Question

The tungsten isotope \({ }_{74}^{162} \mathrm{~W}\) is radioactive and decays by converting a proton to a neutron. (a) Where is this atom likely to lie in the band of stability? (b) Write two nuclear reactions that describe this decay process. (c) Which type of decay is represented by each reaction you wrote in part (b)?

Short answer

(a) The tungsten isotope \({ }_{74}^{162}\mathrm{W}\) has a neutron-to-proton ratio of \(\frac{88}{74} \approx 1.19\), indicating that it lies above the band of stability and is neutron-rich. (b) Two nuclear reactions for the decay process: 1. Beta-minus decay: \({ }_{74}^{162}\mathrm{W} \rightarrow { }_{75}^{162}\mathrm{Re} + { }_{-1}^{0}\beta + \bar{\nu}_{e}\) 2. Electron Capture: \({ }_{74}^{162}\mathrm{W} + { }_{-1}^{0}e \rightarrow { }_{73}^{162}\mathrm{Ta} + \nu_{e}\) (c) Decay types: 1. Beta-minus decay for the first reaction. 2. Electron capture for the second reaction.

Step-by-step solution

(a) Determine position relative to the band of stability

The band of stability is a region on the neutron-proton (N-Z) chart where stable isotopes are most likely to be found. For isotopes with atomic number less than about 20, the proton-to-neutron ratio is close to 1:1, while for heavier elements, this ratio increases gradually to about 1:1.5. To determine where the given tungsten isotope \({ }_{74}^{162}\mathrm{W}\) lies within the band or on one of its edges, we must compare its neutron-to-proton ratio to that of the average ratio for the stable isotopes of tungsten. Expressing the neutron-to-proton ratio mathematically, we have: Neutron-to-proton ratio = \(\frac{\text{Number of neutrons}}{\text{Number of protons}}\) Since the atomic number (Z) represents the number of protons and the mass number (A) represents the sum of protons and neutrons, we can calculate the number of neutrons (N) by subtracting Z from A: \(N = A - Z\) For the given tungsten isotope, we have: \(N = 162 - 74 = 88\) Now, we can find the neutron-to-proton ratio: \(\frac{N}{Z} = \frac{88}{74} \approx 1.19\) The neutron-to-proton ratio for the given tungsten isotope is greater than 1, indicating that it lies above the band of stability. This means that the isotope is neutron-rich and is likely to undergo beta decay.

(b) Write two nuclear reactions that describe the decay process

Since we know that the given tungsten isotope will undergo a decay process that converts a proton into a neutron, we can write the nuclear reactions for the two possible beta decays: beta-minus decay and electron capture. 1. Beta-minus decay: \({ }_{74}^{162}\mathrm{W} \rightarrow { }_{75}^{162}\mathrm{Re} + { }_{-1}^{0}\beta + \bar{\nu}_{e}\) 2. Electron Capture: \({ }_{74}^{162}\mathrm{W} + { }_{-1}^{0}e \rightarrow { }_{73}^{162}\mathrm{Ta} + \nu_{e}\)

(c) Identify the type of decay for each reaction in part (b)

1. The first reaction: \({ }_{74}^{162}\mathrm{W} \rightarrow { }_{75}^{162}\mathrm{Re} + { }_{-1}^{0}\beta + \bar{\nu}_{e}\) In this reaction, a proton is converted to a neutron inside the tungsten nucleus, and a beta-minus particle (electron) and an electron antineutrino are emitted. This type of decay is called beta-minus decay. 2. The second reaction: \({ }_{74}^{162}\mathrm{W} + { }_{-1}^{0}e \rightarrow { }_{73}^{162}\mathrm{Ta} + \nu_{e}\) In this reaction, a proton in the tungsten nucleus captures an orbital electron, converting into a neutron and emitting an electron neutrino. This type of decay is called electron capture.

Key concepts

Band of Stability

In the realm of nuclear chemistry, the band of stability is akin to a safe harbor where atomic nuclei reside in a balanced state. Picture a chart where we plot neutrons against protons. The band represents the zone where stable isotopes, free from radioactive decay, naturally exist.

For lighter elements, the neutron-to-proton ratio for stable isotopes hovers around the 1:1 mark, but as we ascend the periodic table to beefier elements, this ratio climbs, approaching 1:1.5. This shift is due to the need for additional neutrons to counterbalance the increasing proton-proton repulsion in larger nuclei.

If an isotope falls outside this snug band, it's inherently unstable and undergoes radioactive decay to return to stability. The tungsten isotope in question, with a neutron-to-proton ratio of approximately 1.19, sits above the band, indicating an excess of neutrons, which means it's prone to decay towards stability.

Beta-Minus Decay

Think of beta-minus decay as nature's way of restoring balance within a neutron-heavy nucleus. During this process, a neutron is transformed into a proton, while a negatively charged beta particle, which is essentially an electron, and an antineutrino are cast off.

\[ _{\text{Z}}^{\text{A}}X \rightarrow _{\text{Z}+1}^{\text{A}}Y + _{-1}^{0}\beta + \bar{u}_e \]

This allows the atomic nucleus to move down towards a more stable neutron-to-proton ratio. In the case of the isotope \( _{74}^{162}\mathrm{W} \) afflicting with extra neutrons, beta-minus decay nudges it closer to the band of stability by increasing the proton count and thus adjusting the ratio.

Electron Capture

Electron capture is another intriguing method by which an unstable nucleus can reach for stability. When a nucleus, gluttonous for protons, ensnares one of its own orbiting electrons, this electron merges with a proton and conjures a neutron and a neutrino.

\[ _{\text{Z}}^{\text{A}}X + _{-1}^{0}e \rightarrow _{\text{Z}-1}^{\text{A}}Y + u_e \]

It's a sort of cosmic alchemy, transmuting the identity of the atom by reducing the number of protons. For our \( _{74}^{162}\mathrm{W} \) isotope, electron capture is a pathway away from neutron surfeit, lowering the neutron-to-proton ratio and steering the nucleus towards the safety of the band of stability.

Neutron-to-Proton Ratio

The neutron-to-proton ratio is the critical gauge for nuclear stability. This ratio helps us predict the type of decay an unstable nucleus might undergo to achieve a balance in line with the band of stability. It's computed simply by dividing the number of neutrons by the number of protons (N/Z).

A ratio that's too high suggests a surplus of neutrons, often leading to beta decay to even things out. Conversely, a ratio that's too low indicates a lack of neutrons, which could result in positron emission or electron capture.

In essence, this ratio is the nucleus's ledger, with nature insisting on balanced books. For \( _{74}^{162}\mathrm{W} \) sporting a neutron-to-proton ratio of 1.19, it's like a financial red flag, triggering a process (either beta-minus decay or electron capture in this case) to recalibrate and ultimately stabilize.