Question

The nucleus resulting from \({ }_{92} \mathrm{U}^{238}\) after successive emission of two \(\alpha\) and four \(\beta\) particle is (a) \({ }_{90} \mathrm{Th}^{230}\) (b) \({ }_{92} \mathrm{U}^{230}\) (c) \({ }_{88} \mathrm{Ra}^{230}\) (d) \({ }_{94} \mathrm{Pu}^{230}\)

Short answer

The resulting nucleus is \(^{230}_{92}\mathrm{U}\), option (b).

Step-by-step solution

Understanding Alpha Decay

In an \(\alpha\) decay, the nucleus loses 2 protons and 2 neutrons, resulting in a decrease by 4 in the mass number (A) and a decrease by 2 in the atomic number (Z).

Calculating Alpha Decay Results

After two \(\alpha\) decays, the original nucleus \(^{238}_{92}\mathrm{U}\) will decrease its mass number by 8 (4 per \(\alpha\) decay) and its atomic number by 4 (2 per \(\alpha\) decay). The resulting nucleus after two \(\alpha\) emissions is \(^{230}_{88}\mathrm{Th}\).

Understanding Beta Decay

In a \(\beta\) decay, a neutron is transformed into a proton. This increases the atomic number (Z) by 1 without changing the mass number (A).

Calculating Beta Decay Results

After four \(\beta\) decays, the nucleus \(^{230}_{88}\mathrm{Th}\) will increase its atomic number by 4 (1 per \(\beta\) decay) while the mass number remains unchanged. Thus, the resulting nucleus is \(^{230}_{92}\mathrm{U}\).

Identifying the Result

Given the options, the nucleus that matches the calculated result \(^{230}_{92}\mathrm{U}\) is option (b).

Key concepts

Alpha Decay

Alpha decay is a process by which an unstable nucleus releases an alpha particle, consisting of 2 protons and 2 neutrons. This means that each time an atom undergoes alpha decay, it loses 2 protons and 2 neutrons. As a result, the mass number (total number of protons and neutrons) decreases by 4 and the atomic number (number of protons) decreases by 2.

Imagine an atom as a small planetary system. The nucleus is the core, holding protons and neutrons tightly bound. When alpha decay occurs, it's similar to a tiny planet (the alpha particle) shooting off into space, leaving the rest of the system behind. Due to this emission, the identity of the atom changes because the number of protons determines the type of element.

In the provided exercise, the original uranium nucleus, denoted as \(^{238}_{92} \mathrm{U}\), undergoes alpha decay twice. Therefore, the subtraction results in:

• Mass number: 238 - 8 = 230
• Atomic number: 92 - 4 = 88

The new element after these changes is thorium represented by \(^{230}_{88} \mathrm{Th}\). The alpha decay leads to a simpler, lighter nucleus.

Beta Decay

In beta decay, a neutron in an unstable nucleus transforms into a proton, releasing an electron, called a beta particle, and an antineutrino. This increase in protons leads to a change in the element, although the mass number remains unchanged because the balance of nucleons – protons and neutrons – does not alter.

Think of beta decay as a simple swap. A neutron, one of the stable pieces of the atomic puzzle, shifts roles and becomes a proton. This increases the atomic number by 1, but since neutrons and protons have the same mass, the overall mass number does not change.

In our exercise, thorium \(^{230}_{88}\mathrm{Th}\) undergoes four beta decays. Each decay increases the atomic number by 1:

• Atomic number: 88 + 4 = 92

The mass number remains at 230. Thus, the final nucleus returns to uranium, \(^{230}_{92}\mathrm{U}\). Beta decay provides an insightful glimpse into how elements transmute by changing the number of protons while retaining the nucleon's total count.

Mass Number

The mass number, symbolized as \(A\), denotes the sum of protons and neutrons in an atom's nucleus. It's a crucial identifier of an atom's composition because these particles account for the bulk of the nucleus's mass.

Consider the nucleus as a well-packed cluster of blocks, each block representing either a proton or a neutron, adding to its mass. The mass number helps determine isotopes, atoms that possess the same atomic number but different mass numbers due to varying neutron counts.

In practice, tracking the mass number during nuclear reactions, like the ones in the exercise, aids in conserving mass balance during decay. For example, \(^{238}_{92}\mathrm{U}\) has a mass number of 238. After undergoing two alpha decays reducing the mass by 8, it becomes \(^{230}_{88}\mathrm{Th}\), maintaining the rule of conservation of nucleons. Hence, regardless of the reaction type, monitoring the mass number provides insights into atomic stability and the element's identity.

Atomic Number

The atomic number, represented by \(Z\), is a foundational concept in chemistry and physics, signifying the number of protons in an atom's nucleus. It essentially defines an element and its place on the periodic table.

Think of the atomic number as a name badge for elements. Regardless of what else changes – like the atom's mass or neutron count – the atomic number gives the element its identity. For instance, uranium always has 92 protons, making \(^{238}_{92}\mathrm{U}\) different from other elements, despite variations in mass number.

During nuclear decay processes, the atomic number changes signify transformation from one element to another. In our example, two alpha decays decrease uranium's atomic number from 92 to 88, turning it into thorium. Then, four beta decays increase it back to 92, switching it back to uranium \(^{230}_{92}\mathrm{U}\). These adjustments in atomic number illustrate how understanding this value elucidates the alchemy of nuclear transmutations.