Question

Complete and balance the following nuclear equations by supplying the missing particle: (a) ${}^{25}{}_8^2\mathrm{Cf}+{}_5^{10}\mathrm{B}⟶3_0^1\mathrm{n}+?$ (b) ${}_1^2\mathrm{H}+{}_2^3\mathrm{He}⟶{}_2^4\mathrm{He}+?$

Short answer

(a) The missing particle is ${}_{13}^{32}\mathrm{Al}$, so the balanced equation is: ${}^{25}{}_8^2\mathrm{Cf}+{}_5^{10}\mathrm{B}⟶3_0^1\mathrm{n}+{}_{13}^{32}\mathrm{Al}$ (b) The missing particle is ${}_1^1\mathrm{H}$, so the balanced equation is: ${}_1^2\mathrm{H}+{}_2^3\mathrm{He}⟶{}_2^4\mathrm{He}+{}_1^1\mathrm{H}$

Step-by-step solution

Analyze Equation (a)

For equation (a), we have: ${}^{25}{}_8^2\mathrm{Cf}+{}_5^{10}\mathrm{B}⟶3_0^1\mathrm{n}+?$ In this equation, we'll first count the number of protons and neutrons present on the left side of the equation. Protons (Left side): 8 + 5 = 13 Neutrons (left side): (25 - 8) + (10 - 5) = 17 + 5 = 22 Now, let's analyze the numbers on the right side: Protons (Right side): 0 + ? (as there are no protons in a neutron) Neutrons (Right side): 3 * 1 + ?

Find the Missing Particle for Equation (a)

To balance the equation, the number of protons and neutrons on both sides of the equation should be equal. Protons (Left side) = Protons (Right side): 13 = 0 + x x = 13 Neutrons (Left side) = Neutrons (Right side): 22 = 3 * 1 + y 22 = 3 + y y = 19 So, the missing particle has 13 protons and 19 neutrons. Thus, the missing particle will have a mass number of 13+19=32 and an atomic number of 13, which corresponds to the element Al (Aluminum). Therefore, the completed and balanced equation for (a) is: ${}^{25}{}_8^2\mathrm{Cf}+{}_5^{10}\mathrm{B}⟶3_0^1\mathrm{n}+{}_{13}^{32}\mathrm{Al}$

Analyze Equation (b)

For equation (b), we have: ${}_1^2\mathrm{H}+{}_2^3\mathrm{He}⟶{}_2^4\mathrm{He}+?$ As before, we'll first count the number of protons and neutrons present on the left side of the equation: Protons (Left side): 1 + 2 = 3 Neutrons (Left side): (2 - 1) + (3 - 2) = 1 + 1 = 2 On the right side: Protons (Right side): 2 + ? Neutrons (Right side): (4 - 2) + ?

Find the Missing Particle for Equation (b)

Again, we will balance the equation by ensuring the number of protons and neutrons are equal on both sides of the equation. Protons (Left side) = Protons (Right side): 3 = 2 + x x = 1 Neutrons (Left side) = Neutrons (Right side): 2 = (4-2) + y 2 = 2 + y y = 0 So, the missing particle has 1 proton and 0 neutrons. This corresponds to a mass number of 1+0=1 and an atomic number of 1, which corresponds to the element H (Hydrogen). Therefore, the completed and balanced equation for (b) is: ${}_1^2\mathrm{H}+{}_2^3\mathrm{He}⟶{}_2^4\mathrm{He}+{}_1^1\mathrm{H}$

Key concepts

Balancing Nuclear Equations

Nuclear equations, like regular chemical equations, need to be balanced. This means both sides of the equation should have the same number of protons and neutrons. Balancing nuclear equations is crucial for respecting the law of conservation of mass and charge. Each nuclear species involved in a reaction is represented by the notation $${}_Z^{A}X$$Here, $A$ is the mass number (total number of protons and neutrons), $Z$ is the atomic number (protons), and $X$ is the element symbol. To balance a nuclear equation:

• Count the total number of protons and neutrons on each side.
• Ensure both sides have equal numbers of protons and neutrons.
• Identify any missing particles or elements needed to balance the equation.

This balancing process helps us find missing parts of a nuclear equation, like we did in the problem where we determine the presence of Aluminum $\left({}_{13}^{32}\mathrm{Al}\right)$ and Hydrogen $\left({}_1^1\mathrm{H}\right)$ as the products.

Protons and Neutrons

Protons and neutrons are subatomic particles found in the nucleus of an atom. Together, they define the atomic mass and the properties of elements:

• **Protons (P):** They carry a positive charge and determine the element's identity. The number of protons equals the atomic number $Z$.
• **Neutrons (N):** They are neutral particles contributing to the atomic mass. Neutrons can vary in number, forming isotopes of the same element.

The mass number $A$ is calculated by the sum of protons and neutrons:$$A=Z+N$$In nuclear reactions, the number of protons and neutrons can change, but the overall balance is maintained. This understanding helps us find the missing particles that balance equations as seen in the provided exercise, where calculating the number of protons and neutrons was essential for completing the equations.

Nuclear Reactions

Nuclear reactions involve changes in an atom’s nucleus and can transform one element into another. Unlike chemical reactions that affect electrons, nuclear reactions involve:

• **Fission:** Splitting a heavy nucleus into two lighter nuclei.
• **Fusion:** Combining two light nuclei to form a heavier nucleus.
• **Decay:** A radioactive process by which an unstable nucleus loses energy.

In the given exercise, we dealt with scenarios where different particles interacted to form a product while conserving the total number of protons and neutrons. This is why we applied laws of conservation in determining the missing particles in the equations, ensuring that the sum of protons and neutrons remained consistent on both sides of a nuclear equation. Understanding these concepts allows us to predict the outcome and products of nuclear reactions accurately.