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 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 above two decays. (The mass of the electron is \(5.4857 \times 10^{-4}\) amu. \()\) (b) Starting with one mole of \({ }^{90} \mathrm{Sr}\), calculate the number of moles of \({ }^{90} \mathrm{Sr}\) 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 \((\mathrm{b})\)

Understanding mass defect in nuclear reactions is crucial for grasping the principles of nuclear energy and stability. When it comes to nuclear decay, like in the decay of Sr-90 to Y-90, not all of the original mass is conserved as mass. Some of it is converted into energy, according to Einstein's famous equation, \( E = mc^2 \). The mass defect is the difference in mass between the original nucleus and the resulting particles, including any emitted beta particles.

The precise calculation of the mass defect involves determining the mass of the original atom and subtracting the masses of the resulting atoms and any emitted particles. We account for everything, down to the mass of electrons, to ensure accuracy. In our textbook example, the decay of Sr-90 to Y-90 and then to Zr-90, we would determine the mass defect by subtracting the total mass of the decay products from the original mass of the Sr-90. This information allows us to calculate how much energy is released during the decays.

In step-by-step fashion:

• Find the initial and final masses of the reaction.
• Calculate the mass defect by subtracting the sum of the final masses from the original mass.
• Convert the mass defect from atomic mass units (amu) to kilograms, taking into account the conversion factor (1 amu equals \(1.66054 \times 10^{-27}\) kg).
• Apply Einstein's energy-mass equivalence to find the energy released.

Keep in mind that the mass of an electron, though seemingly trivial, must also be included to ensure precision in our calculations for both decays.

Radioactive decay is a random process at the level of single atoms, yet it exhibits a predictable pattern over a large number of nuclei, described by a property known as half-life. The half-life of a radioactive isotope is the time it takes for half of the sample to decay. This fundamental concept allows us to predict the decay pattern over time, which is essential for nuclear waste management, radiocarbon dating, and nuclear medicine.

For instance, if we start with one mole of Sr-90, its half-life of 28.1 years tells us that after this period, only half of the original moles will remain. The decay continues such that after another 28.1 years, only a quarter of the original substance would be left, and so on. By applying the formula \(N_t = N_0(0.5)^{\frac{t}{t_{1/2}}}\), we can predict the amount of radioactive material remaining after any given time.

To connect this concept to our earlier exercise:

• Start with the initial quantity of the substance (1 mole in our case).
• Use the half-life of the substance to calculate the fraction remaining after a specific time (1 year for our example).
• From this, we can deduce the fraction that has decayed, which is necessary for calculating the heat release in the final step.

Understanding half-life provides a framework for predicting how much radioactive material will remain after a certain period, which is crucial in a variety of applications.

The energy release in nuclear decay is an awe-inspiring display of nature's power. Every time an unstable nucleus sheds its excess energy to become more stable, through processes like alpha decay, beta decay, or gamma emission, a certain amount of energy is released. This energy can be calculated accurately using the mass defect concept we discussed earlier and is foundational to nuclear physics and chemistry, as well as practical applications like nuclear power generation.

As derived from Einstein's famous equation, the energy released (\(E\)) from the decay is directly proportional to the mass defect (\(\Delta m\)), multiplied by the speed of light squared (\(c^2\)). To complete our example with Sr-90:

• Calculate the total energy released for each decay from Sr-90 to Y-90, and then to Zr-90.
• Multiply by Avogadro’s number to calculate the energy released for one mole of Sr-90 decay.
• Calculate the heat generated using the number of moles that decayed over a year.

Regarding the heat release for the decayed Sr-90, it's important to see energy in the broader context of its effects on the surroundings. The decay not only results in stable isotopes but also releases heat, impacting the environment where nuclear waste is stored. Managing this heat output is a critical aspect of nuclear waste disposal. Through these insights into the energy released, we gain a deeper understanding of the power of nuclear decay and the importance of safely harnessing and storing the resulting materials.