Question

For which one of the following sets of four quantum numbers, an electron will have the highest energy? 

• A. \( 3 | 2| 1 |  +\frac{1}{2} \)
• B. \(| 4 | 2 | 1 |  +\frac{1}{2}\)
• C. \(| 4 | 1 | 0 |  +\frac{1}{2} \)
• D. \(| 5 | 0 | 0 |  +\frac{1}{2} \)

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 

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) \( n = 4, \, l = 2, \, m = 1, \, s = +\frac{1}{2} \).