Problem 3 - Chemistry for Allied Health

Question

Fill in the missing numbers in each equation. a. \({ }^{196} \mathrm{~Pb}+{ }_{-1}^{0} \mathrm{e} \rightarrow=\mathrm{Tl}\) b. \({ }_{15}^{28} \mathrm{P} \rightarrow=\mathrm{Si}+{ }_{1}^{0} \mathrm{e}\) c. \({ }_{88}^{226} \mathrm{Ra} \rightarrow=\mathrm{Rn}+{ }_{2}^{4} \alpha\) d. \({ }_{30}^{73} \mathrm{Zn} \rightarrow=\mathrm{Ga}+{ }_{-1}^{0} \mathrm{e}\)

Step-by-Step Solution

Answer

a) \(_{81}^{196} \mathrm{Tl}\), b) \(_{14}^{28} \mathrm{Si}\), c) \(_{86}^{222} \mathrm{Rn}\), d) \(_{31}^{73} \mathrm{Ga}\).

Step 1: Understand the Concept

The problem involves nuclear reactions where elements change due to emission or absorption of particles. We solve this by balancing the atomic and mass numbers on both sides of the equation.

Step 2: Solve for Reaction (a)

The reaction involves electron capture: \[^{196} \mathrm{~Pb} + _{-1}^{0} \mathrm{e} \rightarrow _{81}^{196} \mathrm{Tl}\] Since an electron is captured, it decreases the atomic number by 1, leading from lead (Pb) to thallium (Tl). The mass number remains 196.

Step 3: Solve for Reaction (b)

The reaction involves beta decay: \[_{15}^{28} \mathrm{P} \rightarrow _{14}^{28} \mathrm{Si} + _{1}^{0} \mathrm{e}\] In beta decay, a neutron is converted into a proton, increasing the atomic number from 15 (P) to 14 (Si), but the mass number remains the same at 28.

Step 4: Solve for Reaction (c)

The reaction involves alpha decay: \[_{88}^{226} \mathrm{Ra} \rightarrow _{86}^{222} \mathrm{Rn} + _{2}^{4} \alpha\] An alpha particle (\(_{2}^{4} \alpha\)) consists of 2 protons and 2 neutrons, so the atomic number decreases by 2 and the mass number by 4, resulting in radon (Rn) with 86 protons and a mass number of 222.

Step 5: Solve for Reaction (d)

The reaction involves beta decay: \[_{30}^{73} \mathrm{Zn} \rightarrow _{31}^{73} \mathrm{Ga} + _{-1}^{0} \mathrm{e}\] Beta decay increases the atomic number by 1 as a neutron is converted into a proton, changing zinc (Zn) with atomic number 30 to gallium (Ga) with atomic number 31, while the mass number remains 73.

Key Concepts

Electron Capture

Electron capture is a fascinating nuclear process where an atomic nucleus draws in one of its orbiting electrons. This process typically occurs in proton-rich isotopes. Let's see how it works:

• The nuclide captures an electron, usually from the innermost "K" shell.
• This results in a proton being transformed into a neutron.
• The atomic number decreases by one because you lose a proton, but the mass number stays the same, as a neutron has taken the place of the proton.

In reaction (a), electron capture transforms lead (Pb) into thallium (Tl) because the atomic number drops from 82 to 81, while maintaining the mass number of 196.
Remember, when balancing electron capture reactions, ensure the atomic number on the product side is reduced by one compared to the reactant side.

Beta Decay

Beta decay is a nuclear decay process where a beta particle, which is essentially an electron, is released. This occurs in neutron-rich isotopes. Here's a simple breakdown:

• A neutron transforms into a proton, emitting a beta particle (an electron) in the process.
• The atomic number increases by one as the neutron becomes a proton, but the mass number remains the same.

In reactions (b) and (d), beta decay kicks in. For reaction (b) with phosphorus (P), a neutron becomes a proton, transforming it into silicon (Si) as the atomic number goes from 15 to 14. In reaction (d), zinc (Zn) undergoes a similar process to become gallium (Ga), where the atomic number changes from 30 to 31.

Alpha Decay

Alpha decay is a type of nuclear decay where an alpha particle is emitted. Alpha particles are quite heavy as they contain:

• 2 protons and 2 neutrons, making them equivalent to a helium nucleus.
• This emission subtracts 4 units from the mass number and 2 units from the atomic number of the original nucleus.

In reaction (c), radium (Ra) undergoes alpha decay to become radon (Rn). The atomic number decreases from 88 to 86, and the mass number decreases from 226 to 222. Alpha decay is common in heavy elements and leads to substantial changes in the nucleus.

Balancing Atomic Numbers

Balancing atomic numbers in nuclear reactions is crucial. Ensuring the atomic count is accurate helps conserve charge. Here's how you do it:

• Sum the atomic numbers of all particles on both sides of the reaction.
• For particles like electrons in beta decay or positron emission (which are not appeared in this content), adjust accordingly.

In reaction (a), an electron lowers the atomic count, and in reaction (b) and (d), beta particles increase it by changing neutrons to protons. The atomic numbers should align with these transformations.

Balancing Mass Numbers

Balancing mass numbers is equally important and ensures the conservation of matter in nuclear reactions. Keep in mind that:

• The mass number equals the total number of protons and neutrons, so any shift in numbers like in alpha decay needs adjustment.
• Ensure that the mass numbers on both sides of the reaction are equal.

For example, in reaction (c), when radium emits an alpha particle, the mass number decreases by 4, matching the requirements on both sides of the equation. Remember to meticulously adjust any time a reaction suggests a change in the number of neutrons or protons.