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

Expert verified
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

01

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.
02

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
03

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
04

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
05

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.

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