Question

The atomic masses of nitrogen-14, titanium- 48 , and xenon129 are \(13.999234 \mathrm{u}, 47.935878 \mathrm{u},\) and \(128.904779 \mathrm{u},\) respectively. For each isotope, calculate (a) the nuclear mass, (b) the nuclear binding energy, (c) the nuclear binding energy per nucleon.

Short answer

(a) Nuclear mass: - Nitrogen-14: \(2.32405 \times10^{-26} kg\) - Titanium-48: \(7.95868 \times10^{-26} kg\) - Xenon-129: \(2.139192 \times10^{-25} kg\) (b) Nuclear binding energy: - Nitrogen-14: \(1.0805 \times10^{-11} J\) - Titanium-48: \(7.153 \times 10^{-11} J\) - Xenon-129: \(3.1851 \times10^{-10} J\) (c) Nuclear binding energy per nucleon: - Nitrogen-14: \(7.7179 \times10^{-13} J/nucleon\) - Titanium-48: \(1.4906 \times 10^{-12} J/nucleon\) - Xenon-129: \(2.469 \times 10^{-12} J/nucleon\)

Step-by-step solution

Convert atomic mass to kilograms

For each isotope, let's convert their atomic masses in unified atomic mass units (u) to their equivalent mass in kilograms (kg) using the conversion factor: 1 u = 1.6605 x 10^{-27} kg. For nitrogen-14, \(Mass = 13.999234 u \times 1.6605 \times10^{-27} kg/u = 2.3255 \times10^{-26} kg\) For titanium-48, \(Mass = 47.935878 u \times 1.6605 \times10^{-27} kg/u = 7.9672 \times10^{-26} kg\) For xenon-129, \(Mass = 128.904779 u \times 1.6605 \times10^{-27} kg/u = 2.1410 \times10^{-25} kg\)

Calculate nuclear mass

For each isotope, we need to find the nuclear mass by subtracting the mass of their electrons from their respective atomic masses. We know the mass of an electron is approximately 9.1094 x 10^{-31} kg. Nuclear mass of nitrogen-14: \(Nuclear\:Mass = Atomic\:Mass - (14 \times Electron\:Mass)\) \(Nuclear\:Mass =2.3255 \times10^{-26} kg - (14 \times 9.1094\times10^{-31}kg) = 2.32405 \times10^{-26} kg\) Nuclear mass of titanium-48: \(Nuclear\:Mass = Atomic\:Mass - (48 \times Electron\:Mass)\) \(Nuclear\:Mass = 7.9672 \times10^{-26} kg - (48\times 9.1094\times10^{-31} kg) = 7.95868 \times10^{-26} kg\) Nuclear mass of xenon-129: \(Nuclear\:Mass = Atomic\:Mass - (129\times Electron\:Mass)\) \(Nuclear\:Mass = 2.1410 \times10^{-25} kg - (129\times 9.1094\times10^{-31} kg) = 2.139192 \times10^{-25} kg\)

Calculate nuclear binding energy

To find the nuclear binding energy, we need to find the mass defect for each isotope first, and then multiply by the speed of light squared (using Einstein's mass-energy equivalence equation: E = mc^2). Mass defect for nitrogen-14: \(Mass\:Defect = (7 × 1.6736\times 10^{-27} kg + 7 × 1.675\times 10 ^{-27} kg) - 2.32405 \times10^{-26} kg = 1.2005\times 10^{-28} kg\) Nuclear binding energy for nitrogen-14: \(Binding\:Energy = Mass\:Defect \times (3\times10^8 m/s)^2\) \(Binding\:Energy = 1.2005\times 10^{-28} kg \times(3\times10^8 m/s)^2 = 1.0805\times 10^{-11} J\) Mass defect for titanium-48: \(Mass\:Defect = (22 × 1.6736\times 10^{-27} kg + 26 × 1.675\times 10 ^{-27} kg) - 7.95868 \times10^{-26} kg = 7.948 \times 10^{-28} kg\) Nuclear binding energy for titanium-48: \(Binding\:Energy = Mass\:Defect \times (3\times10^8 m/s)^2\) \(Binding\:Energy = 7.948 \times 10^{-28} kg \times(3\times10^8 m/s)^2 = 7.153 \times 10^{-11} J\) Mass defect for xenon-129: \(Mass\:Defect = (54 × 1.6736\times 10^{-27} kg + 75 × 1.675\times 10 ^{-27} kg) - 2.139192 \times10^{-25} kg = 3.539 \times 10^{-27} kg\) Nuclear binding energy for xenon-129: \(Binding\:Energy = Mass\:Defect \times (3\times10^8 m/s)^2\) \(Binding\:Energy = 3.539 \times 10^{-27} kg \times(3\times10^8 m/s)^2 = 3.1851 \times 10^{-10} J\)

Calculate nuclear binding energy per nucleon

Finally, we find the nuclear binding energy per nucleon by dividing the nuclear binding energy by the total number of nucleons (protons + neutrons). Nuclear binding energy per nucleon for nitrogen-14: \(\frac{Binding\:Energy}{Nucleons} = \frac{1.0805 \times10^{-11} J}{14} = 7.7179 \times10^{-13} J/nucleon\) Nuclear binding energy per nucleon for titanium-48: \(\frac{Binding\:Energy}{Nucleons} = \frac{7.153 \times 10^{-11} J}{48} = 1.4906\times 10^{-12} J/nucleon\) Nuclear binding energy per nucleon for xenon-129: \(\frac{Binding\:Energy}{Nucleons} = \frac{3.1851\times 10^{-10} J}{129} = 2.469 \times 10^{-12} J/nucleon\) Summary: (a) Nuclear mass: - Nitrogen-14: \(2.32405 \times10^{-26} kg\) - Titanium-48: \(7.95868 \times10^{-26} kg\) - Xenon-129: \(2.139192 \times10^{-25} kg\) (b) Nuclear binding energy: - Nitrogen-14: \(1.0805 \times10^{-11} J\) - Titanium-48: \(7.153 \times 10^{-11} J\) - Xenon-129: \(3.1851 \times10^{-10} J\) (c) Nuclear binding energy per nucleon: - Nitrogen-14: \(7.7179 \times10^{-13} J/nucleon\) - Titanium-48: \(1.4906 \times 10^{-12} J/nucleon\) - Xenon-129: \(2.469 \times 10^{-12} J/nucleon\)

Key concepts

Nuclear Mass

The term "nuclear mass" refers to the mass of an atomic nucleus, which consists of protons and neutrons, collectively known as nucleons. An atomic nucleus takes up a tiny portion of an atom, but it contains almost all of its mass. To find the nuclear mass, one needs to subtract the mass of electrons from the atomic mass since electrons have negligible mass compared to nucleons.

• The atomic mass includes both the mass of the nucleus and the electrons revolving around it.
• Typically, the mass of an electron is about 9.1094 x 10^{-31} kg, a very small value when compared to nucleons.
• In calculations, the mass of electrons is negligible, but for precise nuclear mass calculations, it must be accounted for.

This step ensures that calculations about atomic interactions focus exclusively on the dense nucleus, ignoring the minor contribution of electrons.

Binding Energy

Nuclear binding energy is the energy required to disassemble a nucleus into its component protons and neutrons. It represents the energy that holds the nucleus together. This binding energy originates from the strong nuclear force and is critical for the stability of the nucleus.

• Higher binding energy means a more stable nucleus.
• This energy can be released in nuclear reactions, such as fission and fusion.

To calculate the nuclear binding energy, one must calculate the mass defect, then apply Einstein’s mass-energy equivalence principle (with the formula \(E = mc^2\)). This concept highlights why nuclear reactions release so much energy—nuclear forces are incredibly strong.

Mass Defect

Mass defect is the difference between the mass of the nucleus and the sum of the masses of its individual nucleons when they are isolated and free. In a bound system, nucleons weigh less than the sum of their individual masses because energy is released when they bind, which correlates to a loss in mass.

• The mass defect illustrates that energy and mass are interchangeable.
• This lost mass is converted into binding energy when nucleons combine into a nucleus.

The concept of mass defect highlights the efficiency of nuclear force in tapping into the intrinsic energies within particles, thereby stabilizing the nucleus.

Nucleon

Nucleons include both protons and neutrons, the key building blocks of an atomic nucleus. Each element's unique properties are chiefly determined by its number of protons (also called the atomic number).

• Protons carry a positive charge, while neutrons are neutral.
• Neutrons contribute to the nuclear stability by mediating the electrostatic repulsion among positively charged protons.

The arrangement and number of nucleons define both the mass and the identity of an element. The balance and interaction between these particles within the nucleus determine the element’s nuclear properties and its overall stability.

Einstein's Mass-Energy Equivalence

Einstein's mass-energy equivalence is a fundamental concept grounded in the equation \(E = mc^2\). This principle establishes a direct relationship between mass and energy, proposing they can be converted into one another. It suggests that a small amount of mass can be converted into a significant amount of energy due to the large value of the speed of light squared \((c^2)\).

• This equation explains the enormous energy unleashed in nuclear reactions.
• It provides a deeper understanding of how mass can "disappear" as energy is released through nuclear processes.

By linking mass and energy, Einstein’s principle has broadened the conventional understanding of physical phenomena and is vital for modern physics, including the understanding of nuclear energy and atomic particles.