It is known that the possible values for the magnetic quantum number m range between -9 and +9. Determine the orbital quantum number and the smallest possible value of the principle quantum number.
l =
n =
10
To determine the orbital quantum number (l) and the smallest possible value of the principle quantum number (n), we need to understand the relationship between these quantum numbers and the allowed values for the magnetic quantum number (m).
The magnetic quantum number (m) represents the orientation of the orbital in a magnetic field and can range from -l to +l, where l is the orbital quantum number. This means that the possible values for m depend on the value of l.
Given that the possible values for m range between -9 and +9, we can infer that the value of l must be such that |l| ≤ 9. The smallest possible value for l would then be -9.
Now, the relationship between l and the principle quantum number (n) is that l can have integer values from 0 to (n-1). Therefore, we need to find the smallest possible value for n that is consistent with the value of l.
Since we have determined that l = -9, we can set up the following inequality:
0 ≤ l ≤ (n-1)
0 ≤ -9 ≤ (n-1)
To find the smallest possible value for n, we need to solve the inequality:
-9 ≤ (n-1)
By adding 1 to both sides of the inequality, we get:
-9 + 1 ≤ n
-8 ≤ n
Therefore, the smallest possible value for the principle quantum number (n) is -8.
In summary:
l = -9
n = -8