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Chapter 21: Problem 11

How does Einstein's equation, \(E=m c^{2},\) enable us to calculate nuclear binding energy?

Short Answer

Expert verified
Einstein's equation, \(E=mc^{2}\), is used to calculate nuclear binding energy by substituting the mass defect of a nucleus (the difference between the total mass of the neutrons and protons and the mass of the nucleus) into the formula. The resulting energy value, typically in Joules, is converted to electron volts.

Step by step solution

01

Understand Einstein's equation

Einstein's equation states that energy (\(E\)) is equal to mass (\(m\)) times the speed of light (\(c\)) squared (\(c^{2}\)). The equation implies that mass and energy are interchangeable.
02

Understand nuclear binding energy

Nuclear binding energy is the energy required to disassemble a nucleus into its constituent protons and neutrons. This energy can be calculated based on differences in the mass of the nucleus and its constituents.
03

Apply Einstein鈥檚 equation

To calculate the nuclear binding energy, one needs to know the mass of the nucleus (\(m_{nucleus}\)), and the sum of the individual neutrons and protons (\(m_{sum}\)). The mass defect (\(螖m\)) is determined by subtracting the mass of the nucleus from the mass sum (\(螖m = m_{sum} - m_{nucleus}\)). Then by using Einstein鈥檚 equation (\(E=mc^{2}\)), we substitute the mass defect into the formula to get the nuclear binding energy.
04

Conversion to appropriate energy unit

The result obtained from step 3 is usually in Joules (J). However, in nuclear physics, binding energy is often expressed in electron volts (eV). Hence, the energy obtained in Joules should be converted to electron volts using the conversion factor \(1 eV = 1.6 \times 10^{-19} J\).

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Most popular questions from this chapter

(a) Calculate the energy released when a \({ }^{238} \mathrm{U}\) isotope decays to \({ }^{234} \mathrm{Th} .\) The atomic masses are given by: \(^{238} \mathrm{U}: 238.0508 \mathrm{amu} ;{ }^{234} \mathrm{Th}: 234.0436 \mathrm{amu} ;{ }^{4} \mathrm{He}\) 4.0026 amu. (b) The energy released in (a) is transformed into the kinetic energy of the recoiling \({ }^{234} \mathrm{Th}\) nucleus and the \(\alpha\) particle. Which of the two will move away faster? Explain.

Why is it preferable to use nuclear binding energy per nucleon for a comparison of the stabilities of different nuclei?

Why is strontium-90 a particularly dangerous isotope for humans?

The radioactive isotope \({ }^{238} \mathrm{Pu},\) used in pacemakers, decays by emitting an alpha particle with a half-life of 86 yr. (a) Write an equation for the decay process. (b) The energy of the emitted alpha particle is \(9.0 \times 10^{-13} \mathrm{~J}\), which is the energy per decay. Assume that all the alpha particle energy is used to run the pacemaker, calculate the power output at \(t=0\) and \(t=10 \mathrm{yr}\). Initially \(1.0 \mathrm{mg}\) of \({ }^{238} \mathrm{Pu}\) was present in the pacemaker (Hint: After \(10 \mathrm{yr}\), the activity of the isotope decreases by 8.0 percent. Power is measured in watts or \(\mathrm{J} / \mathrm{s}\).).

To detect bombs that may be smuggled onto airplanes, the Federal Aviation Administration (FAA) will soon require all major airports in the United States to install thermal neutron analyzers. The thermal neutron analyzer will bombard baggage with low-energy neutrons, converting some of the nitrogen- 14 nuclei to nitrogen- \(15,\) with simultaneous emission of \(\gamma\) rays. Because nitrogen content is usually high in explosives, detection of a high dosage of \(\gamma\) rays will suggest that a bomb may be present. (a) Write an equation for the nuclear process. (b) Compare this technique with the conventional X-ray detection method.
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