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Chapter 29: Problem 62

What is the average binding energy per nucleon for ${ }_{11}^{23} \mathrm{Na} ?$

Short Answer

Expert verified
Question: Calculate the average binding energy per nucleon for the sodium isotope with atomic number 11 and mass number 23. Answer: To find the average binding energy per nucleon for the sodium isotope, follow these steps: 1. Calculate the mass defect: (11 * 1.007276 u + 12 * 1.008665 u) - 22.98977 u 2. Calculate the binding energy: Mass defect * 931.5 MeV/c^2 3. Calculate the average binding energy per nucleon: Binding energy / (11 protons + 12 neutrons)

Step by step solution

01

Gather given information

We are given the element as sodium (Na), with atomic number 11 and mass number 23.
02

Calculate the mass defect

The mass defect is the difference between the mass of a nucleus and the sum of masses of its individual protons and neutrons. Let's first find the masses of protons, neutrons, and the nucleus: Mass of 1 proton = 1.007276 u Mass of 1 neutron = 1.008665 u Mass of the nucleus of \(^{23}_{11}\mathrm{Na}\) = 22.98977 u (from periodic table) Now, we calculate the mass defect: Mass defect = (Mass of 11 protons + Mass of 12 neutrons) - Mass of the nucleus Mass defect = (11 * 1.007276 u + 12 * 1.008665 u) - 22.98977 u
03

Calculate the binding energy

The binding energy is the energy equivalent of the mass defect. To find it, we use the famous Einstein's equation, E=mc^2, where E is the energy, m is the mass, and c is the speed of light. In atomic mass units, we can use a conversion factor to find the energy in million electron volts (MeV) as follows: 1 u = 931.5 MeV/c^2 Now we can find the binding energy: Binding energy = Mass defect * 931.5 MeV/c^2
04

Calculate the average binding energy per nucleon

Finally, we'll find the average binding energy per nucleon by dividing the binding energy by the total number of nucleons (protons and neutrons): Average binding energy per nucleon = Binding energy / (Total number of nucleons) Average binding energy per nucleon = Binding energy / (11 protons + 12 neutrons)

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Most popular questions from this chapter

(a) What is the mass defect of the 'H atom due to the binding energy of the electron (in the ground state)? (b) Should we worry about this mass defect when we calculate the mass of the \(^{1} \mathrm{H}\) nucleus by subtracting the mass of one electron from the mass of the \(^{1} \mathrm{H}\) atom?
Radon gas (Rn) is produced by the \(\alpha\) decay of radium 226 88 28 A How many neutrons and how many protons are present in the nucleus of the isotope of Rn produced by this decay? (b) In the air in an average size room in a student basement apartment in Ithaca, NY, there are about \(10^{7}\) Rn nuclei. The Rn nucleus itself is radioactive; it too decays by emitting an \(\alpha\) particle. The half-life of Rn is 3.8 days. How many \(\alpha\) particles per second are emitted by decaying Rn nuclei in the room?
In this problem, you will verify the statement (in Section 29.4) that the \(^{14} \mathrm{C}\) activity in a living sample is \(0.25 \mathrm{Bq}\) per gram of carbon. (a) What is the decay constant \(\lambda\) for \(^{14} \mathrm{C} ?\) (b) How many \(^{14} \mathrm{C}\) atoms are in \(1.00 \mathrm{g}\) of carbon? One mole of carbon atoms has a mass of \(12.011 \mathrm{g},\) and the relative abundance of \(^{14} \mathrm{C}\) is \(1.3 \times 10^{-12} .\) (c) Using your results from parts (a) and (b), calculate the \(^{14} \mathrm{C}\) activity per gram of carbon in a living sample.
Identify the daughter nuclide when \(_{19}^{40} \mathrm{K}\) decays via \(\beta^{-}\) decay.
Calculate the maximum kinetic energy of the \(\beta\) particle when $_{19}^{40} \mathrm{K}\( decays via \)\beta^{-}$ decay.
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