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Chapter 19: Problem 85

Given the following information: Mass of proton \(=1.00728 \mathrm{u}\) Mass of neutron \(=1.00866 \mathrm{u}\) Mass of electron \(=5.486 \times 10^{-4} \mathrm{u}\) Speed of light \(=2.9979 \times 10^{8} \mathrm{m} / \mathrm{s}\) Calculate the nuclear binding energy of \(_{12}^{24} \mathrm{Mg},\) which has an atomic mass of 23.9850 \(\mathrm{u}\) .

Short Answer

Expert verified
The nuclear binding energy of \(_{12}^{24} \mathrm{Mg}\) can be found by first calculating the mass defect, which is the difference between the sum of individual masses of protons, neutrons, and electrons and the actual atomic mass. Convert the mass defect to kg, then use Einstein's mass-energy equivalence equation, \(E=mc^2\), to find the nuclear binding energy in Joules. Finally, convert the nuclear binding energy to MeV by dividing the energy in Joules by 1.602 × 10^{-13} Joules/MeV.

Step by step solution

01

Find the number of protons, neutrons, and electrons

Since it is \(_{12}^{24} \mathrm{Mg}\), it has 12 protons, and as the atomic mass number is 24, the number of neutrons is 24 - 12 = 12. As it is a neutral atom, it also has 12 electrons.
02

Calculate the sum of individual masses

Using the given values for the mass of proton, neutron, and electron, we can calculate the sum of the individual masses. Sum of individual masses = (12 protons × mass of proton) + (12 neutrons × mass of neutron) + (12 electrons × mass of electron) Sum of individual masses = (12 × 1.00728 u) + (12 × 1.00866 u) + (12 × 5.486 × 10^{-4} u)
03

Calculate the mass defect

Subtract the given atomic mass of \(_{12}^{24} \mathrm{Mg}\) from the sum of individual masses to find the mass defect. Mass defect = Sum of individual masses - Atomic mass
04

Convert the mass defect in kilograms

We need to convert the mass defect from atomic mass units (u) to kilograms (kg) before we apply Einstein's mass-energy equivalence equation. One atomic mass unit (u) is equal to \(1.660539 \times 10^{-27}\) kg. Mass defect (kg) = Mass defect (u) × 1.660539 × 10^{-27} kg/u
05

Calculate the nuclear binding energy

Now, we have the mass defect in kilograms, and we can apply Einstein's mass-energy equivalence equation, \(E=mc^2\), to find the nuclear binding energy. Nuclear binding energy = Mass defect (kg) × (Speed of light)^2 Nuclear binding energy (Joules) = Mass defect (kg) × (2.9979 × 10^8 m/s)^2
06

Convert nuclear binding energy to MeV

Finally, convert the nuclear binding energy from Joules to mega electron volts (MeV). One electron volt (eV) is equal to 1.602 × 10^{-19} Joules, so 1 MeV = 1.602 × 10^{-13} Joules. Nuclear binding energy (MeV) = Nuclear binding energy (Joules) / 1.602 × 10^{-13} Joules/MeV

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Most popular questions from this chapter

Many transuranium elements, such as plutonium-232 , have very short half- lives. (For \(^{232} \mathrm{Pu}\) , the half-life is 36 minutes.) However, some, like protactinium- 231 (half-life \(=3.34 \times 10^{4}\) years), have relatively long half-lives. Use the masses given in the following table to calculate the change in energy when 1 mole of \(^{232} \mathrm{Pu}\) nuclei and 1 mole of \(^{231} \mathrm{Pa}\) nuclei are each formed from their respective number of protons and neutrons. (Since the masses of \(^{232} \mathrm{Pu}\) and \(^{231} \mathrm{Pa}\) are atomic masses, they each include the mass of the electrons present. The mass of the nucleus will be the atomic mass minus the mass of the electrons.)

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