Question

\({ }_{92} \mathrm{U}^{238}\) emits \(8 \alpha\) particles and \(6 \beta\) particle. The neutron/ 92 proton ratio in the product nucleus is (a) \(60 / 41\) (b) \(62 / 41\) (c) \(61 / 62\) (d) \(61 / 40\)

Short answer

The neutron/proton ratio is \(62/41\), so the answer is (b).

Step-by-step solution

Understanding Alpha Decay

In an alpha decay, an atom emits an alpha particle \((\alpha)\), which consists of 2 protons and 2 neutrons. This reduces the atomic number by 2 and the mass number by 4. The given problem states that 8 alpha particles are emitted. Thus, the reduction in atomic number will be \(2 \times 8 = 16\) and in mass number \(4 \times 8 = 32\).

Understanding Beta Decay

In a beta decay, a neutron converts into a proton, and an electron (beta particle) is emitted. This increases the atomic number by 1, with no change in mass number. The problem specifies the emission of 6 beta particles, thus resulting in an increase in atomic number by \(1 \times 6 = 6\).

Calculating Changes in Nucleus

We start with \({ }_{92} \text{U}^{238}\). Since 8 alpha particles reduce the atomic number by 16 and the mass number by 32, initially the new nucleus becomes \({ }_{76} \cdot^{206}\). Adding the 6 beta decays, the atomic number increases to \({ }_{82} \cdot^{206}\). The final product nucleus after all decays is \({ }_{82} \cdot^{206}\).

Calculating Neutron/Proton Ratio

To find the neutron/proton ratio, identify the number of neutrons in the final nucleus: \(\text{Neutrons} = \text{Mass number} - \text{Protons} = 206 - 82 = 124\). The neutrons to protons ratio is calculated as \(\frac{124}{82}\). To simplify, divide by their greatest common divisor of 2, giving \(\frac{62}{41}\).

Key concepts

Neutron/Proton Ratio

The neutron/proton ratio is a critical factor in the stability of an atomic nucleus. To better understand this, let's explore what this ratio represents. In an atomic nucleus, protons and neutrons are bound together by the strong nuclear force. Protons are positively charged, which means they repel each other due to their like charges. Neutrons, however, are neutral and help to hold the nucleus together by offsetting this repulsion.

As atoms become heavier, they generally need more neutrons than protons to remain stable. This is because adding neutrons without increasing the number of positively charged protons can help balance the nuclear forces at play. To calculate the neutron/proton ratio, follow these two steps:

• Count the number of neutrons by finding the difference between the mass number and the atomic number of the element.
• Divide the number of neutrons by the number of protons. This fraction is the neutron/proton ratio.

This ratio helps predict nuclear stability. For example, in the exercise, after all decay processes, the neutron count was 124 while the proton count was 82, resulting in a neutron/proton ratio of \(\frac{124}{82}\), simplified to \(\frac{62}{41}\). This tells us more about how the element behaves post-decay.

Atomic Number Reduction

Understanding atomic number reduction is key to grasping alpha decay processes. In nuclear chemistry, the atomic number defines the number of protons in an atom's nucleus, which is essential because it determines the chemical element's identity itself. When an atom undergoes alpha decay, it emits an alpha particle, consisting of 2 protons and 2 neutrons, which results in a decrease in the atomic number.

For every alpha particle emission, the atomic number reduces by 2. In the given problem, where 8 alpha particles were emitted, the atomic number was reduced by \(2 \times 8 = 16\). Starting with Uranium, having an atomic number of 92, the atomic number was reduced from 92 to 76 after all alpha particles emissions. This significant reduction influences not only which element the atom turns into but also its place on the periodic table, moving it to a different category of elements.

Mass Number Reduction

Alongside atomic number reduction, mass number reduction is another significant outcome of alpha decay. Let's dive deeper into this concept. The mass number of an atom is the sum of the protons and neutrons in its nucleus, as these particles roughly account for the total mass of the nucleus.

In alpha decay, when an atom releases an alpha particle, it loses 2 protons and 2 neutrons. This means a reduction of 4 in the mass number for each alpha particle emitted. For the given scenario where 8 alpha particles were emitted, the calculation is straightforward: the mass number reduces by \(4 \times 8 = 32\). Starting from a mass number of 238 in Uranium, the mass number drops to 206 in the resultant nucleus.
This reduction in mass number indicates a significant change in the atom's composition, as it's shedding parts of its nucleus, making it transform into a different isotope with less mass. Such transitions are critical in understanding how matter changes and behaves on an atomic level.