Nuclear waste disposal is one of the major concerns of the nuclear industry. In choosing a safe and stable environment to store nuclear wastes, consideration must be given to the heat released during nuclear decay. As an example, consider the \(\beta\) decay of \({ }^{90} \mathrm{Sr}\) \((89.907738 \mathrm{amu})\) $$ { }_{38}^{90} \mathrm{Sr} \longrightarrow{ }_{39}^{90} \mathrm{Y}+{ }_{-1}^{0} \beta \quad t_{\frac{1}{2}}=28.1 \mathrm{yr} $$ The \({ }^{90} \mathrm{Y}(89.907152 \mathrm{amu})\) further decays as follows: $$ { }_{39}^{90} \mathrm{Y} \longrightarrow{ }_{40}^{90} \mathrm{Zr}+{ }_{-1}^{0} \beta \quad t_{\frac{1}{2}}=64 \mathrm{~h} $$ Zirconium-90 (89.904703 amu) is a stable isotope. (a) Use the mass defect to calculate the energy released (in joules) in each of the preceding two decays. (The mass of the electron is \(5.4857 \times\) \(10^{-4}\) amu. ( b) Starting with 1 mole of \({ }^{90}\) Sr, calculate the number of moles of \(9^{9}\) Sr that will decay in a year. (c) Calculate the amount of heat released (in kilojoules) corresponding to the number of moles of \({ }^{90} \mathrm{Sr}\) decayed to \({ }^{90} \mathrm{Zr}\) in \((\mathrm{b})\)

Step by step solution

01

Calculate Energy Release Using Mass Defect

To find the energy released during the decay, use the mass defect equation \( \Delta E = \Delta m \times c^2 \), where \( \Delta m \) is the mass defect which equals to the mass of reactants minus the mass of products, and \( c \) is the speed of light in vacuum. The energy will be converted to joules.

02

Compute the Number of Moles of Sr-90 Decay in a Year

This calculation will be based on the half-life of Sr-90. Use the equation \( N = N_0 \times (0.5) ^ {(t/t_{1/2})} \) where \( N \) is the decayed amount, \( N_0 \) is the initial amount, \( t \) is the time passed, and \( t_{1/2} \) is the half-life. The time should be given in years.

03

Compute Heat Released

Finally, the amount of heat released is the energy release per decay times the number of decays. If we use Avogadro's number, we get the energy release in Joules that is first converted to kJ (since 1 kJ is 1000 Joules) and then this amount of energy is mulitplied by the number of moles decayed (from step 2) to find the total amount of heat released.

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