm{A}}+2 \beta^{+}\) The number of neutrons in element \(L\) is (a) 146 (b) 144 (c) 142 (d) 140

Step-by-Step Solution

Answer

The number of neutrons in element L is 144.

Step 1: Determine first reaction product

From the reaction \[{}_{92}M^{238} \rightarrow y^{x}+2{}_{2}He^4\]we know that the total mass and atomic numbers must be conserved. The helium nucleus (alpha particle) has 4 mass units and 2 protons, making two helium nuclei contribute 8 mass units and 4 protons.Thus, the remaining nucleus on the right must be:\[y = 92 - 2 \times 2 = 88\] (atomic number)\[x = 238 - 2 \times 4 = 230\] (mass number)Therefore, we have \[{}_{88}^{230}N\].

Step 2: Analyze second reaction

The second reaction is given as \[_{y}N^{x} \rightarrow {}_{B}L^{A}+2 \beta^+ \]This reaction represents beta-plus decay, where two positive beta particles (or positron emissions) each decrease the atomic number by 1. Therefore, the new atomic number is:\[B = y - 2 = 88 - 2 = 86\]The mass number remains unchanged, so:\[A = 230\]Thus, we have \[{}_{86}^{230}L\].

Step 3: Calculate number of neutrons in L

To find the number of neutrons, subtract the atomic number from the mass number:\[\text{Number of neutrons in } L = A - B = 230 - 86 = 144\]

Step 4: Validate the solution with options

The neutron number we computed is 144, and this matches one of the given options.

Key Concepts

Atomic Number

The concept of atomic number is fundamental in understanding nuclear reactions. The atomic number refers to the number of protons found in the nucleus of an atom. This number is crucial because it defines the element itself. For example, any atom with 1 proton is hydrogen, and any atom with 92 protons is uranium. Knowing the atomic number allows you to identify the element through the periodic table easily.
In nuclear reactions, the atomic number is a key factor when balancing equations. For each nuclear reaction, the sum of the atomic numbers on one side of the equation must equal the sum on the other side. This conservation helps us predict the results of nuclear transformations.
In the given exercise, the first reaction involves uranium-238 ( _{92}^{238}U ightarrow {}_y^xN+2{}_{2}^{4}He). By conserving the atomic number, we determine that the product nucleus has an atomic number of 88. The atomic number drops by 4 due to the release of two helium nuclei (each contributing 2 protons). The steps regulating atomic number transformations are crucial in both natural and man-made nuclear processes.

Mass Number

Mass number plays a critical role in understanding the identity and transformation of isotopes in nuclear reactions. The mass number is equivalent to the total number of protons and neutrons within an atom's nucleus. It's this total, not just the protons alone, that gives us a full picture of the atom.
For instance, uranium-238 has a mass number of 238, indicating 238 protons and neutrons combined. In nuclear reactions, such as fission or fusion, the mass number of reactants is meticulously conserved. In our example of uranium decay, the mass number of the new element must reflect the original state, subtracting the particles ejected (two helium nuclei in this case). This leads to a mass number calculation of:

• Initial state: 238 (uranium)
• Particles ejected: 2 helium nuclei → 2×4 = 8
• Resulting nucleus: 238 - 8 = 230

Understanding mass number changes is necessary for recognizing how nuclear reactions affect atomic nuclei beyond just their elemental identity.

Beta-Plus Decay

Beta-plus decay is a fascinating process of nuclear transformation, where a proton within the nucleus transforms into a neutron, emitting a beta-plus particle (positron) and a neutrino. Unlike beta-minus decay, which involves electron emission, beta-plus decay results in the atomic number decreasing by one while the mass number remains unchanged. This is because a proton is replaced by a neutron, which doesn't affect the total nucleon count. Beta-plus decay plays a significant role in poshitrons emission tomography (PET scans) in medical imaging. In nuclear equations, it's crucial to recognize that this decay alters the identity of the element involved, not its mass. Within our original step-by-step solution, we explore this in the decay of element N, yielding a new atom L. The atomic number decreases by 1 for each beta-plus decay, resulting in:

• Initial atomic number (Y): 88
• 2 beta-plus decays: 88 - 2 = 86
• Mass number (A): unchanged, 230

This distinction enables us to correctly identify and describe nuclear reactions in detail, leading to precise predictions about the resulting isotopes and elements.