Question

Lanthanum-139 is a stable nuclide but lanthanum-140 is unstable \(\left(t_{1 / 2}=40 \mathrm{hr}\right)\). What rule of thumb concerning nuclear stability is involved?

Short answer

The rule of thumb concerning nuclear stability involved here is that nuclides with a neutron-to-proton ratio higher than the stable range for their size are likely to be unstable. Lanthanum-140's high n/p ratio compared to lanthanum-139 likely makes it unstable.

Step-by-step solution

Identifying the Neutron-to-Proton Ratio

To understand the stability of a nuclide, we need to look at the neutron-to-proton (n/p) ratio. Stable nuclides typically have a n/p ratio that falls within a certain range that provides a balanced nuclear force to counteract the repulsion between protons. Since both isotopes have the same number of protons (being the same element), the stability is likely related to the number of neutrons present.

Comparing Neutron Numbers

Lanthanum-139 and lanthanum-140 have different numbers of neutrons. Since their atomic numbers are the same (57 protons for lanthanum), Lanthanum-139 has 139 - 57 = 82 neutrons, whereas lanthanum-140 has 140 - 57 = 83 neutrons.

Applying the Rule of Thumb

A common rule of thumb for stability is that for lighter elements (atomic number less than 20), the n/p ratio should be approximately 1:1, while for larger elements it ranges up to about 1.5:1. As atomic numbers increase, the number of neutrons needed for stability increases, and the n/p ratio becomes greater than 1. Since lanthanum is a larger element, having an extra neutron may push lanthanum-140 out of the range of stability, making it an unstable nuclide.

Rule of Thumb Involved

The rule of thumb involved here is that nuclides with a higher neutron-to-proton ratio than is stable for its size (atomic number) are likely to be unstable. In this case, lanthanum-140 is unstable likely due to having one more neutron than lanthanum-139, which is stable, indicating that lanthanum-140's n/p ratio is too high.

Key concepts

Neutron-to-Proton Ratio

The neutron-to-proton ratio is a fundamental factor that determines the stability of an atomic nucleus. It is the number of neutrons (n) within an atomic nucleus divided by the number of protons (p). For lighter atoms (those with an atomic number less than 20), this ratio should be close to 1:1 for the atom to be stable. As elements get heavier, they require a greater number of neutrons to counterbalance the increasing electrostatic repulsion between protons, and the ratio shifts. Thus, the n/p ratio becomes critical in predicting the stability of a nuclide.

For example, in stable nuclides of heavier elements, the ratio tends to range up to about 1.5:1. If the neutron-to-proton ratio falls outside of the range expected for a given element's atomic number, the nucleus may be unstable, leading to radioactivity or decay. This explains why isotopes of the same element with different neutron numbers can have varying stabilities.

Understanding Neutron Balance

A balanced neutron-to-proton ratio ensures that the strong nuclear force is able to maintain the integrity of the nucleus against the repulsive force of the positively charged protons. Any significant deviation from the ideal n/p ratio can make a nucleus more prone to radioactive decay as a means to achieve stability.

Nuclide Stability

Nuclide stability is about whether a particular nuclide is resistant to changing its structure or composition over time. At the center of this stability is the interplay between the strong nuclear force, which acts to hold the nucleons (protons and neutrons) together, and the electrostatic force that causes like charges (protons) to repel one another.

In stable nuclides, these forces are balanced in such a way that the nucleus remains intact without underdoing radioactive decay. A variety of factors contribute to a nuclide's stability, including the neutron-to-proton ratio, nuclear shell model, and the total number of nucleons.

Significance of Neutron-Proton Equilibrium

While the neutron-to-proton ratio is pivotal, other aspects such as the arrangement of nucleons and the presence of 'magic numbers'—specific numbers of neutrons or protons that complete a nuclear shell—are also crucial for predicting stability. Nuclides that fall within the zone of stability are less likely to undergo radioactive decay, whereas those outside this zone—either neutron-deficient or neutron-rich—are more prone to decay as they attempt to reach a more stable state.

Unstable Isotopes

Unstable isotopes, also known as radioisotopes, are variations of elements that have an unstable atomic nucleus and tend to undergo radioactive decay. The instability often results from an imbalance between the number of protons and neutrons, leading to a neutron-to-proton ratio that is not optimal for that particular element.

The instability of these isotopes causes them to release energy in the form of radiation as they decay into more stable forms. This radioactive decay can happen through various processes such as alpha decay, beta decay, or gamma radiation.

Manifestations of Instability

Radioisotopes have many applications in medicine, industry, and scientific research, where they are used for imaging, treatment, and tracing chemical and biological processes. The half-life of an isotope, which is the time it takes for half of a sample of the isotope to decay, is a critical property that characterizes its rate of decay and level of radioactivity. Isotopes like lanthanum-140, with a short half-life of 40 hours, clearly demonstrate this concept of instability by changing to more stable configurations over relatively short periods.