Problem 16 - Fundamentals of Physics

Question

What is the binding energy per nucleon of the europium isotope \(\frac{152}{63} \mathrm{Eu}\) ? Here are some atomic masses and the neutron mass. $$\begin{array}{lr}\frac{152}{63} \mathrm{Eu} & 151.921742 \mathrm{u} \\\\\mathrm{n} & 1.008665 \mathrm{u}\end{array} { }^{1} \mathrm{H} \quad 1.007825 \mathrm{u}$$

Step-by-Step Solution

Answer

The binding energy per nucleon of \( \frac{152}{63}\mathrm{Eu} \) is approximately 8.22 MeV.

Step 1: Identify Given Data

We are given the atomic mass of the europium isotope, \( \frac{152}{63} \text{Eu} = 151.921742 \text{ u} \), hydrogen atom \( {}^{1} \text{H} = 1.007825 \text{ u} \), and neutron mass \( \text{n} = 1.008665 \text{ u} \). The europium isotope contains 63 protons and 89 neutrons because 152 - 63 = 89.

Step 2: Calculate Mass of Separated Nucleons

To find the mass of the nucleons if they were separate, calculate the mass of 63 protons and 89 neutrons:\[\text{Mass of protons = } 63 \times 1.007825\, \text{u} = 63.492975\, \text{u} \]\[\text{Mass of neutrons = } 89 \times 1.008665\, \text{u} = 89.770185\, \text{u} \]\[\text{Total mass of separated nucleons} = 63.492975 + 89.770185 = 153.26316\, \text{u} \]

Step 3: Calculate Mass Defect

The mass defect is the difference between the mass of separated nucleons and the actual mass of the nucleus: \[ \text{Mass defect} = 153.26316\, \text{u} - 151.921742\, \text{u} = 1.341418\, \text{u} \]

Step 4: Convert Mass Defect to Energy

To convert the mass defect into energy, use the relation \( E = \, \Delta m \times 931.5 \text{ MeV/u} \): \[ E = 1.341418\, \text{u} \times 931.5\, \text{MeV/u} = 1249.641053\, \text{MeV} \]

Step 5: Calculate Binding Energy per Nucleon

The binding energy per nucleon is found by dividing the total binding energy by the number of nucleons in the nucleus (152 for europium-152):\[ \text{Binding energy per nucleon} = \frac{1249.641053\, \text{MeV}}{152} = 8.21868\, \text{MeV/nucleon} \]

Key Concepts

Mass Defect

The concept of mass defect is pivotal in understanding nuclear binding energy. When subatomic particles like protons and neutrons are bound together in a nucleus, the overall mass of the nucleus is slightly less than the sum of individual masses of the separated nucleons. This discrepancy in mass is known as the mass defect.

In the context of the europium isotope \(^ {152}_{63} \text{Eu}\), we calculated the mass defect by first determining the mass of the separate nucleons, which are the 63 protons and 89 neutrons, and then subtracting the observed atomic mass of the europium isotope. The computation was:

• Mass of separate protons: 63.492975 u
• Mass of separate neutrons: 89.770185 u
• Total mass of separated nucleons: 153.26316 u

The actual mass of the europium isotope was given as 151.921742 u. Therefore, the mass defect is the difference: 153.26316 u - 151.921742 u = 1.341418 u.
This mass defect represents the conversion of mass into energy that helps to bind the nucleons in the nucleus.

Nucleons

Nucleons are the collective term for protons and neutrons, which are the building blocks of atomic nuclei. Each element's properties are significantly defined by its nucleon makeup.

For the europium isotope \(^ {152}_{63} \text{Eu}\), the numerals provide a wealth of information:

• Protons: 63 (indicating the atomic number, which defines the type of element)
• Neutrons: 152 - 63 = 89 (obtained by subtracting the number of protons from the mass number)

Together, these 152 nucleons form the stable yet energetically dense core of the europium nucleus.

In nuclear physics, the binding energy per nucleon is a critical measure because it indicates how strongly the nucleons are held together. For \(^ {152}_{63} \text{Eu}\), we computed the binding energy per nucleon, using it to understand the overall nuclear stability and the energy dynamics involved in nuclear reactions.

Europium Isotope

Isotopes are variations of elements that contain the same number of protons but differ in the number of neutrons. Europium has several isotopes, with differences in neutron count influencing their nuclear properties.

The specific europium isotope discussed here is \(^ {152}_{63} \text{Eu}\), characterized by having 152 nucleons: 63 protons and 89 neutrons. The mass number (152) and atomic number (63) are central to defining this isotope's identity.

This isotope of europium forms part of studies in nuclear chemistry due to its stability and the nuclear binding energy involved. The computations of mass defect and resulting binding energy provide insights into its nuclear structure and properties. Understanding these principles is crucial for applications in nuclear energy, medical imaging, and radioactive dating, where isotopes play significant roles.