Problem 23 - Chemistry: An Atoms-Focused Approach

Question

Calculate the neutron-to-proton ratios for each of the following and predict the modes of decay for the following radioactive isotopes: \((\mathrm{a})^{47} \mathrm{Sc} ;(\mathrm{b})^{89} \mathrm{Zr} ;(\mathrm{c})^{230} \mathrm{Th}\).

Step-by-Step Solution

Answer

a. \(^{47}Sc\) b. \(^{89}Zr\) c. \(^{230}Th\) To predict the mode of decay for the given isotopes, we first calculated their neutron-to-proton ratios: a. \(^{47}Sc\) has a neutron-to-proton ratio of approximately 1.24. b. \(^{89}Zr\) has a neutron-to-proton ratio of approximately 1.23. c. \(^{230}Th\) has a neutron-to-proton ratio of approximately 1.56. Using the general guidelines for predicting decay modes: a. \(^{47}Sc\) will most likely undergo beta-minus decay. b. \(^{89}Zr\) will most likely undergo beta-minus decay. c. \(^{230}Th\) will most likely undergo alpha decay since it is a heavy isotope (A > 210).

Step 1: Calculate the neutron-to-proton ratios for each isotope

We will use the given isotopes and the neutron-to-proton ratio formula to compute the ratio for each isotope. (a) \(^{47}Sc\) - Mass number (A) = 47 - Atomic number (Z) = 21 (Scandium) $$ \text{Neutron-to-Proton Ratio} = \frac{(47-21)}{21} = \frac{26}{21} \approx 1.24 $$ (b) \(^{89}Zr\) - Mass number (A) = 89 - Atomic number (Z) = 40 (Zirconium) $$ \text{Neutron-to-Proton Ratio} = \frac{(89-40)}{40} = \frac{49}{40} \approx 1.23 $$ (c) \(^{230}Th\) - Mass number (A) = 230 - Atomic number (Z) = 90 (Thorium) $$ \text{Neutron-to-Proton Ratio} = \frac{(230-90)}{90} = \frac{140}{90} \approx 1.56 $$ Now that we have the neutron-to-proton ratios for each isotope, let's predict their modes of decay.

Step 2: Predict the modes of decay for each isotope

We will use the general guidelines for predicting decay modes discussed in the analysis section to determine the decay mode for each isotope. (a) \(^{47}Sc\) - Neutron-to-Proton Ratio \(\approx 1.24 > 1\) - It will most likely undergo beta-minus decay. (b) \(^{89}Zr\) - Neutron-to-Proton Ratio \(\approx 1.23 > 1\) - It will most likely undergo beta-minus decay. (c) \(^{230}Th\) - Neutron-to-Proton Ratio \(\approx 1.56 > 1\) - Since the isotope is heavy (A > 210), it will most likely undergo alpha decay. In conclusion, the isotope \(^{47}Sc\) and \(^{89}Zr\) will most likely undergo beta-minus decay, while the isotope \(^{230}Th\) will most likely undergo alpha decay.

Key Concepts

Neutron-to-Proton Ratio

The neutron-to-proton ratio is a critical concept in understanding nuclear stability and decay modes. It is defined by the number of neutrons ( ) divided by the number of protons ( p) in a nucleus. This ratio helps predict how an isotope will likely decay. Typically, for lighter elements, a ratio close to 1:1 ensures stability. This means an equal number of neutrons and protons. However, as elements get heavier, more neutrons are necessary to keep the nucleus stable.

For example, in the solution:

• Scandium ( ^{47}Sc): Mass number (A) is 47, and atomic number (Z) is 21, resulting in (rac{26}{21}) ≈ 1.24
• Zirconium ( ^{89}Zr): A is 89, Z is 40, giving (rac{49}{40}) ≈ 1.23
• Thorium ( ^{230}Th): A is 230, Z is 90, resulting in (rac{140}{90}) ≈ 1.56

These ratios reflect their relative stability and suggest possible decay modes needed to reach a more stable state.

Beta-minus Decay

Beta-minus decay is a decay mode occurring in unstable isotopes with an excess of neutrons compared to protons. This decay process involves the transformation of a neutron into a proton, with the emission of an electron (beta particle) and an antineutrino. As a result, the atomic number increases by one, while the mass number stays the same, effectively converting one element into the next higher one in the periodic table.

Examining the exercise:

• Isotopes such as ^{47}Sc and ^{89}Zr, with neutron-to-proton ratios slightly greater than one, indicate instability due to a surplus of neutrons. This situation makes beta-minus decay a probable mode of decay.

The general rule is that isotopes with n/p ratios greater than 1 but not extremely high (like ^{47}Sc's 1.24 and ^{89}Zr's 1.23) often undergo beta-minus decay to achieve a more stable balance by transforming excess neutrons into protons.

Alpha Decay

Alpha decay is a common decay mode for heavy isotopes, especially those with a mass number higher than 210. It involves the emission of an alpha particle, which is essentially a helium nucleus composed of 2 protons and 2 neutrons. This decay reduces the mass number by 4 and the atomic number by 2, leading to the formation of a new element that is lighter and often more stable.

In the case of the exercise, ^{230}Th has a high neutron-to-proton ratio of approximately 1.56. Coupled with its large mass number, these characteristics make alpha decay more likely. This decay helps the heavy nucleus to shed weight and often achieves greater stability by moving towards the balanced band of stability.

Alpha decay usually leads to the creation of another radioactive isotope, which may undergo further decay processes until a stable form is reached. This process exemplifies how heavy, neutron-rich isotopes, such as ^{230}Th, stabilize by emitting alpha particles.