Problem 14 - Fundamentals of Physics

Question

What is the binding energy per nucleon of the americium isotope \({ }_{95}^{244} \mathrm{Am} ?\) Here are some atomic masses and the neutron mass. $$\begin{array}{lr}{\underline{\phantom{xx}}}_{95}^{244} \mathrm{Am} & 244.064279 \mathrm{u} \\\\\mathrm{n} & 1.008665 \mathrm{u}\end{array} { }^{1} \mathrm{H} \quad 1.007825 \mathrm{u}$$

Step-by-Step Solution

Answer

The binding energy per nucleon of \( {}_{95}^{244} \mathrm{Am} \) is approximately 12.82 MeV/nucleon.

Step 1: Calculate the Mass Defect

First, determine the total mass of the nucleus if it were composed of free protons and neutrons. The atomic number of americium \( {}_{95}^{244} \mathrm{Am} \) is 95, meaning there are 95 protons. The number of neutrons can be found by subtracting the atomic number from the mass number: \( 244 - 95 = 149 \) neutrons. The mass of separated nucleons is \( 95 \times 1.007825 + 149 \times 1.008665 = 247.422425 \) u. The mass defect \( \Delta m \) is the difference between this total mass and the actual atomic mass of the isotope:\[ \Delta m = 247.422425 - 244.064279 = 3.358146 \text{ u} \]

Step 2: Convert Mass Defect to Energy

The binding energy \( E \) can be calculated using Einstein's equation \( E=\Delta mc^2 \). Here, we convert the mass defect from atomic mass units (u) into energy (MeV) using the conversion factor \( 1 \text{ u} = 931.5 \text{ MeV}/c^2 \):\[ E = 3.358146 \, \text{u} \times 931.5 \, \frac{\text{MeV}}{\text{u}} = 3128.68 \, \text{MeV} \]

Step 3: Calculate Binding Energy per Nucleon

The binding energy per nucleon is found by dividing the total binding energy by the number of nucleons (which is the mass number of the isotope, 244 in this case):\[ \frac{E}{A} = \frac{3128.68 \, \text{MeV}}{244} = 12.82 \, \text{MeV/nucleon} \]

Key Concepts

Mass Defect

The concept of mass defect is a cornerstone in the study of nuclear physics, particularly when analyzing binding energy. Mass defect refers to the difference between the total mass of a nucleus composed of free nucleons (protons and neutrons) and the actual mass of the nucleus when these nucleons are bound together in an atom. This mass difference is crucial as it represents the energy needed to hold the nucleus together.

When nucleons bind to form a nucleus, some mass is essentially converted into energy—which is relinquished in the process—thus explaining the reduced mass. The equation for mass defect is formally given by:

• Total mass of separated nucleons: Sum of the individual masses of protons and neutrons.
• Measured atomic mass: Actual mass of the isotope.
• Mass defect (\( \Delta m \)) = Total mass of separated nucleons - Measured atomic mass.

This loss of mass potential energy provides insight into the stability of the nucleus, making mass defect a critical measurement for understanding binding energies.

Nucleons

Nucleons are the collective term for the particles found in an atom's nucleus, namely protons and neutrons. These nucleons are the fundamental building blocks of atomic nuclei. Each element is characterized by its unique combination and number of protons, while the number of neutrons can vary even within a single element, leading to different isotopes.

Nucleons are critically important in defining the properties and behavior of a nucleus.

• Protons carry a positive charge and determine the chemical identity of an element (the atomic number).
• Neutrons are neutral; they contribute to the atomic mass but do not influence the electric charge.

In nuclear reactions and calculations of binding energy, the concept of nucleons is essential, as both types are counted to determine the mass number and understand the energy dynamics within the nucleus.

Conversion Factor

A conversion factor is used in physics to transform one set of units into another, facilitating calculations and interpretations of physical phenomena. In the context of nuclear physics, the conversion factor allows us to relate mass in atomic mass units (u) to energy in mega-electron volts (MeV), connecting mass with energy through Einstein's famous equation, \(E=mc^2\).

The specific conversion factor used for this transformation is:

• 1 atomic mass unit (u) = 931.5 MeV/\(c^2\).

This constant is derived from the energy equivalence of mass, allowing for easy and accurate computations of nuclear binding energies. Applying this conversion factor enables the transformation of mass measurements into energy units, revealing the amount of energy that binds nucleons within an atom.

Atomic Mass Units

The atomic mass unit (u), also known as the unified atomic mass unit or Dalton (Da), is a standard unit of mass that quantifies mass on an atomic or molecular scale. This unit allows scientists to conveniently compare the masses of different atoms and subatomic particles.

Defined as one-twelfth the mass of a carbon-12 atom,

• 1 u approximately equals 1.66053906660 × 10\(^{-27}\) kilograms.

Atomic mass units are crucial for calculating the mass of atoms in chemical compounds and in nuclear physics. With this unit, scientists can account for the extremely small masses of nucleons and other particles in a tangible way, assisting in precise calculations and comparisons across diverse scientific queries.