Question

Calculate the binding energy per nucleon for the following nuclei: (a) \({ }_{7}^{14} \mathrm{~N}\) (nuclear mass, \(13.999234\) amu); (b) \({ }^{48} \mathrm{Ti}\) (nuclear mass, \(47.935878\) amu); (c) xenon-129 (atomic mass, \(128.904779\) amu).

Short answer

The binding energy per nucleon is 7.524 MeV for \(N(7,14)\), 7.675 MeV for \(Ti(22,48)\), and 15.842 MeV for \(Xe(54,129)\).

Step-by-step solution

Number of Protons and Neutrons

Number of Protons (Z) = 7 Number of Neutrons (N) = 14 - 7 = 7 (b) For Ti(22,48):

Number of Protons and Neutrons

Number of Protons (Z) = 22 Number of Neutrons (N) = 48 - 22 = 26 (c) For Xe(54,129):

Number of Protons and Neutrons

Number of Protons (Z) = 54 Number of Neutrons (N) = 129 - 54 = 75 Now we proceed to calculate the mass defect (∆m) for each nucleus. (a) Mass defect for N(7,14):

Calculating Mass Defect

∆m = (7 * mass of proton + 7 * mass of neutron) - 13.999234 amu ∆m = (7 * 1.007276 + 7 * 1.008665) - 13.999234 ∆m = 0.113038 amu (b) Mass defect for Ti(22,48):

Calculating Mass Defect

∆m = (22 * mass of proton + 26 * mass of neutron) - 47.935878 amu ∆m = (22 * 1.007276 + 26 * 1.008665) - 47.935878 ∆m = 0.395689 amu (c) Mass defect for Xe(54,129):

Calculating Mass Defect

∆m = (54 * mass of proton + 75 * mass of neutron) - 128.904779 amu ∆m = (54 * 1.007276 + 75 * 1.008665) - 128.904779 ∆m = 2.193766 amu Next, we convert the mass defect into energy (binding energy) using E=∆m*c^2. As 1 amu is equal to 931.5 MeV/c², we can write E=∆m*931.5 MeV. (a) Binding energy for N(7,14):

Calculating Binding Energy

E = 0.113038 * 931.5 MeV E = 105.3375 MeV (b) Binding energy for Ti(22,48):

Calculating Binding Energy

E = 0.395689 * 931.5 MeV E = 368.4135 MeV (c) Binding energy for Xe(54,129):

Calculating Binding Energy

E = 2.193766 * 931.5 MeV E = 2044.45 MeV Finally, we calculate the binding energy per nucleon by dividing the binding energy by the number of nucleons. (a) Binding energy per nucleon for N(7,14):

Calculating Binding Energy per Nucleon

E/nucleon = 105.3375 MeV / 14 E/nucleon = 7.524 MeV (b) Binding energy per nucleon for Ti(22,48):

Calculating Binding Energy per Nucleon

E/nucleon = 368.4135 MeV / 48 E/nucleon = 7.675 MeV (c) Binding energy per nucleon for Xe(54,129):

Calculating Binding Energy per Nucleon

E/nucleon = 2044.45 MeV / 129 E/nucleon = 15.842 MeV Thus, the binding energy per nucleon is 7.524 MeV for N(7,14), 7.675 MeV for Ti(22,48), and 15.842 MeV for Xe(54,129).

Key concepts

Nuclear Mass

The nuclear mass refers to the total mass of the protons and neutrons in an atomic nucleus. This is often slightly less than the sum of their individual masses due to the energy binding them together, known as binding energy.
Understanding nuclear mass is crucial because it allows us to calculate the mass defect, an essential step in determining the binding energy per nucleon.

• In the nucleus, each proton and neutron contribute a specific amount of mass.
• The nuclear mass given in atomic mass units (amu), helps accurately calculate the mass defect.
• For nitrogen-14, the nuclear mass is 13.999234 amu. For titanium-48, it's 47.935878 amu, while for xenon-129, it's 128.904779 amu.

These values are pivotal for further calculations in nuclear physics.

Mass Defect

The mass defect is an intriguing concept in nuclear physics. It is the difference between the calculated sum of individual masses of protons and neutrons and the actual nuclear mass of an atom. The mass defect provides insight into the energy required to hold the nucleus of an atom together.
Let's break it down further:

• The mass defect is calculated by subtracting the nuclear mass from the sum of the masses of protons and neutrons.
• For example, the nitrogen-14 nucleus has a mass defect of 0.113038 amu.
• This difference arises due to the conversion of mass into binding energy during nucleus formation.

Understanding mass defect is crucial for grasping why nuclei are stable and the amount of energy needed for nuclear reactions.

Protons and Neutrons

Protons and neutrons are the building blocks of atomic nuclei, often referred to as nucleons. These subatomic particles play a vital role in the composition and stability of the nucleus.
Here are a few key points to remember:

• Protons are positively charged, while neutrons have no charge.
• The number of protons (Z) defines the element and its atomic number.
• Neutrons contribute to the mass and stability of the nucleus.
• For instance, nitrogen-14 has 7 protons and 7 neutrons. Titanium-48 contains 22 protons and 26 neutrons, and xenon-129 comprises 54 protons and 75 neutrons.

The balance of protons and neutrons is essential for nuclear stability and affects phenomena like radioactivity and isotope formation.

Energy Conversion

The principle of energy conversion is central to understanding nuclear binding energy, derived from Einstein's famous equation, \[E=\Delta m \cdot c^2 \]which shows how mass can be converted into energy. This concept is pivotal in nuclear physics and chemistry.
Here's a simple breakdown:

• The mass defect, when expressed in amu, can be converted to energy using the conversion factor: 1 amu = 931.5 MeV/c².
• The energy calculated is the binding energy, indicating how much energy holds the nucleus together.
• For nitrogen-14, the converted binding energy is 105.3375 MeV, which, when divided by the number of nucleons, yields the binding energy per nucleon of 7.524 MeV.

This conversion is key to understanding nuclear reactions, such as those occurring in the sun or nuclear reactors, where small amounts of mass can produce vast amounts of energy.