Balancing Nuclear Equations Practice Worksheet with fill-in-the-blank reactions and questions on radioactive decay.

Problem Analysis:

The task involves completing nuclear equations and identifying the type of decay (alpha, beta, or gamma) for each reaction. Additionally, there are questions about specific scenarios involving radioactive decay.

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Step-by-Step Solution:

#### 1. General Approach to Completing Nuclear Equations:
In nuclear reactions, the principle of conservation of mass number (A) and atomic number (Z) must be satisfied. For any reaction:
- Mass number (A): The sum of the mass numbers on the left side must equal the sum on the right side.
- Atomic number (Z): The sum of the atomic numbers on the left side must equal the sum on the right side.

Additionally:
- Alpha decay: Emits an alpha particle (${}_{2}^{4}\text{He}$).
- Beta decay:
- Beta-minus decay: Emits a beta particle (${}_{-1}^{0}\text{e}$), converting a neutron into a proton.
- Beta-plus decay: Emits a positron (${}_{+1}^{0}\text{e}$), converting a proton into a neutron.
- Gamma decay: Involves the emission of a gamma photon ($\gamma$), which does not change the mass number or atomic number.

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#### 2. Solving Each Equation:

##### (a) ${}_{88}^{226}\text{Ra} \rightarrow {}_{?}^{?}\text{X} + {}_{2}^{4}\text{He}$
- Alpha decay: The daughter nucleus is formed by subtracting 4 from the mass number and 2 from the atomic number.
- New mass number: $226 - 4 = 222$
- New atomic number: $88 - 2 = 86$
- Element with atomic number 86 is radon (Rn).
- Equation: ${}_{88}^{226}\text{Ra} \rightarrow {}_{86}^{222}\text{Rn} + {}_{2}^{4}\text{He}$
- Type of decay: Alpha decay

##### (b) ${}_{5}^{1}\text{H} \rightarrow {}_{?}^{?}\text{X} + {}_{-1}^{0}\text{e}$
- Beta-minus decay: The daughter nucleus is formed by adding 1 to the atomic number (since a neutron converts to a proton) and keeping the mass number the same.
- New atomic number: $5 + 1 = 6$
- New mass number: $1$ (unchanged)
- Element with atomic number 6 is carbon (C).
- Equation: ${}_{5}^{1}\text{H} \rightarrow {}_{6}^{1}\text{C} + {}_{-1}^{0}\text{e}$
- Type of decay: Beta-minus decay

##### (c) ${}_{93}^{239}\text{Np} \rightarrow {}_{?}^{?}\text{X} + {}_{0}^{1}\text{n}$
- Neutron emission: The daughter nucleus is formed by subtracting 1 from the mass number and keeping the atomic number the same.
- New mass number: $239 - 1 = 238$
- New atomic number: $93$ (unchanged)
- Element with atomic number 93 is neptunium (Np).
- Equation: ${}_{93}^{239}\text{Np} \rightarrow {}_{93}^{238}\text{Np} + {}_{0}^{1}\text{n}$
- Type of decay: Neutron emission

##### (d) ${}_{4}^{7}\text{Be} + {}_{2}^{4}\text{He} \rightarrow {}_{?}^{?}\text{X} + {}_{0}^{1}\text{n}$
- Nuclear fusion: Combine the mass numbers and atomic numbers of the reactants, then account for the emitted neutron.
- Total mass number: $7 + 4 = 11$, and subtract 1 for the neutron: $11 - 1 = 10$
- Total atomic number: $4 + 2 = 6$
- Element with atomic number 6 is carbon (C).
- Equation: ${}_{4}^{7}\text{Be} + {}_{2}^{4}\text{He} \rightarrow {}_{6}^{10}\text{C} + {}_{0}^{1}\text{n}$
- Type of decay: Fusion (not a decay process)

##### (e) ${}_{1}^{1}\text{H} + {}_{1}^{1}\text{H} \rightarrow {}_{?}^{?}\text{X} + {}_{0}^{0}\gamma$
- Fusion: Combine the mass numbers and atomic numbers of the reactants.
- Total mass number: $1 + 1 = 2$
- Total atomic number: $1 + 1 = 2$
- Element with atomic number 2 is helium (He).
- Equation: ${}_{1}^{1}\text{H} + {}_{1}^{1}\text{H} \rightarrow {}_{2}^{2}\text{He} + {}_{0}^{0}\gamma$
- Type of decay: Fusion (not a decay process)

##### (f) ${}_{3}^{7}\text{Li} + {}_{1}^{0}\text{n} \rightarrow {}_{?}^{?}\text{X} + {}_{2}^{4}\text{He}$
- Fission-like reaction: Combine the mass numbers and atomic numbers of the reactants, then account for the emitted alpha particle.
- Total mass number: $7 + 1 = 8$, and subtract 4 for the alpha particle: $8 - 4 = 4$
- Total atomic number: $3 + 0 = 3$, and subtract 2 for the alpha particle: $3 - 2 = 1$
- Element with atomic number 1 is hydrogen (H).
- Equation: ${}_{3}^{7}\text{Li} + {}_{0}^{1}\text{n} \rightarrow {}_{1}^{4}\text{H} + {}_{2}^{4}\text{He}$
- Type of decay: Fission-like (not a decay process)

##### (g) ${}_{7}^{14}\text{N} + {}_{1}^{1}\text{H} \rightarrow {}_{?}^{?}\text{X} + {}_{?}^{?}\text{Y}$
- Proton capture: Combine the mass numbers and atomic numbers of the reactants.
- Total mass number: $14 + 1 = 15$
- Total atomic number: $7 + 1 = 8$
- Element with atomic number 8 is oxygen (O).
- Equation: ${}_{7}^{14}\text{N} + {}_{1}^{1}\text{H} \rightarrow {}_{8}^{15}\text{O} + \gamma$
- Type of decay: Proton capture (not a decay process)

##### (h) ${}_{38}^{90}\text{Sr} \rightarrow {}_{?}^{?}\text{X} + {}_{-1}^{0}\text{e}$
- Beta-minus decay: The daughter nucleus is formed by adding 1 to the atomic number and keeping the mass number the same.
- New atomic number: $38 + 1 = 39$
- New mass number: $90$ (unchanged)
- Element with atomic number 39 is yttrium (Y).
- Equation: ${}_{38}^{90}\text{Sr} \rightarrow {}_{39}^{90}\text{Y} + {}_{-1}^{0}\text{e}$
- Type of decay: Beta-minus decay

##### (i) ${}_{92}^{239}\text{U} \rightarrow {}_{?}^{?}\text{X} + {}_{2}^{4}\text{He}$
- Alpha decay: The daughter nucleus is formed by subtracting 4 from the mass number and 2 from the atomic number.
- New mass number: $239 - 4 = 235$
- New atomic number: $92 - 2 = 90$
- Element with atomic number 90 is thorium (Th).
- Equation: ${}_{92}^{239}\text{U} \rightarrow {}_{90}^{235}\text{Th} + {}_{2}^{4}\text{He}$
- Type of decay: Alpha decay

##### (j) ${}_{82}^{210}\text{Pb} \rightarrow {}_{?}^{?}\text{X} + {}_{-1}^{0}\text{e}$
- Beta-minus decay: The daughter nucleus is formed by adding 1 to the atomic number and keeping the mass number the same.
- New atomic number: $82 + 1 = 83$
- New mass number: $210$ (unchanged)
- Element with atomic number 83 is bismuth (Bi).
- Equation: ${}_{82}^{210}\text{Pb} \rightarrow {}_{83}^{210}\text{Bi} + {}_{-1}^{0}\text{e}$
- Type of decay: Beta-minus decay

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#### 3. Additional Questions:

##### (k) What particles are left when Potassium-38 undergoes beta decay?
- Beta decay: Potassium-38 (${}_{19}^{38}\text{K}$) decays by emitting a beta particle (${}_{-1}^{0}\text{e}$). This increases the atomic number by 1, forming calcium-38 (${}_{20}^{38}\text{Ca}$).
- Particles left: Calcium-38 (${}_{20}^{38}\text{Ca}$) and a beta particle (${}_{-1}^{0}\text{e}$).

##### (l) What particles are left when Polonium-210 undergoes beta decay?
- Beta decay: Polonium-210 (${}_{84}^{210}\text{Po}$) decays by emitting a beta particle (${}_{-1}^{0}\text{e}$). This increases the atomic number by 1, forming astatine-210 (${}_{85}^{210}\text{At}$).
- Particles left: Astatine-210 (${}_{85}^{210}\text{At}$) and a beta particle (${}_{-1}^{0}\text{e}$).

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Final Answers:

1. ${}_{88}^{226}\text{Ra} \rightarrow {}_{86}^{222}\text{Rn} + {}_{2}^{4}\text{He}$ (Alpha decay)
2. ${}_{5}^{1}\text{H} \rightarrow {}_{6}^{1}\text{C} + {}_{-1}^{0}\text{e}$ (Beta-minus decay)
3. ${}_{93}^{239}\text{Np} \rightarrow {}_{93}^{238}\text{Np} + {}_{0}^{1}\text{n}$ (Neutron emission)
4. ${}_{4}^{7}\text{Be} + {}_{2}^{4}\text{He} \rightarrow {}_{6}^{10}\text{C} + {}_{0}^{1}\text{n}$ (Fusion)
5. ${}_{1}^{1}\text{H} + {}_{1}^{1}\text{H} \rightarrow {}_{2}^{2}\text{He} + {}_{0}^{0}\gamma$ (Fusion)
6. ${}_{3}^{7}\text{Li} + {}_{0}^{1}\text{n} \rightarrow {}_{1}^{4}\text{H} + {}_{2}^{4}\text{He}$ (Fission-like)
7. ${}_{7}^{14}\text{N} + {}_{1}^{1}\text{H} \rightarrow {}_{8}^{15}\text{O} + \gamma$ (Proton capture)
8. ${}_{38}^{90}\text{Sr} \rightarrow {}_{39}^{90}\text{Y} + {}_{-1}^{0}\text{e}$ (Beta-minus decay)
9. ${}_{92}^{239}\text{U} \rightarrow {}_{90}^{235}\text{Th} + {}_{2}^{4}\text{He}$ (Alpha decay)
10. ${}_{82}^{210}\text{Pb} \rightarrow {}_{83}^{210}\text{Bi} + {}_{-1}^{0}\text{e}$ (Beta-minus decay)
11. Calcium-38 (${}_{20}^{38}\text{Ca}$) and a beta particle (${}_{-1}^{0}\text{e}$)
12. Astatine-210 (${}_{85}^{210}\text{At}$) and a beta particle (${}_{-1}^{0}\text{e}$)

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Final Boxed Answers:

$$
\boxed{
\begin{aligned}
1. & \quad {}_{88}^{226}\text{Ra} \rightarrow {}_{86}^{222}\text{Rn} + {}_{2}^{4}\text{He} \quad (\text{Alpha decay}) \\
2. & \quad {}_{5}^{1}\text{H} \rightarrow {}_{6}^{1}\text{C} + {}_{-1}^{0}\text{e} \quad (\text{Beta-minus decay}) \\
3. & \quad {}_{93}^{239}\text{Np} \rightarrow {}_{93}^{238}\text{Np} + {}_{0}^{1}\text{n} \quad (\text{Neutron emission}) \\
4. & \quad {}_{4}^{7}\text{Be} + {}_{2}^{4}\text{He} \rightarrow {}_{6}^{10}\text{C} + {}_{0}^{1}\text{n} \quad (\text{Fusion}) \\
5. & \quad {}_{1}^{1}\text{H} + {}_{1}^{1}\text{H} \rightarrow {}_{2}^{2}\text{He} + {}_{0}^{0}\gamma \quad (\text{Fusion}) \\
6. & \quad {}_{3}^{7}\text{Li} + {}_{0}^{1}\text{n} \rightarrow {}_{1}^{4}\text{H} + {}_{2}^{4}\text{He} \quad (\text{Fission-like}) \\
7. & \quad {}_{7}^{14}\text{N} + {}_{1}^{1}\text{H} \rightarrow {}_{8}^{15}\text{O} + \gamma \quad (\text{Proton capture}) \\
8. & \quad {}_{38}^{90}\text{Sr} \rightarrow {}_{39}^{90}\text{Y} + {}_{-1}^{0}\text{e} \quad (\text{Beta-minus decay}) \\
9. & \quad {}_{92}^{239}\text{U} \rightarrow {}_{90}^{235}\text{Th} + {}_{2}^{4}\text{He} \quad (\text{Alpha decay}) \\
10. & \quad {}_{82}^{210}\text{Pb} \rightarrow {}_{83}^{210}\text{Bi} + {}_{-1}^{0}\text{e} \quad (\text{Beta-minus decay}) \\
11. & \quad \text{Calcium-38 (}_{{20}}^{{38}}\text{Ca)} \text{ and a beta particle (}_{{-1}}^{{0}}\text{e)} \\
12. & \quad \text{Astatine-210 (}_{{85}}^{{210}}\text{At)} \text{ and a beta particle (}_{{-1}}^{{0}}\text{e)}
\end{aligned}
}
$$