Question

The binding energy Per nucleon of \({ }_{8} \mathrm{O}^{16}\) is \(7.97 \mathrm{MeV}\) and that of \(_{8} \mathrm{O}^{17}\) is \(7.75 \mathrm{MeV}\) The energy \((\mathrm{in}-\mathrm{MeV})\) required to remove a neutron from \(_{8} \mathrm{O}^{17}\) is (A) \(3.65\) (B) \(7.86\) (C) \(3.52\) (D) \(4.23\)

Short answer

The energy required to remove a neutron from \(_{8}\mathrm{O}^{17}\) is \(4.23 \mathrm{MeV}\).

Step-by-step solution

Calculate total binding energies for both isotopes

We are given the binding energy per nucleon for both \(_{8}\mathrm{O}^{16}\) and \(_{8}\mathrm{O}^{17}\). We need to calculate the total binding energy for each isotope by multiplying the binding energy per nucleon by the total number of nucleons: Total binding energy for \(_{8}\mathrm{O}^{16}\): \(BE_{16} = 16 \times 7.97 \mathrm{MeV}\) Total binding energy for \(_{8}\mathrm{O}^{17}\): \(BE_{17} = 17 \times 7.75 \mathrm{MeV}\)

Find the difference in binding energies

Now we need to find the difference in total binding energies to find the energy required to remove a neutron from \(_{8}\mathrm{O}^{17}\): Energy required = Total binding energy of \(_{8}\mathrm{O}^{17}\) - Total binding energy of \(_{8}\mathrm{O}^{16}\) \(E = BE_{17} - BE_{16}\)

Calculate the energy required to remove the neutron

Substitute the values from Step 1 into the equation from Step 2: \(E = (17 \times 7.75 \mathrm{MeV}) - (16 \times 7.97 \mathrm{MeV})\) Now perform the calculations to find the energy required to remove the neutron: \(E = 131.75 \mathrm{MeV} - 127.52 \mathrm{MeV} = 4.23 \mathrm{MeV}\)

Choose the correct option

Now that we have determined the energy required to remove a neutron from \(_{8}\mathrm{O}^{17}\) is \(4.23 \mathrm{MeV}\), we can choose the correct option from the given choices. In this case, the correct answer is: (D) \(4.23 \mathrm{MeV}\)

Key concepts

Isotopes

Isotopes are different forms of the same element that have the same number of protons but different numbers of neutrons. This means that isotopes of an element have the same atomic number but different mass numbers. For example, in the original exercise, we have oxygen isotopes:

• \(_{8}\mathrm{O}^{16}\), which has 8 protons and 8 neutrons.
• \(_{8}\mathrm{O}^{17}\), which has 8 protons and 9 neutrons.

Despite these differences in neutrons, isotopes still have similar chemical properties because they rely on electrons and protons in the atom. However, their nuclei can behave differently, which is why they have different physical properties such as stability and radioactive decay rates.
Isotopes are crucial in various applications, including medical imaging, archaeological dating, and energy production. Understanding how isotopes differ is essential for students grappling with nuclear chemistry and physics.

Nuclear Physics

Nuclear physics is a branch of physics that studies the constituents and interactions of atomic nuclei. It encompasses a variety of phenomena, such as nuclear decay, fission, and fusion. In the context of the problem we are exploring:

• Nuclear processes are heavily influenced by nuclear binding energy, which is the energy required to separate a nucleus into its component protons and neutrons.
• The binding energy per nucleon is a useful way to understand the stability of a nucleus; higher binding energy per nucleon generally implies a more stable nucleus.
• Understanding binding energies helps scientists to comprehend various processes, like the ones in stars, and why certain isotopes are more prevalent in nature.

Students should view nuclear physics not just as numbers and equations, but as a fundamental explanation of why matter behaves the way it does. From supporting life on Earth through radioactive processes to paving the way for advances in medical treatments, nuclear physics is integral to life as we know it.

Neutron Removal Energy

Neutron removal energy is the energy required to detach a neutron from an atomic nucleus. In the given exercise, we calculate this energy for an isotope of oxygen, \(_{8} \mathrm{O}^{17}\). Here's how to think about it:

• The total binding energy of a nucleus keeps the nucleus intact by holding nucleons together.
• When a neutron is removed, the change in total binding energy provides the energy required for this process.
• In simpler terms, it's the energy it takes to overcome the nuclear forces keeping the neutron bound in the nucleus.

For the oxygen isotope problem, the energy needed to remove the neutron from \(_{8} \mathrm{O}^{17}\) depends on the difference in total binding energy with and without the neutron. This particular exercise shows how specific calculations can highlight general principles governing subatomic particles. Understanding neutron removal energy is thus essential for exploring concepts related to nuclear stability and reactions.