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Chapter 10: Problem 47

(a) Write the complete \(\beta^{-}\) decay equation for \(^{90} \mathrm{Sr}\), a major waste product of nuclear reactors. (b) Find the energy released in the decay.

Short Answer

Expert verified
The complete β⁻ decay equation for Strontium-90 is \(^{90}_{38}Sr => ^{90}_{39}Y + ^0_{-1}e + \bar{v}_e\). The energy released can be calculated using the mass-energy conservation principle and would be E=Δm.c², where Δm is the mass difference before and after the decay and c is the speed of light.

Step by step solution

01

Understand Beta Decay

Beta decay is a type of radioactive decay in which a beta particle, which is an electron or a positron, is emitted. In the case of β⁻ decay, a neutron in the nucleus transforms into a proton, an electron, and an electron antineutrino. This increases the atomic number by 1.
02

Write the Beta Decay Equation for Sr-90

The β⁻ decay equation for Strontium-90 can be written as follows: \(^{90}_{38}Sr => ^{90}_{39}Y + ^0_{-1}e + \bar{v}_e\) where \(^{90}_{38}Sr\) is Strontium-90, \(^{90}_{39}Y\) is Yttrium-90, ^0_{-1}e is the beta particle (electron), and \(\bar{v}_e\) is the electron antineutrino.
03

Calculate the Energy Released in the Decay

The energy released in the decay can be calculated using the principle of conservation of mass-energy, which states that the total energy before and after the decay remains the same. First, the difference in mass before and after the decay (Δm) needs to be calculated. This is given by Δm = (mass of \(^{90}_{38}Sr\)) - ((mass of \(^{90}_{39}Y\)) + mass of electron)). Then, the energy (E) released is given by the equation E=Δm.c², where c is the speed of light. The masses must be in equivalent energy units (such as MeV/c² or kg if c is given in m/s) while calculating Δm.

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Most popular questions from this chapter

Why is the number of neutrons greater than the number of protons in stable nuclei that have an \(A\) greater than about 40? Why is this effect more pronounced for the heaviest nuclei?

If two nuclei are to fuse in a nuclear reaction, they must be moving fast enough so that the repulsive Coulomb force between them does not prevent them for getting within \(R \approx 10^{-14} \mathrm{m}\) of one another. At this distance or nearer, the attractive nuclear force can overcome the Coulomb force, and the nuclei are able to fuse. (a) Find a simple formula that can be used to estimate the minimum kinetic energy the nuclei must have if they are to fuse. To keep the calculation simple, assume the two nuclei are identical and moving toward one another with the same speed \(v\). (b) Use this minimum kinetic energy to estimate the minimum temperature a gas of the nuclei must have before a significant number of them will undergo fusion. Calculate this minimum temperature first for hydrogen and then for helium. (Hint: For fusion to occur, the minimum kinetic energy when the nuclei are far apart must be equal to the Coulomb potential energy when they are a distance \(R\) apart.)

This problem demonstrates that the binding energy of the electron in the ground state of a hydrogen atom is much smaller than the rest mass energies of the proton and electron. (a) Calculate the mass equivalent in u of the \(13.6-\mathrm{eV}\) binding energy of an electron in a hydrogen atom, and compare this with the known mass of the hydrogen atom. (b) Subtract the known mass of the proton from the known mass of the hydrogen atom. (c) Take the ratio of the binding energy of the electron (13.6 eV) to the energy equivalent of the electron's mass (0.511 \(\mathrm{MeV}\) ). (d) Discuss how your answers confirm the stated purpose of this problem.

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