Question

For hydrogen atom, the orbital/s with lowest energy is/are:

• A. (B), (C), and (D) only
• B. (A) only
• C. (B) only
• D. (A) and (E) only

Hint:

The \( 1s \) orbital is the lowest energy orbital for the hydrogen atom.

Answer 1

The Correct Option is A

The energy of orbitals in an atom depends on the principal quantum number (n) and the angular momentum quantum number (l). For hydrogen atoms, the energy of orbitals is primarily determined by the value of the principal quantum number (n) alone, as all orbitals of the same n value have the same energy. The only exception is for multi-electron atoms, where orbital shapes and electron-electron repulsion come into play, but this is not the case for hydrogen.

Step 1: Understand the Energy Levels of Orbitals

The energy of an orbital in a hydrogen atom is given by the formula:

\[ E = - \frac{13.6 \, \text{eV}}{n^2} \] where \(n\) is the principal quantum number (1 for the K-shell, 2 for the L-shell, etc.).

In hydrogen, the orbitals with lower values of \(n\) will have lower (more negative) energy. Therefore, the energy of orbitals decreases as \(n\) decreases.

Step 2: Analyze Each Orbital

Option (A): 4s

The 4s orbital belongs to the n=4 shell. It has a relatively higher energy compared to orbitals with a lower value of n. Hence, this orbital does not have the lowest energy in the hydrogen atom.

Option (B): 3p_x

The 3p orbital belongs to the n=3 shell. This orbital is lower in energy than the 4s orbital, as \(n=3\) is less than \(n=4\). In hydrogen, all 3p orbitals (including 3p_x, 3p_y, 3p_z) have the same energy, so the 3p_x orbital has lower energy than 4s. Therefore, it is a valid candidate for the lowest energy orbital.

Option (C): 3d_x²−y²

The 3d_x²−y² orbital belongs to the n=3 shell. While 3d orbitals generally have a higher energy than 3p orbitals, they still belong to the same n=3 shell. Therefore, this orbital also has lower energy than the 4s orbital, making it a valid candidate for the lowest energy orbital in hydrogen.

Option (D): 3d_z²

Similar to 3d_x²−y², the 3d_z² orbital also belongs to the n=3 shell. It has the same energy as the other 3d orbitals and is lower in energy than the 4s orbital. Hence, this orbital is another valid candidate for the lowest energy orbital in hydrogen.

Option (E): 4p_z

The 4p_z orbital belongs to the n=4 shell, which has higher energy than orbitals from the n=3 shell. Thus, this orbital does not have the lowest energy in hydrogen.

Step 3: Conclusion

Based on the analysis, the orbitals with the lowest energy for a hydrogen atom are:

• 3p_x (Option B)
• 3d_x²−y² (Option C)
• 3d_z² (Option D)

Therefore, the correct answer is (B), (C), and (D) only.

Answer 2

Step 1: Understand the hydrogen atom's energy levels.
In a hydrogen atom, the energy of an orbital depends primarily on its principal quantum number \( n \), and to some extent on the angular momentum quantum number \( l \). The orbitals with the lowest energy are those with the lowest \( n \) and \( l \) values.

Step 2: Analyze the orbitals.
- 4s orbital (A): The \( 4s \) orbital has \( n = 4 \) and \( l = 0 \). The 4s orbital is farther from the nucleus than the 3p, 3d orbitals, and thus generally has a higher energy.
- 3p_x orbital (B): The \( 3p \) orbital has \( n = 3 \) and \( l = 1 \). The \( 3p \) orbitals are at a lower energy compared to the 4s orbital, but still higher than 3d orbitals for hydrogen.
- 3d_x²-y² orbital (C): The \( 3d_{x^2-y^2} \) orbital has \( n = 3 \) and \( l = 2 \). The 3d orbitals are of lower energy than the 4s orbital, but they are higher in energy compared to 3p orbitals.
- 3d_z² orbital (D): The \( 3d_{z^2} \) orbital, like the 3d_x²-y² orbital, has \( n = 3 \) and \( l = 2 \). This orbital is also lower in energy compared to the 4s orbital but has similar energy to the other 3d orbitals.
- 4p_z orbital (E): The \( 4p_z \) orbital has \( n = 4 \) and \( l = 1 \). The 4p orbital is generally higher in energy compared to the 3p and 3d orbitals for hydrogen.

Step 3: Conclusion.
The orbitals with the lowest energy are those with the lowest principal quantum numbers and lower angular momentum quantum numbers.
Thus, the correct answer is (B), (C), and (D) only.

Final Answer:
\[ \boxed{\text{(B), (C), and (D) only}}. \]