Question

Quantum numbers arise naturally from the mathematics used to describe the possible states of an electron in an atom. The four quantum numbers, the principal quantum number (n),(n), the angular momentum quantum number (?),(?), the magnetic quantum number (m?),(m?), and the spin quantum number (ms)(ms) have strict rules which govern the possible values.

Identify all allowable combinations of quantum numbers for an electron.

1. n=3, ?=-2, m? = -2, ms = +1/2

2. n=3, ?=2, m? = -2, ms = -1/2

3. n=5, ?=4, m? = -1, ms = -1/2

4. n=2, ?=0, m? = 0, ms =1

5. n=1, ?=1, m? = -1, ma = -1/2

6. n=4, ?=3 m? = 4, ms = -1/2

Answer 1

Quantum numbers are the numbers which give complete information of electrons such as position, energy.....

There are 4 quantum numbers.

1) Principal quantum number:

i) It is proposed by Bohr & is denoted by "n"

ii) It has values 1,2,3,4,5........so on

iii) It gives energy of electron and size of the orbit.

2) Azimuthal quantum number:

i) It is proposed by Sommerfeld & is denoted by "l"

For a given principal quantum number, there are n values of Azimuthal quantum numbers.

ii) It has values 0,1,2,3,4,5........(n-1).

iii) It gives shape of the orbital and number of sub-shells present in a main shell.

If l = 0, S-orbital (spherical shape); l =1, p- orbital (dumbbell shape)......

3) Magnetic quantum number:

i) It is proposed by Lande & is denoted by "m or m​​​​​​l"

For a given l, there are (2l+1) m values

ii) It has values from -l, -(l-1),.......-3,-2,-1, 0, +1, +2,+3,......+(l-1), +l.

iii) It gives orientation of orbital in space

4) Spin quantum number:

It is proposed by Ullenbeck and Goldsmith.

It is denoted by "s or m​​​​​​s"

It has 2 values +1/2 & -1/2.

It gives spin of the electron.

Option-1:

Given that n = 3

if n = 3, l has values from 0, 1, 2.

Therefore l = 2 is acceptable.

For l = 2, there are 5 (2*2+1) m values.

Those m values are -2,-1,0, +1, +2.

Therefore m = -2 is acceptable.

m​​​​​​s or s value +1/2 is acceptable.

But given that l = -2.

So it is not acceptable

Hence option-1 is not an allowable combination.

Option-2:

Given that n = 3

if n = 3, l has values from 0, 1, 2.

Therefore l = 2 is acceptable.

For l = 2, there are 5 (2*2+1) m values.

Those m values are -2,-1,0, +1, +2.

Therefore m = -2 is acceptable.

m​​​​​​s or s value -1/2 is acceptable.

Hence option-2 is allowable combination.

Option-3:

Given that n = 5

if n = 5, l has values from 0, 1, 2, 3, 4

Therefore l = 4 is acceptable.

For l = 4, there are 9 (2*4+1) m values.

Those m values are -4, -3, -2,-1,0, +1, +2, +3, +4

Therefore m​​​​​​l = -1 is acceptable.

m​​​​​​s or s value -1/2 is acceptable.

Hence option-3 is allowable combination.

Option-4:

Given that n = 2

if n = 2, l has values from 0, 1.

Therefore l = 0 is acceptable.

For l = 0, there is only 1 (2*0+1) m values.

This m value is 0

Therefore m​​​​​​l = 0 is acceptable.

m​​​​​​s or s value 1 is not acceptable.

Hence option-4 is not an allowable combination.

Option-5:

Given that n = 1

if n = 1, l has only one value ie l = 0

Therefore l = 0 is acceptable.

But given that l = 1

So it is not acceptable

Hence option-5 is not an allowable combination.

Option-6:

Given that n = 4

if n = 4, l has values from 0, 1, 2, 3

Therefore l = 3 is acceptable.

For l = 3, there are 7 (2*3+1) m values.

Those m values are -3, -2,-1,0, +1, +2, +3

Therefore m​​​​​​l = 4 is not acceptable

Hence option-6 is not an allowable combination.

Options 2 & 3 are only allowed combinations. The rest are not allowable combinations.

option 1 is not allowed due to l = -2 is not acceptable.

Option-4 is not allowed combination due to s or m​​​​​​s = 1 not acceptable.

Option-5 is not allowed combination due to l = 1 not acceptable (since n = 1)

Option-6 is also a not allowed combination due to m = 4 is not acceptable ( since l=3)