Question

As noted in Section \(11.5,\) carbon- 11 decays to boron. 11 . How do the various factors involved in nuclear stability argue that such a decay is favorable?

Short answer

The decay of carbon-11 to boron-11 is favorable due to a more stable neutron to proton ratio, stronger nuclear forces, and lower energy states achieved after decay. These factors contribute to enhanced nuclear stability of boron-11.

Step-by-step solution

Understand Nuclear Stability

Nuclear stability is primarily determined by the neutron to proton ratio. Stable nuclei have a balanced neutron to proton ratio. Carbon-11 has 6 protons and 5 neutrons, while boron-11 has 5 protons and 6 neutrons, achieving a balanced neutron to proton ratio through beta decay. This is a process where a neutron in the nucleus decays to a proton, an electron, and an electron antineutrino.

Analyzing Nuclear Forces

Nuclear forces come into play in holding the nucleus of an atom together. Creation of boron-11 from carbon-11 involves beta decay where a neutron changes to a proton thereby increasing positive charge in the nucleus. This process is favorable for carbon-11 as it helps to attain stability by enhancing the nuclear forces.

Consider Energy Considerations

Energy considerations are also a critical factor for nuclear stability. Nuclear reactions often favor products with lesser mass and hence, lower energy states. Beta decay results in a nucleus with slightly lower mass and therefore a lower energy state, driving the carbon-11 to boron-11 transformation.

Key concepts

Neutron to Proton Ratio

A vital aspect of nuclear stability is the neutron to proton ratio (n/p ratio), which is the number of neutrons divided by the number of protons in a nucleus. This ratio helps determine if a nucleus will be stable or undergo radioactive decay. For lighter elements, a ratio of about 1:1 is ideal, offering a stable configuration. However, as atomic numbers increase, a higher neutron proportion becomes necessary to balance the repulsive forces between protons, leading to a preferred n/p ratio closer to 1.5:1 in heavier elements. In the case of Carbon-11, it possesses 5 neutrons and 6 protons, resulting in an n/p ratio less than 1. This imbalance prompts the nucleus to undergo beta decay, aiming to correct the ratio by converting a neutron into a proton, thus forming Boron-11. Boron-11 has a more stable neutron to proton ratio, thereby increasing its nuclear stability.

Beta Decay

Beta decay is a radioactive decay process in which a neutron transforms into a proton within an unstable atomic nucleus. This transformation results in the emission of a beta particle, which is a high-energy, high-speed electron or positron, and an antineutrino. This process is crucial in managing the neutron to proton ratio in unstable nuclei. For Carbon-11, experiencing beta decay means that one of its neutrons turns into a proton. This change increases the number of protons to 6, while the neutrons decrease to 5. Consequently, this alteration transforms the element into Boron-11. Through beta decay, the unbalanced Carbon-11 achieves a more stable n/p ratio, making its nucleus more energetically favorable and ensuring stability.

Nuclear Forces

Nuclear forces, also known as strong forces, are the forces that act between the particles inside an atomic nucleus. These are incredibly strong, short-range forces that are responsible for holding protons and neutrons together, overcoming the natural repulsion that would occur between the positively charged protons. In Carbon-11, when it undergoes beta decay, a neutron changes into a proton, which increases the total positive charge of the nucleus. This added proton increases the strength of the nuclear forces because it creates a tighter, more stable packing of particles within the nucleus. The conversion process to Boron-11 enhances these nuclear forces, helping the nucleus become more stable and less prone to further decay.

Energy Considerations

Energy considerations play a fundamental role in the stability of atomic nuclei. Generally, nuclear transformations tend to occur toward configurations that are energetically favorable, often involving a reduction in mass and energy release. In the transition from Carbon-11 to Boron-11 via beta decay, the system moves to a state of lower energy. Even precise changes, like the release of mass-equivalent energy according to Einstein's equation, \( E = mc^2 \), contributes to stabilizing the new nucleus. As Carbon-11 decays to Boron-11, the resulting configuration possesses less mass and thus lower energy, making this transformation energetically favorable. By achieving lower energy states, the nucleus becomes more stable and less likely to undergo further decay.