Question

Strontium-90 is a dangerous byproduct of atomic testing because it mimics the action of calcium in the body. It decays in two beta emissions to give zirconium- 90 (Nuclear mass \(=89.8824 \mathrm{~g}\) ). (a) Write a balanced nuclear reaction for the overall decay of Sr-90. (b) Calculate \(\Delta m\) in grams when one mole of Sr-90 decays to \(\mathrm{Zr}-90 .\) (c) How much energy (in kilojoules) is given off by the decay of \(6.50 \mathrm{mg}\) of \(\mathrm{Sr}-90 ?\)

Short answer

Answer: The energy released by the decay of 6.50 mg of Strontium-90 is approximately 7.157 x 10⁻²⁰ kJ.

Step-by-step solution

Write a balanced nuclear reaction for Sr-90 decay

Strontium-90 undergoes two consecutive beta emissions to form Zirconium-90. In beta emission, an atom releases a beta particle, which is essentially an electron (e⁻) originating from the nucleus. One neutron in the nucleus transforms into a proton, and the atom's mass number does not change, but its atomic number increases by 1. 1. First beta emission: Sr-90 → Y-90 + e⁻ 2. Second beta emission: Y-90 → Zr-90 + e⁻ Combining both reactions: Sr-90 → Zr-90 + 2e⁻

Calculate the decrease in mass when one mole of Sr-90 decays to Zr-90

We need to find the change in mass for the decay process. Given: Nuclear mass of Zirconium-90 (Zr-90) = 89.8824 g/mol Mass of Sr-90 = Mass of Zr-90 + (2 × Mass of electron) We know that the mass of an electron is 0.000548597 g/mol. Mass of Sr-90 = 89.8824 g/mol + (2 × 0.000548597 g/mol) = 89.8835 g/mol Now we can calculate the decrease in mass: Δm = Mass of Sr-90 - Mass of Zr-90 Δm = 89.8835 g/mol - 89.8824 g/mol = 0.0011 g/mol

Calculate the energy released by the decay of 6.50 mg of Sr-90

We will now use the Einstein's mass-energy relation to determine the energy released: E = mc² Here, E = Energy released m = Change in mass c = speed of light in a vacuum (3.00 × 10^8 m/s) First, we need to find the energy released by the decay of 1 mole of Sr-90: E = (0.0011 g/mol) × (3.00 x 10^8 m/s)² E ≈ 9.899 × 10⁻¹³ J/mol Now, we need to find the energy released by the decay of 6.50 mg of Sr-90: Number of moles of Sr-90 = (6.50 mg) / (89,883.5 g/mol) = 7.232 × 10⁻⁵ mol Energy released = (7.232 × 10⁻⁵ mol) × (9.899 × 10⁻¹³ J/mol) Energy released ≈ 7.157 × 10⁻¹⁷ J Since we need the energy in kilojoules, we will convert the energy: Energy released = 7.157 × 10⁻¹⁷ J × (1 kJ / 10^3 J) = 7.157 × 10⁻²⁰ kJ So, the energy released by the decay of 6.50 mg of Sr-90 is approximately 7.157 x 10⁻²⁰ kJ.

Key concepts

Beta Emission

Beta emission is a type of radioactive decay wherein a beta particle, which is a high-energy, high-speed electron or positron, is emitted from an atomic nucleus. This occurs when a neutron in the nucleus is transformed into a proton and an electron. The electron, which is ejected from the nucleus, is the beta particle. Understanding beta decay is vital as it allows students to grasp how certain elements can transmute into others while releasing energy and particles in the process.

In the case of strontium-90, it decays via beta emission, leading to an increase in its atomic number by one. This transformation takes place because one neutron in the nucleus of the strontium atom changes into a proton, which remains in the nucleus, and an electron, which is released. The electron's emission is what we observe as beta radiation. It's important for students to recognize that the mass number of the element does not change during beta decay, only the atomic number does, due to the relative masslessness of the electron in comparison to the nucleus.

Nuclear Reaction Balancing

Balancing a nuclear reaction is much like balancing a chemical equation. However, instead of balancing just the number of each type of atom, we must balance both the number of protons and the number of neutrons, which together account for an atom's mass number and atomic number. When strontium-90 undergoes beta decay, it transforms into yttrium-90, which subsequently decays into zirconium-90. Each beta decay step increases the atomic number by 1 but the mass number remains unchanged.

By writing out the nuclear equation, one can verify that both atomic and mass numbers are balanced on each side of the reaction. The reason for the electrons (beta particles) having negligible mass is that their mass is significantly smaller than that of neutrons or protons, so it does not affect the mass number during balancing.

Mass-Energy Equivalence

Mass-energy equivalence, best encapsulated by Einstein's famous equation, E=mc², is a fundamental principle that underpins the energy release in nuclear reactions. The equation expresses the idea that mass and energy are interrelated; a small amount of mass can be converted into a large amount of energy and vice versa. This concept is critical in nuclear chemistry where the transformation of mass into energy is observed.

When strontium-90 decays to zirconium-90, a slight mass difference (denoted as \(\Delta m\)) occurs due to the mass of the emitted beta particles. Applying mass-energy equivalence, we can calculate the energy released when this mass is converted into energy. The relationship is direct: the greater the mass loss (\(\Delta m\)), the more energy is released during the decay. In nuclear chemistry, this released energy can be measured, and it is imperative for students to understand that even small mass changes can result in significant energy release due to the speed of light squared (\(c^2\)) term in the equation.

Nuclear Chemistry

Nuclear chemistry involves the study of the changes that occur within the nuclei of atoms. The processes often involve radioactive decay, fission, and fusion reactions, which are pivotal in understanding the behavior of elements and isotopes on an atomic level. The decay of strontium-90 is an important topic within nuclear chemistry due to its health implications and how it showcases the principles of nuclear stability and transmutation.

Understanding the nuclear chemistry of strontium-90 is not only academically significant, but it also has real-world implications, as elements like strontium-90 can be harmful due to their radioactive nature and chemical similarity to calcium. In this context, strontium-90's decay pathway through beta emission demonstrates how unstable isotopes regain stability by releasing energy and transforming into stable ones. The study of such transformations is crucial for the safe handling and disposal of radioactive materials and for the utilization of nuclear reactions in medical and energy-producing technologies.