Question

The atomic masses of nitrogen-14, titanium-48, and xenon-129 are 13.999234 amu, 47.935878 amu, and 128.90479 amu, respectively. For each isotope, calculate (a) the nuclear mass, (b) the nuclear binding energy, (c) the nuclear binding energy per nucleon.

Short answer

The nuclear masses of nitrogen-14, titanium-48, and xenon-129 are calculated as 13.999234 - 14 × 0.000549, 47.935878 - 22 × 0.000549, and 128.90479 - 54 × 0.000549 amu, respectively. Their binding energies are obtained by substituting their nuclear masses, the mass of proton (1.007276 amu), and mass of neutron (1.008665 amu) into the binding energy formula. Finally, the binding energy per nucleon can be determined by dividing the binding energy by the total number of nucleons for each isotope.

Step-by-step solution

Calculate Nuclear Mass

First, we will find the nuclear mass of the isotopes using the formula: \(M_nuclear = M_atomic -\) # of electrons × mass of electron (in amu) \((M_e)\) The mass of an electron is approximately 0.000549 amu. For Nitrogen-14: \(M_nuclear\) = 13.999234 amu - (14 × 0.000549) amu For Titanium-48: \(M_nuclear\) = 47.935878 amu - (22 × 0.000549) amu For Xenon-129: \(M_nuclear\) = 128.90479 amu - (54 × 0.000549) amu

Calculate Binding Energy

Next, we calculate the binding energy (BE) of the isotopes using the formula: BE = # of nucleons × mass of proton + # of neutrons × mass of neutron - \(M_nuclear\) The mass of a proton (Mp) is approximately 1.007276 amu, and the mass of a neutron (Mn) is approximately 1.008665 amu. For Nitrogen-14: BE = (7 × 1.007276) + (7 × 1.008665) - \(M_nuclear\) For Titanium-48: BE = (22 × 1.007276) + (26 × 1.008665) - \(M_nuclear\) For Xenon-129: BE = (54 × 1.007276) + (75 × 1.008665) - \(M_nuclear\)

Calculate Binding Energy per Nucleon

Finally, we determine the binding energy per nucleon by dividing the binding energy by the total number of nucleons: For Nitrogen-14: BE per nucleon = BE / 14 For Titanium-48: BE per nucleon = BE / 48 For Xenon-129: BE per nucleon = BE / 129 By following these steps, we will find the nuclear mass, binding energy, and binding energy per nucleon for nitrogen-14, titanium-48, and xenon-129.

Key concepts

Nuclear Mass

Understanding the concept of nuclear mass is fundamental to the study of atomic physics. It refers to the total mass of an atomic nucleus, which is made up of protons and neutrons. These protons and neutrons are collectively known as nucleons.

At the nuclear level, mass can be somewhat counterintuitive because it doesn't include the mass of the electrons orbiting the nucleus. This is because electrons contribute very little to the overall mass of an atom, and their removal allows us to focus on the mass that's central to nuclear reactions and binding energy calculations. To find the nuclear mass, we subtract the combined mass of the electrons from the atomic mass of the element, which is typically given in atomic mass units (amu).

The mass of an electron is approximately 0.000549 amu, very small compared to that of nucleons, but essential for precise calculations. When solving the provided exercises, care must be taken to accurately multiply the electron's mass by the number of electrons in the neutral atom and subtract this from the atomic mass, as shown in step one of the solution.

Atomic Mass Unit (amu)

To comprehend calculations involving nuclear mass and binding energy, grasping the concept of the atomic mass unit (amu) is crucial. One amu is defined as one twelfth (1/12) of the mass of a carbon-12 atom. This standard unit allows chemists and physicists to compare the masses of different atoms on a scale suitable for atomic-sized particles.

The usefulness of amu becomes evident when dealing with nuclear reactions and binding energies, as these calculations involve incredibly tiny measurements. An amu is a convenient unit of measure because it allows us to express the masses of atoms and subatomic particles in a manageable numeric range, rather than in kilograms, which would yield numbers that are impractical to work with due to their small size.

In the context of the provided exercises, understanding that we use amu to measure the nuclear and subatomic masses ensures accurate and relevant calculations. It's this standardization that allows scientists to reliably discuss and compare atomic level measurements across the world.

Binding Energy per Nucleon

Binding energy per nucleon plays a critical role in determining the stability and energy characteristics of an atomic nucleus. It is a measure of the strength of the forces holding a nucleus together, which is important in both natural phenomena and practical applications such as nuclear energy production or medical imaging technologies.

The nuclear binding energy itself represents the energy required to completely disassemble the nucleus of an atom into its separate protons and neutrons. When the binding energy is divided by the number of nucleons — the sum of protons and neutrons — we obtain the binding energy per nucleon. A higher value indicates a more tightly bound and stable nucleus.

The calculations in step three of our exercises involve dividing the total binding energy by the number of nucleons to determine the binding energy per nucleon. This provides insight into the nucleus's stability; for example, nuclei with a higher binding energy per nucleon are generally more stable. It's essential to accurately perform these calculations to understand the energies involved in nuclear transformations and why some elements undergo radioactive decay while others do not.