![For a given value of the principal quantum number n, the azimuthal quantum number l may have all integral values from 0 to (n-1), each of which represents a different sub-level, sub-shell or sub-orbit and are usually denoted by letters s, p, d and f.[The letters s, p, d and f designate old spectral terms, viz, sharp, principal, diffuse and fundamental respectively.]For example, n=1;(n-1)=0; l=0 (s sub-level)n=2;(n-1)=1; l=0 (s sub-level), 1(p sub-level) n=3, (n-1) = 2; l=0(s sub-level), 1(p sub-level), 2(d sub-level)n=4, (n-1) =3; l=0 (s sub-level), 1(p sub-level), 2(d sub-level),3(f sub- level)Value of l: 0 1 2 3 Symbol of sub-shell: s p d fThe energies of the various sub-shell in the same shell are in the order s<p<d<f.](https://image.slidesharecdn.com/quantumnumbers-100507003716-phpapp01/85/Quantum-numbers-7-320.jpg)
Quantum numbers
- 1. QUANTUM NUMBERS The numbers used for completely characterizing each electron of atom are known as “quantum numbers”. Four such quantum numbers are found to be necessary for describing an electron completely.Principal quantum numberAzimuthal quantum numberMagnetic quantum numberSpin quantum number
- 2. 1. PRINCIPAL QUANTUM NUMBER (n):It was proposed by Bohr.
- 3. This number gives the average distance of the electron from the nucleus and corresponds to the principal energy level to which the electron belongs.Thus it gives an idea of the position of the electron around the nucleus.The energy of electron and volume of electron cloud are dependent upon this number.
- 4. Higher the principal quantum number, greater is its distance from the nucleus, greater is its size and also higher is its energy.
- 5. Although theoretically its value may be from 1 to ∞, only values from 1 to 7 have so far been established for atoms of the known elements.These are designated either as 1,2,3,4,5,6 and 7 or as K,L,M,N,O,P and Q respectively.The maximum number of electrons in n principal quantum number is given by 2n22. AZIMUTHAL QUANTUM NUMBER (l):It was proposed by Sommerfeld.
- 6. This number denotes the sub-level (orbital) to which the electron belongs and also determines the shape of the orbital and the energy associated with the angular momentum of the electron.
- 7. For a given value of the principal quantum number n, the azimuthal quantum number l may have all integral values from 0 to (n-1), each of which represents a different sub-level, sub-shell or sub-orbit and are usually denoted by letters s, p, d and f.[The letters s, p, d and f designate old spectral terms, viz, sharp, principal, diffuse and fundamental respectively.]For example, n=1;(n-1)=0; l=0 (s sub-level)n=2;(n-1)=1; l=0 (s sub-level), 1(p sub-level) n=3, (n-1) = 2; l=0(s sub-level), 1(p sub-level), 2(d sub-level)n=4, (n-1) =3; l=0 (s sub-level), 1(p sub-level), 2(d sub-level),3(f sub- level)Value of l: 0 1 2 3 Symbol of sub-shell: s p d fThe energies of the various sub-shell in the same shell are in the order s<p<d<f.
- 8. It is important to note that as we go up into the higher energy levels the sub-shells start overlapping, eg. 3d sub-level is higher than the 4s sub-level and thus the energy of 3d sub-level is higher than the 4s sub-level.1s1p2s2p3p3s3d4p4f4d4s5d5s5p6s6p6dMemory - aid for remembering order of increasing energy7s
- 9. AZIMUTHAL QUANTUM NUMBER (CONTINUED)The maximum number of electrons which can be held by these sub-levels is given by 2(2l+1).
- 10. Thus sub-level (where l=0) can have 2 electrons, p can have 6, d can have 10 and f can have 14 electrons.3. MAGNETIC QUANTUM NUMBER (m or ml):It was proposed by Lande to explain Zeeman and Stark effects.
- 11. This quantum number determines the preferred orientations of orbitals in space.
- 12. This number gives number of orbitals in sub-orbit.
- 13. The value of m is dependent upon the value of l and can take only integral value from +l to -l through zero.
- 14. In general, for each value of l there will be (2l+1) values of m.For l=0 (s sub-shell), m=1; i.e., 0. Hence there is only one orientation for the s sub-shell.For l=1 (p sub-shell), m=3; i.e.; -1, 0, +1. Hence three orientations are possible for the p sub-shell. The three corresponding orbitals are written as px, py and pz.For l=2 (d sub-shell), m=5; i.e.; -2, -1, 0, +1, +2. Hence d sub-shell can have five different orientations, and orbitals corresponding to these are dxy, dyz, dzx, dx2-y2, and dz2. For l=3 (f sub-shell), m=7; i.e.; -3, -2, -1, 0, 1, 2, 3. Hence seven f orbitals are possible.
- 15. 4. SPIN QUANTUM NUMBER (s or ms):It was proposed by Uhlenbeck and Goudsmit.
- 16. This quantum arises due to the spinning of the electron about its own axis.
- 17. This spin can be clockwise represented by +1/2 or ; or anticlockwise represented by -1/2 or .
- 18. In case the two electrons possess the same sign of spin quantum number, these are known to have parallel spin or , while the two electrons having the opposite sign of spin quantum number are known to have antiparallel, opposite or paired up spins( ).
- 19. Remember that this quantum number is independent of other quantum numbers.Spin quantum number is the only quantum number that has non-integral values.Thus, the four quantum numbers completely describe the position of an electron in an atom by defining its principal energy level (n), sub-level (l), the orientation of the orbital (m) and the direction of the spin (s). Thus, they make it possible to identify an electron in an atom by giving its complete address.
- 20. The set of four quantum numbers for the two electrons of helium are as shown below: n l m s 1 0 0 +1/2 1 0 0 -1/2The set of quantum numbers for the third electron of lithium are as follows: n l m s 2 0 0 +1/2
- 21. Relationship between the various Quantum numbersThe first shell’s {1st principal energy level (n=1)} azimuthal quantum number (l)= (n-1)=0. It therefore has one sub-level i.e. 1s. Its sub-level has one orbital and therefore has magnetic quantum number (m)= (-l,…,0,…+l)=0 The second shell’s {2nd principal energy level (n=2)} azimuthal quantum number (l)=from 0 to (n-1)=1 i.e. o (2s) and 1 (2p). It therefore has two sub-levels, i.e. 2s and 2p (total = 2).Its 2s sub-level has one orbital {magnetic quantum number (m)=0}. Its 2p orbital has three orbitals {magnetic quantum number (m) = -1, 0, +1 }. The second sub-shell, therefore has two sub-levels which have total fourorbitals [=n2 =22 =4]. The third shell’s {3rd principal energy level (n=3)} azimuthal quantum number (l)=0 to 2 i.e. 0(3s), 1(3p), 2(3d). It therefore has total three sub-levels. Its 3s sub-level has one orbital {magnetic quantum number (m)=0}. Its 3p sub-level has three orbitals {magnetic quantum number (m)=-1, 0, +1}. Its 3d sub-level has five orbitals {magnetic quantum number (m)= -2, -1, 0, +1, +2}. The third sub-level therefore has three sub-levels which have total nine orbitals [=n2 =32 =9].The fourth shell’s {3rd principal energy level (n=4)} azimuthal quantum number (l)=from 0 to 3 i.e. 0 (4s), 1(4p), 2(4d), 3(4f). It therefore has total four sub-levels. Its 4s sub-level has one orbital {magnetic quantum number (m)=0}. Its 4p sub-level has three orbitals {magnetic quantum number (m)=-1, 0, +1}. Its 4d sub-level has five orbitals {magnetic quantum number (m)= -2, -1, 0, +1, +2}.Its 4f sub-level has seven orbitals {magnetic quantum number (m)= -3, -2, -1, 0, +1, +2, +3}.The fourth sub-level therefore has four sub-levels which have total sixteenorbitals [n2 =42 =16]
- 26. The maximum number of electrons in n principal quantum number is given by 2n2For a given value of the principal quantum number n, the azimuthal quantum number l may have all integral values from 0 to (n-1)The maximum number of electrons which can be held by these sub-levels is given by 2(2l+1).In general, for each value of l there will be (2l+1) values of mFor l=2 (d sub-shell), m=5; i.e.; -2, -1, 0, +1, +2. Hence d sub-shell can have five different orientations, and orbitals corresponding to these are dxy, dyz, dzx, dx2-y2, and dz2.