Question

In a multi-electron atom, which of the following orbitals described by the three quantum members will have the same energy in the absence of magnetic and electric fields?

(a) n = 1, l = 0, m = 0

(b) n = 2, l = 0, m = 0

(c) n = 2, l = 1, m = 1

(d) n = 3, l = 2, m = 1

(e) n = 3, l = 2, m = 0

• A. (b) and (c)
• B. (a) and (b)
• C. (d) and (e)
• D. (c) and (d)

Answer 1

Answer: (A), (C)

(d) and (e)

Explanation for correct answers:

Option C (d) and (e)

• In the absence of magnetic and electric fields, the orbitals defined by magnetic quantum numbers degenerate.
• So, the energy of the orbital, in the absence of magnetic and electric fields depends on the $(\mathrm{n}+\mathrm{l})$ value.
• Higher the $(\mathrm{n}+\mathrm{l})$ value, the larger the energy of the orbital.
• Orbitals with the same ‘$\mathrm{n}$’ and ‘$(\mathrm{n}+\mathrm{l})$’ values are degenerate and have the same energy.
• In the given combinations both ‘(d)’ and ’(e)' the ‘$\mathrm{n}$’ value is 3 that is, ‘(d)’ and ’(e)' has the same ‘$\mathrm{n}$’ and ‘$(\mathrm{n}+\mathrm{l})$' values.
• So, have the same energy in the absence of magnetic and electric fields.
• Hence option (A) is correct.

Explanation for incorrect answers:

Option A (b) and (c)

• In the given combinations both ‘(b)’ and ’(c)' the ‘$\mathrm{n}$’ value is 2 that is, ‘(d)’ and ’(e)' have the same ‘$\mathrm{n}$’ and ‘$(\mathrm{n}+\mathrm{l})$’ values.
• Thus, they have a different energy in the absence of magnetic and electric fields.
• Hence option (C) is incorrect.

Option B (a) and (b)

• In the given combinations both ‘(a)’ and ’(b)' the ‘$\mathrm{n}$’ value is 1 and 2 respectively that is, ‘(a)’ and ’(b)' has different ‘$\mathrm{n}$’ and ‘$(\mathrm{n}+\mathrm{l})$’ values
• So, have a different energy in the absence of magnetic and electric fields.
• Hence option B is incorrect.

Option D (c) and (d)

• In the given combinations both ‘(c)’ and ’(d)' the ‘$\mathrm{n}$’ value is 2 and 3 respectively that is, ‘(c)’ and ’(d)' has different ‘$\mathrm{n}$’ and ‘$(\mathrm{n}+\mathrm{l})$’ values
• Thus, they have a different energy in the absence of magnetic and electric fields.
• Hence option D is incorrect.