Problem 76 - Chemistry: An Atoms-Focused Approach

Question

Gadolinium-153 and samarium-153 both have the same mass number. Why might \(^{153}\) Gd decay by electron capture whereas \(^{153} \mathrm{Sm}\) decays by emitting \(\beta\) particles?

Step-by-Step Solution

Answer

Answer: Gadolinium-153 decays by electron capture because it has more protons relative to its neutrons, and converting a proton into a neutron through electron capture helps it achieve a more stable ratio. On the other hand, Samarium-153 decays by emitting beta particles because it has more neutrons relative to its protons, and converting a neutron into a proton through beta decay brings it closer to a stable ratio.

Step 1: Understanding decay processes

There are two main types of decay processes we will consider for this problem: electron capture and beta decay. In electron capture, an electron from the inner shell of an atom is captured by the nucleus, causing a proton to convert into a neutron, and releasing an energy in the form of a neutrino. The resulting nuclide will have one fewer proton and one more neutron. $$p^+ + e^- \to n + \nu_e$$ In beta decay, an unstable nucleus undergoes a transformation where a neutron converts into a proton, releasing a beta particle (an electron) and an antineutrino. The resulting nuclide will have one more proton and one fewer neutron. $$n \to p^+ + e^- + \bar{\nu}_e$$

Step 2: Considering atomic structure of isotopes

Gadolinium (Gd) and Samarium (Sm) both have a mass number of 153. The atomic number of Gadolinium is 64, and for Samarium, it is 62. This means that Gadolinium-153 has 64 protons and 89 neutrons, while Samarium-153 has 62 protons and 91 neutrons. The difference between the two isotopes lies in their proton and neutron numbers.

Step 3: Relating decay processes to isotope stability

Isotope stability depends on the ratio of neutrons to protons. If an isotope has too many neutrons relative to protons, it can undergo beta decay to increase its proton count. If it has too many protons relative to neutrons, it can undergo electron capture to increase its neutron count. In the case of Gadolinium-153, it has more protons than Samarium-153, which means it has a higher proton-neutron ratio. To become more stable, it undergoes electron capture to convert a proton into a neutron, resulting in a closer to stable ratio. For Samarium-153, it has more neutrons than Gadolinium-153 and a lower proton-neutron ratio. To become more stable, it undergoes beta decay to convert a neutron into a proton, increasing its proton-neutron ratio.

Step 4: Conclusion

In conclusion, Gadolinium-153 decays by electron capture because it has more protons relative to its neutrons, and converting a proton into a neutron through electron capture helps it achieve a more stable ratio. On the other hand, Samarium-153 decays by emitting beta particles because it has more neutrons relative to its protons, and converting a neutron into a proton through beta decay brings it closer to the stable ratio.

Key Concepts

Electron Capture

Imagine an electron from an atom's inner shells being pulled into the nucleus. This is what happens during electron capture. The captured electron combines with a proton, transforming it into a neutron.
This transformation comes with the release of a tiny neutrino. The change within the nucleus during electron capture is quite specific:

• Proton count decreases by one.
• Neutron count increases by one.
• The overall atom becomes a different element with a lower atomic number.

Why does this happen? It typically occurs in isotopes that have more protons relative to neutrons. Electron capture helps balance the ratio, aiming for nuclear stability.

Beta Decay

Beta decay can be thought of as a rearrangement process within the nucleus. In this form of decay, it's the neutron that changes. It becomes a proton by emitting a beta particle, which is actually an electron.
This transformation also emits an antineutrino. Let’s see what this does to the atom:

• Proton count increases by one.
• Neutron count decreases by one.
• The atom changes into a different element with a higher atomic number.

Beta decay is common in isotopes with a surplus of neutrons. By turning some neutrons into protons, the atom moves towards a more balanced neutron-to-proton ratio.

Nuclear Stability

Every isotope strives for a perfect neutron-to-proton ratio that grants stability.
The forces inside the nucleus are most effective when this ratio is correct. When an isotope has too many neutrons or too many protons, it can become unstable.
This instability predisposes it to spontaneous radioactive decay. Isotopes with more protons often undergo electron capture to build more neutrons. Those with more neutrons may perform beta decay to boost their proton numbers. Both processes aim for that "sweet spot" of nuclear stability.

Isotopes

Isotopes are variants of a chemical element that share the same number of protons but differ in their number of neutrons. This similarity gives isotopes the same atomic number but a different mass number.
For example, Gadolinium-153 and Samarium-153 both hold a mass number of 153 but differ in their neutron counts. This difference in neutron numbers is key:

• Gadolinium-153 has 64 protons and 89 neutrons.
• Samarium-153 has 62 protons and 91 neutrons.

This neutron-proton variation influences the type of radioactive decay each isotope will undergo to reach nuclear stability.