Question

How much energy must be supplied to break a single \({ }^{21}\) Ne nucleus into separated protons and neutrons if the nucleus has a mass of 20.98846 amu? What is the nuclear binding energy for \(1 \mathrm{~mol}\) of \({ }^{21} \mathrm{Ne} ?\)

Short answer

The energy required to break a single Ne-21 nucleus into separated protons and neutrons is 167.312 MeV, and the nuclear binding energy for 1 mole of Ne-21 is \( 1.007 \times 10^{26} \) MeV.

Step-by-step solution

Calculate the mass of separated protons and neutrons

Ne-21 has an atomic number of 10 and a mass number of 21. This means there are 10 protons and 11 neutrons in its nucleus. The mass of a proton is approximately 1.007276 amu and the mass of a neutron is approximately 1.008665 amu. Mass of 10 protons = 10 * 1.007276 = 10.07276 amu Mass of 11 neutrons = 11 * 1.008665 = 11.095315 amu Total mass of separated protons and neutrons = 10.07276 + 11.095315 = 21.168075 amu

Calculate the mass defect of the nucleus

The mass defect of the nucleus can be calculated by subtracting the mass of the nucleus from the total mass of separated protons and neutrons. Mass defect = 21.168075 - 20.98846 = 0.179615 amu

Calculate the energy required to break the nucleus

We can use the mass-energy equivalence formula to find the energy required to break the nucleus: Energy = Mass defect * c^2 * conversion factor from amu to MeV where c is the speed of light and the conversion factor is 931.5 MeV/c^2. Energy = 0.179615 * (931.5 MeV/c^2) = 167.312 MeV

Convert the energy to nuclear binding energy for 1 mole of Ne-21

To find the nuclear binding energy for 1 mole of Ne-21, we can use Avogadro's number (6.022 x 10^23): Nuclear binding energy for 1 mole = 167.312 MeV * 6.022 x 10^23 = \( 1.007 \times 10^{26} \) MeV The energy required to break a single Ne-21 nucleus into separated protons and neutrons is 167.312 MeV, and the nuclear binding energy for 1 mole of Ne-21 is \( 1.007 \times 10^{26} \) MeV.

Key concepts

Mass Defect

The concept of mass defect is central to understanding nuclear stability and the energy that holds a nucleus together. Simply put, it is the difference in mass between a nucleus and its constituent protons and neutrons when they are separated.

In the context of the Ne-21 nucleus from the exercise, we calculate the mass defect by adding the mass of the individual protons and neutrons and then subtracting the mass of the nucleus itself. This difference represents the 'missing' mass, which has been converted into binding energy to hold the nucleus together, according to Einstein's mass-energy equivalence principle.

Mathematically, we expressed this as:

• Mass of separated constituents (protons and neutrons) - Mass of the nucleus = Mass defect.
• In numbers: 21.168075 amu (total mass of separate nucleons) - 20.98846 amu (mass of the nucleus) = 0.179615 amu (mass defect).

Understanding this mass defect is crucial because it directly relates to the nuclear binding energy, a measure of the nucleus's stability. The greater the mass defect, the higher the binding energy, indicating a more stable nucleus.

Mass-Energy Equivalence

Building on the idea of mass defect, mass-energy equivalence is a foundational concept in physics encapsulated by Einstein’s famous equation:

E = mc^2

Where

• E is energy,
• m is mass, and
• c is the speed of light in a vacuum.

This equation reveals that mass can be converted into energy and vice versa. In the nuclear realm, the mass defect represents the mass that has been converted into energy, binding the nucleus together. To determine the energy equivalent of the mass defect, we use the above formula, with a conversion factor to transition from atomic mass units (amu) to energy units like megaelectronvolts (MeV).

In our exercise's case, the mass defect is first calculated and then transformed into energy using the mass-energy equivalence formula. The result is the energy required to overcome the nucleus's binding energy and separate the Ne-21 nucleus into individual protons and neutrons, amounting to 167.312 MeV.

Avogadro's Number

When we operate with individual atoms or nuclei, we often want to scale up to macroscopic quantities to understand the bulk properties or equivalence in lab measurements. That’s where Avogadro's number comes into play. It signifies the number of atoms or molecules in one mole of a substance, which is precisely

6.022 x 10^23.

This quantity is named after the Italian scientist Amedeo Avogadro, who first realized the importance of the number of particles in a given amount of gas in determining its properties.

In our textbook exercise, after finding the binding energy for one Ne-21 nucleus, we multiply this value by Avogadro's number to find the total nuclear binding energy for one mole of Ne-21. This calculation helps us relate the microscopic energy scales of individual atomic nuclei to the macroscopic scales involved in chemical amounts regularly used in laboratories. For Ne-21, multiplying the binding energy (167.312 MeV) for one nucleus by Avogadro's number gives us an extensive energy value in mole scale, demonstrating the immense energy involved in nuclear binding.