1. www.soran.edu.iq Inorganic chemistry Assistance Lecturer Amjad Ahmed Jumaa  Quantum numbers of an atomic orbital.  Let us learn the following concepts Labeling an atomic orbital.  Counting the number of orbitals associated with a principal quantum number.  Counting the number of electrons in a principal level. 1

2. www.soran.edu.iq  Quantum numbers of an atomic orbital 1-The principal quantum number (n): is a positive integer (1, 2, 3, and so forth), it indicates the relative size of the orbital and therefore the relative distance from the nucleus. It specifies the energy level of the hydrogen atom: the higher the (n) value, the higher the energy level. When the electron occupies an orbital with n=1, the hydrogen atom is in its ground state and has lower energy than when the electron occupies the n=2, orbital (first excited state).

3. www.soran.edu.iq 2-The angular momentum quantum number ( l ): is an integer from 0 to n – 1. It is related to the shape of the orbital, and is sometimes called the orbital shape quantum number. Note that is, ( n ) limits (l).for an orbital with n = 1, l can have a value of only 0. For orbitals with n =2, l can have a value of 0 or 1. For those with n =3, l can be 0, 1, or 2.  Note that: the number of possible (l) values equals the value of (n). 3-The magnetic quantum number ( m l ): is an integer from - l through 0 to +l. it is related to the orientation of the orbital in the space around the nucleus and it is sometimes called the orbital orientation quantum number.

4. www.soran.edu.iq Note that is, l determines the m l values. An orbital with l =0, can have only m l =0, an orbital with l =1, can have any one of three m l values -1, 0, or +1. Thus there are three possible orbitals with l =1, each with its own orientation.

5. www.soran.edu.iq Note that: the number of possible m l values equals the number of orbitals which is (2l+1), for a given l values. Or the number of orbitals= the number of m l =2l+1. The total number of orbitals for a given (n) value is n 2. is (2l+1), for a given l values. Or the number of orbitals= the number of m l =2l+1. The total number of orbitals for a given (n) value is n 2. Solved problem: What values of the angular momentum (l) and magnetic (m l ) quantum numbers are allowed for a principal quantum number (n) Of 3? Solution: Determining l values: for n = 3. l =0, 1, 2. Determining m l for each l value: For l= 0, m l =0 For l=1, m l = -1, 0,+1 For l= 2, m l = -2, -1, 0, +1, +2 There are nine (9) m l values, so there are nine orbitals with n=3. Solution: Determining l values: for n = 3. l =0, 1, 2. Determining m l for each l value: For l= 0, m l =0 For l=1, m l = -1, 0,+1 For l= 2, m l = -2, -1, 0, +1, +2 There are nine (9) m l values, so there are nine orbitals with n=3.

6. www.soran.edu.iq The energy states and orbitals of the atom are described with specific terms and associated with one or more quantum numbers: 1- Level: the atom's energy level or energy shells are given by the ( n ) value: the smaller the ( n ) value, the lower the energy level and the greater the probability of the electron being closer to the nucleus. 2-sublevel: the atom's levels contain sublevels or subshells, which designate the orbital shape, each sublevel has a letter designation: l= 0, is an s sublevel. l =1, is a p sublevel. l= 2, is a d sublevel. l =3, is an f sublevel. The energy states and orbitals of the atom are described with specific terms and associated with one or more quantum numbers: 1- Level: the atom's energy level or energy shells are given by the ( n ) value: the smaller the ( n ) value, the lower the energy level and the greater the probability of the electron being closer to the nucleus. 2-sublevel: the atom's levels contain sublevels or subshells, which designate the orbital shape, each sublevel has a letter designation: l= 0, is an s sublevel. l =1, is a p sublevel. l= 2, is a d sublevel. l =3, is an f sublevel.

7. www.soran.edu.iq Note: sublevels are named by joining the ( n ) value and the letter designation (s, p, d, f). For example: the sublevel (subshell) with n =2 and l =0 is called the 2s sublevel. 3-orbital: each allowed combination of n, l, and m l values specifies one of the atom's orbital. Note: sublevels are named by joining the ( n ) value and the letter designation (s, p, d, f). For example: the sublevel (subshell) with n =2 and l =0 is called the 2s sublevel. 3-orbital: each allowed combination of n, l, and m l values specifies one of the atom's orbital. The three quantum number that describe an orbital, express: Its size (energy), n. and shape, l, and spatial orientation, m l. Now: you can easily give the quantum numbers of the orbitals in any sublevel if you know the: A-sublevel letter designation. B-and the quantum number hierarchy. For example: for 2s sublevel has only one orbital and its quantum number are: n =2, l =0, m l =0. The three quantum number that describe an orbital, express: Its size (energy), n. and shape, l, and spatial orientation, m l. Now: you can easily give the quantum numbers of the orbitals in any sublevel if you know the: A-sublevel letter designation. B-and the quantum number hierarchy. For example: for 2s sublevel has only one orbital and its quantum number are: n =2, l =0, m l =0.

8. www.soran.edu.iq Number of orbital= number of m l value= 2 l +1=2×0+1=1. 3p sublevel has three orbitals: One orbital with n =3, l =1, and m l =-1. Another orbital with n =3, l =1, and m l =0. And the third with n=3, l =1, and m l =+1. Number of orbital for sublevel 3p = number of m l value= 2 l +1=2×1+1=3 orbital. Number of orbital= number of m l value= 2 l +1=2×0+1=1. 3p sublevel has three orbitals: One orbital with n =3, l =1, and m l =-1. Another orbital with n =3, l =1, and m l =0. And the third with n=3, l =1, and m l =+1. Number of orbital for sublevel 3p = number of m l value= 2 l +1=2×1+1=3 orbital. Solved problem: Give the name, magnetic quantum numbers, and number of orbitals for each sublevel with the following quantum numbers: (a) n= 3, l=2 (b) n=2, l=0 (c) n= 5, l= 1 (d) n= 4, l=3.

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10. 10  The number of orbitals = the number of m l values= 2l+1.

11. www.soran.edu.iq Finally, there is a fourth quantum number that tell us the spin of the electron, the electron spin quantum umber ( m s ), which has values of (+ 1/2) or (-1/2), which correspond to the two spinning motion of the electron.  Let us learn the following concepts: Labeling an atomic orbital:  Let us learn the following concepts: Labeling an atomic orbital: To "label" an atomic orbital, you need to specify the three quantum numbers ( n, l, m l ). That gives information about the distribution of electrons in orbital. Remember, ( m s ), tells us the spin of the electron, which tells us nothing about the orbital. Example: List the values of (n, l, m l ), for orbitals in the (2p) subshell.

12. www.soran.edu.iq  Counting the number of orbitals associated with a principal quantum number:  To work this type of problem, you must take into account the energy level (n)The types of orbitals in that energy level (l), and the number of orbitals in a subshell with a particular (l) value (m l ). Example: What is the total number of orbitals associated with the principal quantum number (n=2). Solution: For (n=2), there are only two possible values of (l)( 0 and 1). Thus, there is one (2s) orbital, and there are three (2p) orbitals. For (l=1, there are three possible (m l ) values, -1,0,+1.) Therefore, the total number of orbitals in the n=2 energy level is 1+3=4. Solution: The number given in the designation of the subshell is the principal quantum number, so in this case n =2. For ( p orbitals), l =1. ( m l ) can have integer values from ( - l to +l ). therefore, ( m l ) can be -1,0, and +1. (the three values for ( m l ), correspond to the three ( p ) orbitals.). Solution: The number given in the designation of the subshell is the principal quantum number, so in this case n =2. For ( p orbitals), l =1. ( m l ) can have integer values from ( - l to +l ). therefore, ( m l ) can be -1,0, and +1. (the three values for ( m l ), correspond to the three ( p ) orbitals.).

13. www.soran.edu.iq Tip: the total number of orbitals with a given (n value) is (n 2 ). For example, the total number of orbitals in the (n=2 level) equals (2 2 =4)  Counting the number of electrons in a principal level: To this type of problem, you need to know that the number of orbitals with a particular (l) value is ( 2l+1). Also each orbital can accommodate two electrons. Example: What is the maximum number of electrons that can be present in the principal level for which n=4. Solution: When n=4, l=0,1,2, and 3. The number of orbitals for each (l) value is given by: Example: What is the maximum number of electrons that can be present in the principal level for which n=4. Solution: When n=4, l=0,1,2, and 3. The number of orbitals for each (l) value is given by:

14. www.soran.edu.iq Value of (l)Number of orbitals (2l+1) 01 13 25 37  The total number of orbitals in the principal level n =4 is sixteen. Since each orbital can accommodate two electrons, the maximum number of electrons that can reside in the orbitals is (2 x 16 = 32). the above result can be generalized by the formula (2n 2 ). For example, we have n=4, so (2(4) 2 )= 32).