Problem 32 - Chemistry The Central Science

Question

Write balanced equations for each of the following nuclear reactions: \((\mathbf{a}) _{92}^{238} \mathrm{U}(\mathrm{n}, \gamma)_{92}^{239} \mathrm{U},\) \((\mathbf{b})_{8}^{16} \mathrm{O}(\mathrm{p}, \alpha)_{7}^{13} \mathrm{N},\) \((\mathbf{c})_{8}^{18} \mathrm{O}\left(\mathrm{n}, \beta^{-}\right)_{9}^{19} \mathrm{F}.\)

Step-by-Step Solution

Answer

The balanced nuclear reactions are: a) \(_{92}^{238} \mathrm{U} + \mathrm{n} \rightarrow _{92}^{239} \mathrm{U} + \gamma\) b) \(_{8}^{16} \mathrm{O} + \mathrm{p} \rightarrow _{7}^{13} \mathrm{N} + \alpha\) c) \(_{8}^{18} \mathrm{O} + \mathrm{n} \rightarrow _{9}^{19} \mathrm{F} + \beta^-\)

Step 1: Determine the particles involved in the reaction 2.

Balance atomic numbers 3.

Step 2: Balance mass numbers

Reaction a: \(_{92}^{238} \mathrm{U}(\mathrm{n}, \gamma)_{92}^{239} \mathrm{U}\)

Step 1: Determine the particles involved in the reaction

For this reaction, the particles involved are: - \(_{92}^{238} \mathrm{U}\): Uranium-238 nucleus - \(\mathrm{n}\): Neutron - \(\gamma\): Gamma photon - \(_{92}^{239} \mathrm{U}\): Uranium-239 nucleus

Step 2: Balance Atomic Numbers

Since the atomic number does not change in this reaction (it is a neutron capture process that does not change the element), the atomic number is already balanced: 92 = 92.

Step 3: Balance Mass Numbers

We have the neutron on the left side and the gamma photon on the right side, which do not change the mass number. Therefore, we just need to account for the increase of the mass number by one due to the captured neutron: 238 + 1 = 239. The reaction is balanced.

Step 6: Reaction b: \(_{8}^{16} \mathrm{O}(\mathrm{p}, \alpha)_{7}^{13} \mathrm{N}\)

Step 1: Determine the particles involved in the reaction

Step 7: For this reaction, the particles involved are: - \(_{8}^{16} \mathrm{O}\): Oxygen-16 nucleus - \(\mathrm{p}\): Proton - \(\alpha\): Alpha particle (Helium-4 nucleus, \(_2^4\)He ) - \(_{7}^{13} \mathrm{N}\): Nitrogen-13 nucleus

Step 2: Balance Atomic Numbers

Step 8: We need to add the atomic number of the proton to the atomic number of Oxygen-16 to equal the atomic number of Nitrogen-13: 8 + 1 = 7+2.

Step 3: Balance Mass Numbers

Step 9: We also need to add the mass numbers of Oxygen-16 and proton and subtract the mass number of the alpha particle (Helium-4 nucleus) to find the mass number of Nitrogen-13: 16 + 1 - 4 = 13. The reaction is balanced.

Reaction c: \(_{8}^{18} \mathrm{O}\left(\mathrm{n}, \beta^{-}\right)_{9}^{19} \mathrm{F}\)

Step 1: Determine the particles involved in the reaction

For this reaction, the particles involved are: - \(_{8}^{18} \mathrm{O}\): Oxygen-18 nucleus - \(\mathrm{n}\): Neutron - \(\beta^{-}\): Beta minus (electron) particle - \(_{9}^{19} \mathrm{F}\): Fluorine-19 nucleus

Step 2: Balance Atomic Numbers

Since a neutron gets converted into a proton inside an Oxygen-18 nucleus during beta decay, the atomic number increases by one: 8 + 1 = 9. As Fluorine's atomic number is 9, the atomic numbers are balanced.

Step 3: Balance Mass Numbers

The mass number remains unchanged during beta decay, as one neutron is converted into a proton without any change in the total nucleons count. Therefore, the mass numbers are already balanced: 18 + 1 = 19. The reaction is balanced.

Key Concepts

Neutron Capture

Neutron capture is a type of nuclear reaction where a nucleus absorbs a neutron. This process is common in the atomic world and is critical to the formation of heavier elements in stars through a process called nucleosynthesis. In the given textbook exercise, uranium-238 captures a neutron to become uranium-239. This can be understood as simply adding the mass of the neutron to the original nucleus, since a neutron has no charge and hence does not affect the atomic number of the element.

To visualize this, imagine you have a bag of numbered balls representing protons and neutral balls representing neutrons. During neutron capture, you're adding an extra neutral ball into the bag without changing the numbers on any of the numbered balls. That's essentially what's happening inside a nucleus during neutron capture. The atomic number remains the same, but the mass number increases by one, reflecting the addition of the neutron. Ensuring atomic and mass numbers are balanced is crucial when writing nuclear reaction equations for processes like neutron capture.

Beta Decay

Beta decay represents a more complex nuclear reaction where an unstable atomic nucleus transforms into a more stable one with the emission of beta particles. These beta particles are high-energy, high-speed electrons or positrons which are ejected from the nucleus. During beta minus decay, a neutron is transformed into a proton with the ejection of an electron and an antineutrino.It's a bit like a bank transaction where neutron 'currency' is exchanged into proton 'currency'—the balance in the 'atomic number account' goes up by one, while the 'mass number account' remains the same. In our exercise, an Oxygen-18 nucleus undergoes beta decay to become a Fluorine-19 nucleus. The atomic number increases from 8 to 9 due to the transformation of a neutron into a proton, but the mass number remains unchanged at 18 because the mass of the nucleon overall does not change. This fundamental understanding of beta decay helps students grasp how to balance atomic and mass numbers in nuclear reaction equations.

Balancing Atomic Numbers

Balancing atomic numbers in nuclear reaction equations is akin to ensuring that each side of a seesaw has an equal number of people to stay in balance. The atomic number, which represents the number of protons in a nucleus, must be conserved in a nuclear reaction. In reactions, like the ones provided in the textbook, you verify that the sum of atomic numbers on the reactant side equals the sum on the product side.

For the reaction with Oxygen-16 turning into Nitrogen-13, we add the incoming proton to Oxygen’s atomic number to achieve balance with Nitrogen’s atomic number plus the atomic number of the outgoing alpha particle. This attention to detail is crucial for students to produce correct balanced nuclear equations, as it ensures proper identification of the resulting elements after a reaction has occurred.

Balancing Mass Numbers

When it comes to balancing mass numbers, we apply the principle that the total mass before and after a nuclear reaction must remain the same. You can think of it like ensuring that pieces of a puzzle fit together perfectly, creating a complete and unchanged picture. Mass number, which is the sum of protons and neutrons in the nucleus, needs to be accounted for each step of the way in a nuclear reaction.

In our textbook problem, each step of balancing mass numbers involves tabulating the sum of the mass numbers of all the reactants, and ensuring they equal the sum of the mass numbers of all the products. For instance, in the oxygen to nitrogen transformation, we add the mass of Oxygen-16 and the proton together, then subtract the mass of the alpha particle, which should match the mass number of the resulting Nitrogen-13 nucleus. Students should be meticulous with this step to sustain the law of conservation of mass, which states that mass cannot be created or destroyed in a closed system.