Problem 64 - College Physics

Question

The atomic mass of \(\frac{56}{26} \mathrm{Fe}\) is 55.934939 \(\mathrm{u}\) , and the atomic mass of mass of \(\frac{56}{27}\) Co is 55.939847 \(\mathrm{u}\) (a) Which of these nuclei will decay into the other? (b) What type of decay will occur? (c) How much kinetic energy will the products of the decay have?

Step-by-Step Solution

Answer

(a) \( ^{56}_{27}\mathrm{Co} \) decays into \( ^{56}_{26}\mathrm{Fe} \). (b) Beta-plus decay. (c) 4.57 MeV.

Step 1: Understand the process

For part (a), compare the atomic masses of \( ^{56}_{26}\mathrm{Fe} \) and \( ^{56}_{27}\mathrm{Co} \). Generally, a nucleus will decay into a more stable form which often involves a lower atomic mass.

Step 2: Compare atomic masses

Since \( 55.934939 \mathrm{u} \) for \( ^{56}_{26}\mathrm{Fe} \) is less than \( 55.939847 \mathrm{u} \) for \( ^{56}_{27}\mathrm{Co} \), the \( ^{56}_{27}\mathrm{Co} \) is likely to decay into \( ^{56}_{26}\mathrm{Fe} \) because it results in a decrease in mass, indicating a higher stability.

Step 3: Determine the decay type

Since \( ^{56}_{27}\mathrm{Co} \) is decaying into \( ^{56}_{26}\mathrm{Fe} \), this represents a decrease in atomic number by 1. This is a characteristic of beta-plus decay \((\beta^+\)) wherein a proton is converted into a neutron, emitting a positron and a neutrino.

Step 4: Calculate kinetic energy release

The energy released in the decay is the mass difference between the parent and daughter nucleus, converted to energy using Einstein’s equation \( E = mc^2 \). The mass difference is \( \Delta m = 55.939847\mathrm{u} - 55.934939\mathrm{u} = 0.004908 \mathrm{u} \). Convert the mass difference to energy: \( 0.004908 \mathrm{u} \times 931.5 \mathrm{MeV/u} = 4.57 \mathrm{MeV} \). This energy is released as kinetic energy of the decay products.

Key Concepts

Beta-Plus Decay

Beta-plus decay, or \( \beta^+ \) decay, is a process where a proton in a nucleus is transformed into a neutron. This conversion results in the emission of a positron ( an anti-electron with a positive charge) and a neutrino (a nearly massless, neutral particle). It typically occurs when a nucleus has too many protons and is attempting to achieve stability by converting one of its protons into a neutron. In the context of nuclear decay, beta-plus decay is identified by a decrease in the atomic number by one unit, as seen when \( ^{56}_{27}\mathrm{Co} \) decays into \( ^{56}_{26}\mathrm{Fe} \). The process effectively changes one chemical element into another and is a key mode of radioactive decay for proton-rich isotopes.

Beta-plus decay is essential in nuclear physics because it conserves the total charge while allowing the nucleus to achieve a more stable configuration. After the decay, the daughter nucleus is closer to the optimum balance of protons and neutrons for stability.

Atomic Mass Comparison

A crucial step in predicting nuclear decay is comparing atomic masses of the involved nuclei. A lower atomic mass generally signifies a more stable nucleus. In the example of \( ^{56}_{27}\mathrm{Co} \) and \( ^{56}_{26}\mathrm{Fe} \), we observe that the atomic mass of iron, \( 55.934939 \mathrm{u} \), is less than the atomic mass of cobalt, \( 55.939847 \mathrm{u} \).

This mass discrepancy indicates that \( ^{56}_{27}\mathrm{Co} \) is less stable and will naturally decay into \( ^{56}_{26}\mathrm{Fe} \). By moving to a nucleus with a lower atomic mass, the system releases energy and aligns closer to a minimum energy state. This behavior underpins the spontaneity of the nuclear decay process and helps identify which isotopes are likely to undergo decay to achieve greater nuclear stability.

Energy Conversion in Decay

Energy conversion is integral in nuclear decay processes as it explains the appearance of energy in various forms, mostly kinetic, after the decay event. The transformation of mass into energy is guided by Einstein's celebrated equation: \( E = mc^2 \). Here, \( E \) represents energy, \( m \) is the mass difference between the parent and daughter nucleus, and \( c \) is the speed of light. This principle reflects energy conservation law.

For \( ^{56}_{27}\mathrm{Co} \) decay to \( ^{56}_{26}\mathrm{Fe} \), the mass difference is \( 0.004908 \mathrm{u} \). By converting this mass difference into energy using the conversion factor \( 931.5 \mathrm{MeV/u} \), we find the released energy is approximately \( 4.57 \mathrm{MeV} \). This energy is primarily distributed as kinetic energy among the decay products, including the emitted positron and neutrino, thereby maintaining the balance of energy within the system.

Stability of Nuclei

Nuclear stability is fundamentally about the balance between protons and neutrons in a nucleus. A stable nucleus has a proportion of neutrons and protons that results in the lowest possible energy state. The need to attain or retain such stability often drives nuclear decay, where isotopes shed or transform particles to approach this optimal configuration.

For \( ^{56}_{27}\mathrm{Co} \), possessing an excess of protons, stability is sought by converting a proton into a neutron via beta-plus decay, reducing its proton count and thus moving closer to a more stable state embodied by \( ^{56}_{26}\mathrm{Fe} \). This conversion decreases the atomic number but conserves the overall mass number of the nucleus, further aligning it along the line of stable isotopes known as the "valley of stability" in nuclear physics. By following this reaction path, the nucleus achieves a more energetically favorable configuration.