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 nuclear binding energy for 1 mole of \({ }^{21}\mathrm{Ne}\) can be calculated by first finding the mass defect, which is the difference between the total mass of individual protons and neutrons and the actual mass of the nucleus. Then, convert the mass defect to energy using Einstein's mass-energy equivalence formula, and finally, multiply the energy of a single nucleus by Avogadro's number. Mass defect = ((10 × 1.00728 amu) + (11 × 1.00867 amu)) - 20.98846 amu Energy = Mass defect × \(931.5\) MeV Nuclear binding energy for 1 mole = Energy of a single nucleus × Avogadro's number \(≈ 6.022 \times 10^{23}\)

Step-by-step solution

Calculate the mass defect

The mass defect is the difference between the total mass of individual protons and neutrons and the actual mass of the nucleus. In the case of \({ }^{21}\mathrm{Ne}\), it has 10 protons and 11 neutrons. The mass of 1 proton is approximately \(1.00728\) amu and the mass of 1 neutron is approximately \(1.00867\) amu. So, the total mass of 10 protons and 11 neutrons can be calculated as follows: Total mass of individual particles = (10 × 1.00728 amu) + (11 × 1.00867 amu) Now, we can find the mass defect: Mass defect = Total mass of individual particles - Actual mass of the nucleus Mass defect = ((10 × 1.00728 amu) + (11 × 1.00867 amu)) - 20.98846 amu

Convert mass defect to energy

Now we can calculate the energy required to break a single \({ }^{21}\mathrm{Ne}\) nucleus into separated protons and neutrons using Einstein's mass-energy equivalence formula: Energy = Mass defect × c² In this problem, we'll use atomic mass units (amu) and convert it to energy using MeV (megaelectronvolt). 1 amu = \(931.5\) MeV. Energy = Mass defect × \(931.5\) MeV

Calculate nuclear binding energy for 1 mole of \({ }^{21}\mathrm{Ne}\)

Now that we have calculated the energy required to break a single \({ }^{21}\mathrm{Ne}\) nucleus, let's find the nuclear binding energy for 1 mole of \({ }^{21}\mathrm{Ne}\). We can do that by multiplying the energy of a single nucleus by Avogadro's number, which is approximately \(6.022 \times 10^{23}\). Nuclear binding energy for 1 mole = Energy of a single nucleus × Avogadro's number Now, combining all the steps above, we can calculate the nuclear binding energy for 1 mole of \({ }^{21}\mathrm{Ne}\).

Key concepts

Mass Defect

The concept of the mass defect lies at the heart of nuclear physics. It's the slight difference between the sum of the masses of a nucleus's separate protons and neutrons and its actual mass. You can think of it like this: when you purchase a jigsaw puzzle, the pieces are scattered, but once you put all the pieces together, they seem to take up less space. Similarly, when protons and neutrons combine to form a nucleus, the mass of the nucleus is a bit less than their total individual masses when separate.

To calculate the mass defect, we sum the masses of the individual nucleons (protons and neutrons) and subtract the mass of the nucleus. It may seem puzzling that mass can simply 'disappear', but this missing mass is actually the secret to the energy that holds the nucleus together, known as the 'binding energy'. The larger the mass defect, the more energy is holding the nucleus together, which contributes to its stability.

Einstein's Mass-Energy Equivalence

Einstein's mass-energy equivalence, encapsulated in the world-famous equation E=mc², is a central principle in modern physics. It tells us that mass can be converted into energy, and vice versa. In simple terms, even a tiny amount of mass can be turned into a tremendous amount of energy. This is the key to understanding nuclear processes.

In the context of our problem, when a nucleus is formed from protons and neutrons, the 'lost' mass (mass defect) gets converted into nuclear binding energy, which glues the nucleons together in the nucleus. When we calculate the energy needed to break apart a nucleus, we're essentially figuring out how much energy we'd have to put back in to reverse this conversion and turn the binding energy back into mass.

Atomic Mass Unit (amu)

The atomic mass unit (amu) is the standard unit of mass used to express atomic and molecular weights. It's incredibly tiny—about 1.66 x 10^-24 grams. One amu is defined as one twelfth of the mass of a carbon-12 atom, which has six protons and six neutrons.

Why is this useful? Because the amu provides a scale for comparing the masses of different atoms and molecules. This granular measure allows scientists to work with the microscopic world’s numbers in a practical, though still extraordinarily precise, way. In nuclear physics, it's crucial when we talk about mass defects. Since we are dealing with such minuscule changes in mass, representing them in amu makes the numbers more intuitive.

Avogadro's Number

Avogadro's number, approximately 6.022x10^23, is a fundamental constant and defines the number of constituent particles (usually atoms or molecules) in one mole of a substance. The mole is one of the base units in chemistry, providing a bridge between the atomic scale and the macroscopic scale.

When we talk about nuclear binding energy per mole, we're applying Avogadro's number to transition from the microscopic energy needed to break apart a single nucleus to the macroscopic energy for a mole of nuclei. This allows us to compare energies across different scales, enabling meaningful measurements and calculations in both laboratory and real-world applications. Whenever you're working with macroscopic quantities in chemistry or physics, Avogadro's number is indispensable for connecting our macro world with the atomic and molecular realms.