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Chapter 19: Problem 51

Calculate the binding energy in J/nucleon for carbon-12 (atomic mass \(=12.0000\) u) and uranium-235 (atomic mass \(=\) 235.0439 u). The atomic mass of \(_{1}^{1} \mathrm{H}\) is 1.00782 \(\mathrm{u}\) and the mass of a neutron is 1.00866 u. The most stable nucleus known is \(^{56}\) Fe $(\text { see Exercise } 50)\( . Would the binding energy per nucleon for \)^{56} \mathrm{Fe}$ be larger or smaller than that of \(^{12} \mathrm{C}\) or \(^{235} \mathrm{U}\) ? Explain.

Short Answer

Expert verified
For carbon-12, the mass defect is 0.09876 u, and the binding energy per nucleon is 6.466 × 10^{-13} J/nucleon. For uranium-235, the mass defect is 6.02942 u, and the binding energy per nucleon is 1.264 × 10^{-12} J/nucleon. The binding energy per nucleon for iron-56 is 1.407 × 10^{-12} J/nucleon, which is larger than that of carbon-12 and smaller than that of uranium-235.

Step by step solution

01

Determine the mass defect

For carbon-12, which has 6 protons and 6 neutrons, and uranium-235, which has 92 protons and 143 neutrons, we need to determine the mass defects. We will use the following formula: Mass defect = (Mass of protons + Mass of neutrons) - Atomic mass For carbon-12: Mass defect = ((6 × 1.00782 u) + (6 × 1.00866 u)) - 12.0000 u For uranium-235: Mass defect = ((92 × 1.00782 u) + (143 × 1.00866 u)) - 235.0439 u Calculate the mass defects for both elements.
02

Calculate the binding energy

Next, we will use the mass defects to find the binding energy of both elements. To do this, we have to use the famous Einstein's equation, E=mc², where E is the binding energy, m is the mass defect, and c is the speed of light (approximately 299,792,458 m/s). To convert from atomic mass units (u) to kilograms (kg), use the conversion factor of 1 u = 1.66054 × 10^{-27} kg. For carbon-12: E = ((mass defect in u) × (1.66054 × 10^{-27} kg/u)) × (299,792,458 m/s)² For uranium-235: E = ((mass defect in u) × (1.66054 × 10^{-27} kg/u)) × (299,792,458 m/s)² Calculate the binding energies for both elements.
03

Find the binding energy per nucleon

Now that we have the binding energies, we can find the binding energy per nucleon by dividing the total binding energy by the number of nucleons (protons + neutrons). For carbon-12: Binding energy per nucleon = total binding energy / 12 For uranium-235: Binding energy per nucleon = total binding energy / 235 Calculate the binding energy per nucleon for both elements in J/nucleon.
04

Compare the binding energy per nucleon

Finally, we can compare the binding energy per nucleon of carbon-12 and uranium-235 with the most stable nucleus, iron-56. The binding energy per nucleon for iron-56 is approximately 8.790 MeV/nucleon or 1.407 × 10^{-12} J/nucleon. Compare the binding energy per nucleon values for carbon-12, uranium-235, and iron-56 to determine if the binding energy per nucleon for iron-56 is larger or smaller than that of the other two elements.

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Most popular questions from this chapter

Scientists have estimated that the earth's crust was formed 4.3 billion years ago. The radioactive nuclide \(176 \mathrm{Lu},\) which decays to 176 \(\mathrm{Hf}\) , was used to estimate this age. The half-life of 176 \(\mathrm{Lu}\) is 37 billion years. How are ratios of \(^{176} \mathrm{Lu}\) to 176 \(\mathrm{Hf}\) utilized to date very old rocks?

A living plant contains approximately the same fraction of carbon-14 4 as in atmospheric carbon dioxide. Assuming that the observed rate of decay of carbon-14 4 from a living plant is 13.6 counts per minute per gram of carbon, how many counts per minute per gram of carbon will be measured from a \(15,000\) -year-old sample? Will radiocarbon dating work well for small samples of 10 \(\mathrm{mg}\) or less? (For \(^{14} \mathrm{C}, t_{1 / 2}=5730\) years.)

A small atomic bomb releases energy equivalent to the deto- nation of \(20,000\) tons of TNT; a ton of TNT releases \(4 \times 10^{9} \mathrm{J}\) of energy when exploded. Using \(2 \times 10^{13} \mathrm{J} / \mathrm{mol}\) as the energy released by fission of \(^{235} \mathrm{U}\) , approximately what mass of \(^{235} \mathrm{U}\) undergoes fission in this atomic bomb?

Rubidium- 87 decays by \(\beta\) -particle production to strontium- 87 with a half-life of \(4.7 \times 10^{10}\) years. What is the age of a rock sample that contains 109.7 \mug of \(^{87} \mathrm{Rb}\) and 3.1\(\mu \mathrm{g}\) of $^{87} \mathrm{Sr} ?\( Assume that no \)^{87}$ Sr was present when the rock was formed. The atomic masses for \(^{87}\mathrm{Rb}\) and \(^{87} \mathrm{Sr}\) are 86.90919 \(\mathrm{u}\) and 86.90888 u, respectively.

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 \(\mathrm{u}\) .
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