Question

For which one of the following sets of four quantum numbers, an electron will have the highest energy? % Table of Quantum Numbers \begin{center} \begin{tabular}{|c|c|c|c|c|} \hline & \( n \) & \( l \) & \( m \) & \( s \)
\hline

• A. & 3 & 2 & 1 & \( +\frac{1}{2} \)
• B. & 4 & 2 & 1 & \( +\frac{1}{2} \)
• C. & 4 & 1 & 0 & \( +\frac{1}{2} \)
• D. & 5 & 0 & 0 & \( +\frac{1}{2} \)
  \hline \end{tabular} \end{center}

Hint:

The energy of an electron in an atom is primarily determined by the sum of the principal quantum number (\( n \)) and the azimuthal quantum number (\( l \)). The higher the value of \( n + l \), the higher the energy.

Answer 1

The Correct Option is B

Step 1: Understanding Quantum Numbers
The four quantum numbers are:
\( n \) (Principal quantum number) - determines the energy level.
\( l \) (Azimuthal quantum number) - determines the subshell.
\( m \) (Magnetic quantum number) - determines the orbital orientation.
\( s \) (Spin quantum number) - represents the electron spin.
Step 2: Energy of an Electron
The energy of an electron is primarily determined by the principal quantum number \( n \) and the azimuthal quantum number \( l \).
The general rule is that energy increases as \( n + l \) increases.
If two orbitals have the same \( n + l \) value, the one with the higher \( n \) has higher energy.
Step 3: Calculating \( n + l \) Values | Option | \( n \) | \( l \) | \( n + l \) | |--------|------|------|------| | (a) | 3 | 2 | 5 | | (b) | 4 | 2 | 6 | | (c) | 4 | 1 | 5 | | (d) | 5 | 0 | 5 |
The highest \( n + l \) value is 6, which corresponds to option (B).
Step 4: Determining the Highest Energy
The highest energy corresponds to the highest \( n + l \) value.
If there is a tie, the one with the higher \( n \) value has the highest energy.
Hence, the electron with the highest energy is in option (B) with \( n = 4, l = 2 \). Final Answer: The electron with the highest energy is in (B) \( n = 4, \, l = 2, \, m = 1, \, s = +\frac{1}{2} \).