The nuclear masses of 7Be, 9Be, and 10Be are 7.0147, 9.0100, and 10.0113 amu, respectively. Which of these nuclei has the largest binding energy per nucleon?

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Step 1: The binding energy of a nucleus is the energy required to disassemble a nucleus into its constituent protons and neutrons. It is equivalent to the mass defect of the nucleus, which is the difference between the mass of the nucleus and the sum of the masses of its individual protons and neutrons.

Step 2: To find the binding energy, we first need to calculate the mass defect. The mass defect (Δm) can be calculated using the formula Δm = Zm_p + Nm_n - M, where Z is the number of protons, m_p is the mass of a proton, N is the number of neutrons, m_n is the mass of a neutron, and M is the mass of the nucleus.

Step 3: For each isotope of Beryllium (7Be, 9Be, and 10Be), calculate the mass defect using the given nuclear masses and the known masses of a proton (1.0073 amu) and a neutron (1.0087 amu).

Step 4: Convert the mass defect from atomic mass units (amu) to energy using Einstein's equation E=mc^2, where E is energy, m is mass, and c is the speed of light. Note that 1 amu is equivalent to 931.5 MeV/c^2.

Step 5: Finally, to find the binding energy per nucleon, divide the total binding energy by the number of nucleons (protons + neutrons) in the nucleus. The nucleus with the largest binding energy per nucleon is the most stable.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Binding Energy

Binding energy is the energy required to disassemble a nucleus into its constituent protons and neutrons. It is a measure of the stability of a nucleus; the higher the binding energy, the more stable the nucleus. This energy arises from the strong nuclear force that holds the nucleons together, overcoming the repulsive electromagnetic force between protons.

Binding Energy per Nucleon

Binding energy per nucleon is calculated by dividing the total binding energy of a nucleus by the number of nucleons (protons and neutrons) it contains. This value provides a normalized measure of nuclear stability, allowing for comparison between different nuclei. A higher binding energy per nucleon indicates a more stable nucleus, which is crucial for understanding nuclear reactions and stability.

Mass Defect

Mass defect refers to the difference between the mass of a nucleus and the sum of the individual masses of its constituent nucleons. This 'missing' mass is converted into binding energy according to Einstein's equation, E=mc². Understanding mass defect is essential for calculating the binding energy and, consequently, the binding energy per nucleon for different isotopes.

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In 2010, a team of scientists from Russia and the United States reported creation of the first atom of element 117, which is named tennessine, and whose symbol is Ts. The synthesis involved the collision of a target of ²⁴⁹₉₇Bk with accelerated ions of an isotope which we will denote Q. The product atom, which we will call Z, immediately releases neutrons and forms ²⁹⁴₁₁₇Ts: ²⁴⁹₉₇Bk + Q → Z → ²⁹⁴₁₁₇Ts + 3 ¹₀n (b) Isotope Q is unusual in that it is very long-lived (its half-life is on the order of 1019 yr) in spite of having an unfavorable neutron-to-proton ratio (Figure 21.1). Can you propose a reason for its unusual stability?

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