Question

An unstable nucleus is characterized by (i) \(\mathrm{n} / \mathrm{p}>1\) (ii) low binding energy (iii) high temperature and pressure (iv) high packing fraction (a) (i), (ii) (b) (i), (ii), (iv) (c) (i), (ii), (iii) (d) (i), (ii), (iii), (iv)

Short answer

Option (b): (i), (ii), (iv).

Step-by-step solution

Understand the Properties of an Unstable Nucleus

Nuclei are characterized by their stability, which is determined by factors such as the neutron-to-proton ratio ( frac{n}{p}), binding energy, temperature and pressure, and packing fraction. An unstable nucleus typically has a neutron-to-proton ratio greater than 1, which can lead to instability. Low binding energy implies that the nucleons are not held together tightly, contributing to instability.

Assess Each Option for Instability

Analyze each characteristic in the options: - High neutron-to-proton ratio ( frac{n}{p} > 1) indicates instability due to neutron excess. - Low binding energy means less stability, as the nucleus can undergo decay more easily. - High packing fraction, which relates to the density of nucleons, can also influence instability. Review these in the context of the provided options to determine which combination reflects an unstable nucleus.

Eliminate Unrelated Characteristics

High temperature and pressure ( (iiiig)) are not intrinsic to nuclear instability in the context of this question. Rather, they affect reaction rates and conditions of a different nature, such as in stellar environments, but do not change the fact that an unstable nucleus is characterized by internal factors like frac{n}{p} ratio and binding energy.

Choose the Correct Option

Evaluate the options: - Option (a): Includes (i) frac{n}{p} > 1 and (ii) low binding energy. - Option (b): Includes (i), (ii), and (iv) high packing fraction. - Option (c): Includes high temperature and pressure, which are not direct indicators of instability. - Option (d): Every factor including (iii), which is unrelated. Option (b) is correct as it includes (i), (ii), and additional instability factor (iv).

Key concepts

Neutron-Proton Ratio

The neutron-to-proton ratio is a critical factor in determining nuclear stability. A stable nucleus typically has a neutron-to-proton ratio that is close to 1, especially for light elements. However, as the size of the nucleus increases, the ratio shifts to accommodate more neutrons which help to counteract the repulsive forces between protons.

When an atomic nucleus has a neutron-to-proton ratio significantly greater than one, it often becomes unstable. This excess of neutrons can lead to instability as the energy levels and forces within the nucleus become unbalanced, making the nucleus prone to radioactive decay processes. The decay will often work to correct this imbalance by converting excess neutrons into protons via beta decay, which helps to stabilize the nucleus over time.

• A neutron excess can lead to spontaneous transformations.
• Proton repulsion needs to be countered by additional neutrons.
• Stability is achieved when balance is maintained.

Binding Energy

Binding energy refers to the energy that holds the nucleons (protons and neutrons) together in the nucleus. It represents the energy required to split a nucleus into its individual components. A higher binding energy typically signifies a more stable nucleus as the nucleons are more tightly bound.

Nuclei with low binding energy are more likely to undergo nuclear decay because the forces holding the nucleons together are weaker. In such nuclei, the energy barrier to decay is lower, allowing for transitions that lead the nucleus to a more stable configuration. Therefore, identifying nuclei with low binding energy can help predict their instability and potential for decay.

• Measures the strength of nuclear binding.
• More energy needed to disassemble tighter nuclei.
• Instability arises with lower binding energies.

Packing Fraction

Packing fraction is a concept used to describe how tightly nucleons are packed within a nucleus. It is defined mathematically as the difference between the mass number and the actual mass of the nucleus, normalized by the mass number. In essence, it reflects the mass defect and the density of the nucleus.

A high packing fraction indicates a significant deviation from the ideal mass, which can be a marker of instability. This may result from a high density of nucleons causing internal strain or imperfections in the nuclear forces. When packing fractions are high, it often suggests that a nucleus is not at its most stable, increasing the likelihood for nuclear reactions or decay as it attempts to reach a lower energy state.

• Reflects nucleon density and mass defect.
• Indicates potential for high strain within the nucleus.
• Correlation with instability when elevated.