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Chapter 40: Problem 63

\(^{8} \mathrm{Li}\) is an isotope that has a lifetime of less than one second. Its mass is \(8.022485 \mathrm{u} .\) Calculate its binding energy in \(\mathrm{MeV}\).

Short Answer

Expert verified
Based on the step-by-step solution provided, provide a short answer to the question: To calculate the binding energy of Lithium-8, first determine the number of protons and neutrons, which are 3 and 5, respectively. Then, find the total mass of protons and neutrons using their respective masses in atomic mass units. Next, calculate the mass defect by subtracting the mass of the Lithium-8 nucleus from the sum of the masses of its individual protons and neutrons. Finally, convert the mass defect to energy using the mass-energy equivalence formula E=mc² and the conversion factor 1u = 931.4941 MeV/c². This will give you the binding energy of Lithium-8 in MeV.

Step by step solution

01

Determine the number of protons and neutrons

As we are given Lithium-8, it has 8 nucleons in total, with 3 protons (since Lithium has an atomic number of 3) and the rest being neutrons. So there are 5 neutrons in the nucleus.
02

Calculate the mass of individual protons and neutrons

Knowing the atomic mass unit, which is approximately \(1.660539 \times 10^{-27} \mathrm{kg}\) and using the mass of a proton (\(1.007276 \mathrm{u}\)) and the mass of a neutron (\(1.008665 \mathrm{u}\)), we can calculate the mass of 3 protons and 5 neutrons. Total mass of protons: \(3 \times 1.007276 \mathrm{u}\) Total mass of neutrons: \(5 \times 1.008665 \mathrm{u}\)
03

Calculate the mass defect

Now, we will find the mass defect by subtracting the mass of the Lithium-8 nucleus from the sum of the masses of its individual protons and neutrons. Mass defect = (Total mass of protons + Total mass of neutrons) - Mass of Lithium-8 Mass defect = \((3 \times 1.007276 \mathrm{u} + 5 \times 1.008665 \mathrm{u}) - 8.022485 \mathrm{u}\)
04

Convert mass defect to energy

Using the mass-energy equivalence formula, E=mc², where 'E' is energy, 'm' is the mass defect, and 'c' is the speed of light (\(2.99792458 \times 10^8 \mathrm{m/s}\)). We will also use the conversion factor \(1 \mathrm{u} = 931.4941 \mathrm{MeV/c^2}\) in our calculation. Energy = Mass defect × c² Energy = Mass defect × (\(931.4941 \mathrm{MeV/c^2}\))
05

Calculate the binding energy

Now, plug in the mass defect from step 3 into the energy equation from step 4 and solve for the binding energy. Binding energy = (Mass defect) × (931.4941 MeV/c²) Make sure to include units in the final answer.

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