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 \(20.98846 \mathrm{u}\) ? What is the nuclear binding energy for \(1 \mathrm{~mol}\) of \({ }^{21} \mathrm{Ne}\) ?

Short answer

The energy needed to break a single \({ }^{21}\mathrm{Ne}\) nucleus into separated protons and neutrons is 167.3 MeV. The nuclear binding energy for 1 mole of \({ }^{21}\mathrm{Ne}\) is approximately \(1.007 × 10^{26}\) MeV/mol.

Step-by-step solution

Determine the number of protons and neutrons in the nucleus

The given neon isotope is \({ }^{21}\mathrm{Ne}\). From the notation, we can identify that it has an atomic number (number of protons) of 10, as neon normally has 10 protons. To find the number of neutrons in the isotope, subtract the atomic number from the mass number: Number of Neutrons = Mass number - Atomic number = 21 - 10 = 11 neutrons So, the \({ }^{21}\mathrm{Ne}\) nucleus contains 10 protons and 11 neutrons.

Calculate the total mass of separate protons and neutrons

Using the known atomic mass unit (u) values for a single proton and neutron, find the total mass of separate protons and neutrons. Mass of a proton = 1.007276 u Mass of a neutron = 1.008665 u Total mass of 10 protons = 10 × 1.007276 u = 10.07276 u Total mass of 11 neutrons = 11 × 1.008665 u = 11.095315 u Total mass of separate protons and neutrons = 10.07276 u + 11.095315 u = 21.168075 u

Calculate the mass defect

The mass defect is the difference between the mass of the separate protons and neutrons and the mass of the given neon nucleus. Mass defect = Total mass of separate protons and neutrons - Mass of \({ }^{21}\mathrm{Ne}\) nucleus Mass defect = 21.168075 u - 20.98846 u = 0.179615 u

Convert the mass defect to energy

To compute the energy, needed to break the nucleus into separate protons and neutrons, we use Einstein's mass-energy equivalence equation: Energy = Mass defect × c^2 Here, c is the speed of light, which is approximately \(3 × 10^8\) m/s. Since we are calculating energy in atomic mass units (u), we should convert the speed of light into MeV/c²: 1 u = 931.5 MeV/c² Energy = 0.179615 u × 931.5 MeV/c² = 167.3 MeV

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

The energy calculated in the previous step is for a single nucleus. To find the nuclear binding energy for 1 mole of \({ }^{21}\mathrm{Ne}\), multiply the energy per nucleus by Avogadro's number, approximately \(6.022 × 10^{23}\) particles/mol: Nuclear binding energy for 1 mole = 167.3 MeV/nucleus × \(6.022 × 10^{23}\) nuclei/mol Nuclear binding energy for 1 mole = \(1.007 × 10^{26}\) MeV/mol So, the nuclear binding energy for 1 mole of \({ }^{21}\mathrm{Ne}\) is approximately \(1.007 × 10^{26}\) MeV/mol.

Key concepts

Mass Defect

The mass defect is a key concept in nuclear physics that helps us understand how energy is stored in atomic nuclei. When protons and neutrons come together to form a nucleus, there is a tiny discrepancy between the combined weight of these particles compared to when they are separate. This difference in mass is known as the mass defect. To calculate it, we subtract the actual mass of the nucleus from the total mass of the separate protons and neutrons. For example, in the case of the \(^{21}\text{Ne} \) nucleus, the total mass of 10 protons and 11 neutrons is calculated as 21.168075 atomic mass units (u), while the mass of the nucleus itself is 20.98846 u. Subtracting these gives a mass defect of 0.179615 u. This mass defect is not just lost but converted into energy, according to Einstein's mass-energy equation \(E=mc^2\). Here, the energy equivalent to the mass defect illustrates the binding energy necessary to keep the nucleus intact.

Neon Isotope

In chemistry and physics, isotopes are forms of an element that have the same number of protons but differ in the number of neutrons. When we look at the \(^{21}\text{Ne}\) isotope, we see that it is a particular form of the element neon. This isotope of neon has:

• 10 protons, which defines it as neon in the periodic table.
• 11 neutrons, giving us a mass number of 21 (10 protons + 11 neutrons).

Isotopes vary in stability and can have different properties. The nuclear binding energy of an isotope like \(^{21}\text{Ne}\) tells us about the energy it takes to disassemble the nucleus into separate protons and neutrons. This energy also provides insights into the isotope’s stability. More binding energy typically means a more stable isotope.

Avogadro's Number

Avogadro's number is a fundamental constant that is pivotal in chemistry and physics. It serves as a bridge between the macroscopic and microscopic worlds, allowing us to count atoms by weighing them. Avogadro's number is approximately \(6.022 \times 10^{23}\) and tells us how many particles are in one mole of a substance.In the nuclear binding energy exercise, Avogadro's number helps us convert the energy of a single \(^{21}\text{Ne}\) nucleus into the energy for a mole of such nuclei. By multiplying the single nucleus energy (in this case, 167.3 MeV) by Avogadro's number, we calculate the binding energy of one mole of nuclei. This amount, about \(1.007 \times 10^{26}\) MeV/mol, represents how much energy would be required to completely break apart one mole of \(^{21}Ne\) nuclei into individual protons and neutrons. This calculation underscores the vast scale of nuclear energy compared to everyday chemical reactions.