Certainly! Let's break down the information and the equations step by step to find the correct nuclear reaction that illustrates the fusion of two deuterium nuclei to form helium-3 and an emitted neutron.
1. Deuterium Nucleus Composition:
- A deuterium nucleus consists of one proton ([tex]\(^{1}\mathrm{H}\)[/tex]) and one neutron. The mass number (A) is 2, and the atomic number (Z) is 1. Hence, deuterium is represented as [tex]\(^{2}_1 \mathrm{H}\)[/tex].
2. Nuclear Fusion Process:
- When two deuterium nuclei fuse, they form a helium-3 nucleus and emit a neutron.
- To understand the products:
- Helium-3 is represented as [tex]\(^{3}_2 \mathrm{He}\)[/tex] (with mass number 3 and atomic number 2).
- A neutron is represented as [tex]\(^{1}_0 \mathrm{n}\)[/tex] (with mass number 1 and atomic number 0).
3. Balancing the Equation:
- For the nuclear reaction to be correct, the sum of atomic numbers (protons) and mass numbers (protons + neutrons) on both sides must be equal.
Let's match the given options to these criteria:
- Option 1:
[tex]\(2 \mathrm{H}_2 + \mathrm{O}_2 \rightarrow 2 \mathrm{H}_2\mathrm{O}\)[/tex]
This is a chemical reaction where hydrogen (H) reacts with oxygen (O) to form water (H₂O). This does not involve nuclear fusion of deuterium.
- Option 2:
[tex]\(^{2}_2 \mathrm{H} \rightarrow { }_1^3 \mathrm{H} + { }_1^1 \mathrm{H}\)[/tex]
This suggests one isotope of hydrogen ([tex]\(^{2}_2 \mathrm{H}\)[/tex]) splitting into two other isotopes ([tex]\(^{3}_1 \mathrm{H}\)[/tex] and [tex]\(^{1}_1 \mathrm{H}\)[/tex]). This does not match the requirements for two deuterium nuclei fusing into helium-3.
- Option 3:
[tex]\(2 { }_1^1 \mathrm{H} \rightarrow{ }_1^2 \mathrm{H} + \frac{0}{1} e\)[/tex]
This is not correct for our required reaction. This option describes proton-proton fusion, not deuterium fusion.
- Option 4:
[tex]\(2{ }_1^2 \mathrm{H} \rightarrow { }_2^3 \mathrm{He} + { }_0^1 \mathrm{n}\)[/tex]
Here:
- Left side: [tex]\(2{ }_1^2 \mathrm{H}\)[/tex] (Two deuterium nuclei, having a total mass number of 4 and atomic number of 2)
- Right side: [tex]\( { }_2^3 \mathrm{He}\)[/tex] (Helium-3 nucleus, mass number 3, atomic number 2) and [tex]\({ }_0^1 \mathrm{n}\)[/tex] (neutron, mass number 1, atomic number 0)
- The number of protons on both sides adds up to 2 and the total masses on both sides add up to 4.
- This confirms the nuclear equation, correctly showing the fusion process.
Conclusion:
The correct equation illustrating the fusion of two deuterium nuclei to form helium-3 and an emitted neutron is:
[tex]\[2{ }_1^2 \mathrm{H} \rightarrow { }_2^3 \mathrm{He} + { }_0^1 \mathrm{n}\][/tex]
Hence, the answer is:
[tex]\[4\][/tex]
