Question

Consider the following graph of binding energy per nucleon as a function of mass number a. What does this graph tell us about the relative half-lives of the nuclides? Explain your answer. b. Which nuclide shown is the most thermodynamically stable? Which is the least thermodynamically stable? c. What does this graph tell us about which nuclides undergo fusion and which undergo fission to become more stable? Support your answer.

Short answer

The graph of binding energy per nucleon as a function of mass number reveals that nuclides with higher binding energy per nucleon have longer half-lives. The most thermodynamically stable nuclide has the highest binding energy per nucleon and is shown by the highest peak on the graph, while the least stable has the lowest. Lighter elements, typically on the left side of the graph, are more likely to undergo fusion to achieve higher stability, while heavier elements, on the right side of the graph, are more likely to undergo fission to become more stable.

Step-by-step solution

Understand the graph and binding energy per nucleon

The graph of binding energy per nucleon as a function of mass number gives a general idea of the energy required to separate nucleons (protons and neutrons) from each other in a nuclide. Higher binding energy per nucleon means that more energy is needed to separate the nucleons; therefore, the nucleus is more stable. #a#

Relating binding energy to half-lives

The half-life of a nuclide is a measure of time it takes for half of the substance to decay. Generally, the higher the binding energy per nucleon, the more time it takes for the nucleus to decay, and hence, it has a longer half-life. So, by looking at the graph, we can determine that the nuclides with higher binding energy per nucleon have longer half-lives, while those with lower binding energy per nucleon have shorter half-lives. #b#

Identifying the most and least thermodynamically stable nuclides

The most thermodynamically stable nuclide is the one with the highest binding energy per nucleon, as it requires the most energy to separate the nucleons. On the graph, we can identify this nuclide as the one with the highest peak. Similarly, the least thermodynamically stable nuclide is the one with the lowest binding energy per nucleon, as it requires the least amount of energy to separate the nucleons. We can identify this nuclide on the graph as the one with the lowest value. #c#

Fusion and fission processes for nuclide stability

The graph can help us understand which nuclides are more likely to undergo fusion or fission to become more stable. Fusion is a process in which two lighter nuclei combine to form a heavier nucleus, while fission is a process in which a heavy nucleus splits into two lighter nuclei. Nuclides with lower binding energy per nucleon (lighter elements, usually found on the left side of the graph) are more likely to undergo fusion to achieve higher stability. This is because, during fusion, the resulting heavier nucleus will have a higher binding energy per nucleon, making it more stable. In contrast, nuclides with higher mass numbers and lower binding energy per nucleon (heavier elements, usually found on the right side of the graph) are more likely to undergo fission to become more stable. During fission, the heavy nucleus splits into two lighter nuclei, and the binding energy per nucleon of these lighter nuclei increases, making them both more stable. In summary, the graph of binding energy per nucleon as a function of mass number allows us to determine relative half-lives, thermodynamic stability, and which nuclides are more likely to undergo fusion or fission processes for better stability.

Key concepts

Nuclide Stability

To understand nuclide stability, we look at the binding energy per nucleon. This metric helps us determine how strongly the nucleons (protons and neutrons) are held together within a nucleus. Higher binding energy per nucleon indicates a more stable nuclide because more energy is required to break these bonds. Conversely, a lower binding energy per nucleon suggests a less stable nuclide, as it takes less energy to pull the nucleons apart.

Stability is crucial because it influences how long a nuclide can exist without undergoing radioactive decay. Stable nuclides have long half-lives, while unstable ones decay faster. For example, considering the graph of binding energy, nuclides located at the peak are typically the most stable, often corresponding to elements like iron and nickel on the periodic table. These nuclides resist change and are less likely to participate in nuclear reactions.

Fusion and Fission

Fusion and fission are two fundamental processes that alter nuclide stability. Both processes aim to create more stable nuclei by either combining or splitting existing ones.

**Fusion**: This process involves the merging of two lighter nuclei to form a heavier one. It is favored by nuclides on the lighter, left side of the binding energy graph. When these lighter nuclei combine, the resulting nucleus typically has a higher binding energy per nucleon than the initial pair, thereby increasing stability. Fusion is the energy source behind stars, including our Sun.

**Fission**: In contrast, fission is the splitting of a heavy nucleus into two lighter, more stable ones. This process is more common among heavier nuclides, which reside on the right side of the graph and have relatively low binding energy per nucleon. By splitting, these heavy nuclei increase the overall binding energy of the products, hence moving towards greater stability. Fission is the process used in nuclear reactors to produce energy.

Half-lives of Nuclides

The half-life of a nuclide is a measure of how long it takes for half of a given sample to decay. It's directly influenced by the binding energy per nucleon.

Nuclides with high binding energy per nucleon are typically stable and thus possess longer half-lives. They take longer timescales to decay, since their strong nuclear forces resist disintegration. On the other hand, nuclides with low binding energy per nucleon are usually unstable, possessing shorter half-lives and decaying more rapidly.

Understanding half-lives is essential in fields like nuclear medicine, archaeology, and astrophysics, where accurate age and stability predictions are crucial. For instance, radioactive dating techniques rely on the known half-lifes of specific isotopes to deduce the age of objects, illustrating how this concept plays a vital role across various scientific disciplines.