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For the [tex]n=3[/tex] electron shell, which of the following quantum numbers are valid? Check all that apply.

A. [tex]I=3[/tex]

B. [tex]m=3[/tex]

C. [tex]I=0[/tex]

D. [tex]m=-2[/tex]

E. [tex]I=-1[/tex]

F. [tex]m=2[/tex]

Sagot :

GinnyAnswer
To determine the validity of the given quantum numbers for the electron shell with [tex]\( n = 3 \)[/tex]:

First, we must review the constraints on the quantum numbers:
1. The azimuthal quantum number ([tex]\( I \)[/tex]) must satisfy [tex]\( 0 \leq I \leq n - 1 \)[/tex].
For [tex]\( n = 3 \)[/tex], this means [tex]\( I \)[/tex] can be: [tex]\( 0, 1, \)[/tex] or [tex]\( 2 \)[/tex].

2. The magnetic quantum number ([tex]\( m \)[/tex]) must satisfy [tex]\( -I \leq m \leq I \)[/tex].
For each valid [tex]\( I \)[/tex], let's identify the possible values for [tex]\( m \)[/tex]:
- If [tex]\( I = 0 \)[/tex], then [tex]\( m \)[/tex] can only be [tex]\( 0 \)[/tex].
- If [tex]\( I = 1 \)[/tex], then [tex]\( m \)[/tex] can be [tex]\( -1, 0, \)[/tex] or [tex]\( 1 \)[/tex].
- If [tex]\( I = 2 \)[/tex], then [tex]\( m \)[/tex] can be [tex]\( -2, -1, 0, 1, \)[/tex] or [tex]\( 2 \)[/tex].

Now, let's check each of the provided quantum numbers:

1. [tex]\( I = 3 \)[/tex]
- This is not valid because it does not lie within the range [tex]\( 0 \leq I \leq 2 \)[/tex].

2. [tex]\( m = 3 \)[/tex]
- For any [tex]\( I \)[/tex], [tex]\( m \)[/tex] values range from [tex]\( -I \)[/tex] to [tex]\( I \)[/tex]. Since the highest [tex]\( I \)[/tex] can be [tex]\( 2 \)[/tex], [tex]\( m = 3 \)[/tex] is not valid.

3. [tex]\( I = 0 \)[/tex]
- This is valid because [tex]\( 0 \leq I \leq 2 \)[/tex].

4. [tex]\( m = -2 \)[/tex]
- For [tex]\( I = 2 \)[/tex], [tex]\( m = -2 \)[/tex] is valid as it lies within [tex]\( -2 \leq m \leq 2 \)[/tex].

5. [tex]\( I = -1 \)[/tex]
- This is not valid because [tex]\( I \)[/tex] must be non-negative and [tex]\( -1 \)[/tex] is less than [tex]\( 0 \)[/tex].

6. [tex]\( m = 2 \)[/tex]
- For [tex]\( I = 2 \)[/tex], [tex]\( m = 2 \)[/tex] is valid as it lies within [tex]\( -2 \leq m \leq 2 \)[/tex].

Summarizing our findings, the valid quantum numbers from the given options are:
[tex]\[ I = 0 \][/tex]
[tex]\[ m = -2 \][/tex]
[tex]\[ m = 2 \][/tex]

Thus, the valid quantum numbers for the electron shell [tex]\( n = 3 \)[/tex] from the given list are:
[tex]\[ ([0], [-2, 2]). \][/tex]
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