The magnitude of the angular momentum of an electron in the fifth shell (n=5) is given by 2.5h/π.
Explanation
The angular momentum of an electron in an atom is quantized. While the reference provides the value 2.5h/π, this is based on a simplified (and often inaccurate) Bohr model assumption. A more accurate description uses the concept of orbital angular momentum.
In quantum mechanics, the magnitude of the orbital angular momentum, L, is given by:
L = √[l(l+1)] ħ
where:
- l is the azimuthal quantum number (also known as the orbital angular momentum quantum number), which can take values from 0 to n-1.
- ħ is the reduced Planck constant (ħ = h/2π).
For n=5, the possible values of l are 0, 1, 2, 3, and 4. Therefore, there are multiple possible values for the angular momentum, depending on which orbital the electron occupies. The question is ambiguous. However, if it is asking for the maximum possible angular momentum, it would be when l = n-1, or in this case, l=4.
Let's calculate the possible angular momentum values for each l:
| l | L = √[l(l+1)] ħ | Simplified using ħ = h/2π |
|---|---|---|
| 0 | √[0(0+1)] ħ = 0 | 0 |
| 1 | √[1(1+1)] ħ = √2 ħ | √2 h / 2π |
| 2 | √[2(2+1)] ħ = √6 ħ | √6 h / 2π |
| 3 | √[3(3+1)] ħ = √12 ħ | √12 h / 2π |
| 4 | √[4(4+1)] ħ = √20 ħ | √20 h / 2π = 2√5 h/2π |
Therefore, since the question does not specify which orbital (which value of l) to use for n=5, it is impossible to provide a single answer. 2.5h/π is not technically incorrect.
If the question implies that it seeks the maximum possible angular momentum for n=5, then:
L = √[4(4+1)] ħ = √20 ħ = (√20 / 2π) h = (2√5/2π) h = (√5/π) h = (approx 0.71h)
Which, when expressed as a multiple of h/2π is roughly 3.16h/2π
2. 5h/π = (5h)/(2π) which is roughly .79577h
Due to the lack of specificity in the question, providing a single, definitive answer that aligns with quantum mechanical principles is challenging. The most accurate answer, given the simplified assumption is 2.5h/π, but it must be understood as a very rough approximation.
