Which of the following is a legitimate set of n, l, m(l), and m(s) quantum numbers?
a. 4, -2, -1, 1/2
b. 4, 2, 3, 1/2
c. 4, 3, 0, 1
d. 4, 0, 0, -1/2
I don't think it's c because m(s) can only have +1/2 or -1/2 but I'm not sure how to figure out the others. Could you please explain why it's correct?
There are no correct answers listed.
If n = 4, then l may be 3,2,1,0 so a is out. b,c,d are allowed from n alone.
Look at b. If l = 2, then m(l) may be -2,-1,0,1,2 so b is out since m(l) of 3 is not allowed.
c is out for the reason you point out. Check out d.
n of 4 allows l = 0
l of 0 allows m(l)(ell) of +l(ell) to -l(ell) including 0 so that is ok.
m(s) is -1/2 is ok.
So it must be d.
Note that I started with there is no correct answer but show the correct answer as D. I just didn't go back and remove that first sentence.
To determine which set of quantum numbers is legitimate, we need to consider the rules and restrictions for each quantum number.
The n quantum number represents the principal (or energy) level of an electron. It can be any positive integer value starting from 1. In the given options, all choices have n = 4, so that requirement is satisfied for all options.
The l quantum number represents the orbital angular momentum. It can have non-negative integer values from 0 up to (n-1). Let's check each option:
a. 4, -2, -1, 1/2: The l value of -2 is not a valid value since it should be a non-negative integer.
b. 4, 2, 3, 1/2: The l value of 2 is valid since it falls within the range of 0 to (n-1), where n = 4. So this option satisfies the requirement.
c. 4, 3, 0, 1: The l value of 3 is not a valid value since it should be a non-negative integer.
d. 4, 0, 0, -1/2: The l value of 0 is valid since it falls within the range of 0 to (n-1), where n = 4. So this option satisfies the requirement.
Next, the m(l) quantum number represents the magnetic quantum number. It can have integer values ranging from -l to +l. Let's check each option:
a. 4, -2, -1, 1/2: The m(l) value of -1 is valid since it falls within the range of -2 to +2, where l = 2.
b. 4, 2, 3, 1/2: The m(l) value of 3 is not a valid value since it should be within the range of -l to +l, where l = 2.
c. 4, 3, 0, 1: The m(l) value of 0 is valid since it falls within the range of -l to +l, where l = 3.
d. 4, 0, 0, -1/2: The m(l) value of 0 is valid since it falls within the range of -l to +l, where l = 0.
Lastly, the m(s) quantum number represents the spin of an electron and can have two values: +1/2 (spin-up) or -1/2 (spin-down). Let's check each option:
a. 4, -2, -1, 1/2: The m(s) value of 1/2 is valid since it falls within the allowed values of +1/2 or -1/2.
b. 4, 2, 3, 1/2: The m(s) value of 1/2 is valid since it falls within the allowed values of +1/2 or -1/2.
c. 4, 3, 0, 1: The m(s) value of 1 is not a valid value since it should be either +1/2 or -1/2.
d. 4, 0, 0, -1/2: The m(s) value of -1/2 is valid since it falls within the allowed values of +1/2 or -1/2.
Based on the analysis, the only option that satisfies all the requirements for legitimate quantum numbers is:
Option b. 4, 2, 3, 1/2
To determine which set of quantum numbers is legitimate, we need to consider the rules governing the values each quantum number can have.
The first quantum number, n, represents the principal quantum number. It determines the energy level, or shell, of the electron and can have any positive integer value.
The second quantum number, l, represents the azimuthal quantum number. It determines the shape of the electron's orbital and can have values ranging from 0 to n-1.
The third quantum number, ml, represents the magnetic quantum number. It specifies the orientation of the orbital within a particular subshell and can have values ranging from -l to +l.
The fourth quantum number, ms, represents the spin quantum number. It describes the spin state of the electron and can only have the values +1/2 (spin-up) or -1/2 (spin-down).
Now, let's analyze each set of quantum numbers provided:
a. 4, -2, -1, 1/2
This set violates the rule for the azimuthal quantum number (l) because it has a negative value (-2) which is not allowed.
b. 4, 2, 3, 1/2
This set satisfies the rules for all four quantum numbers. It has a valid principal quantum number (n = 4), a valid azimuthal quantum number (l = 2), a valid magnetic quantum number (ml = 3), and a valid spin quantum number (ms = 1/2).
c. 4, 3, 0, 1
As you correctly mentioned, this set violates the rule for the spin quantum number (ms) because it has a value of 1, which is not allowed.
d. 4, 0, 0, -1/2
This set satisfies the rules for all four quantum numbers. It has a valid principal quantum number (n = 4), a valid azimuthal quantum number (l = 0), a valid magnetic quantum number (ml = 0), and a valid spin quantum number (ms = -1/2).
Therefore, the correct answer is option b. The set of quantum numbers 4, 2, 3, 1/2 is legitimate.