# Shapes of Orbitals

Designation of sub –shell à               s*         p          d          f           g          h          i

Table 5.3 shows the permissible values of l for a given principal quantum number and the corresponding sub-shell notation.

Table 5:3 Sub-shell Notations

n                      l                       Sub-shell notation

1                      0                                  1s

2                      0                                  2s

2                      1                                  2p

3                      0                                 3s

3                      1                                  3p

3                      2                                  3d

4                      0                                  4s

4                      1                                  4p

4                      2                                  4d

4                      3                                  4f

THE MAGNETIC QUANTUM N UMBER (m1)

This quantum number which is denoted by m1 refers to the different orientations of electron cloud in a particular sub-shell. These different orientations are called orbitals. The number of orbitals in a particular sub-shell within a principal energy level is given by the number of values allowed to m1 which in tum depends on the values of l. The possible values of m1 range from +l through 0 to -l, thus making a total of (2l + 1) values. Thus, in a sub-shell, the number of orbitals is equal to (2l + 1).

For l = 0 (i.e., s-sub-shell), m1 can have only one value, m1 = 0. It means that s-sub-shell has only one orbital.

For l = 1 (i.e., p-sub-shell), m1 can have three values, + 1, 0 and -1. This implies that p- sub-shell bas three orbitals.

For l = 2 (i.e., d-sub-shell), m1 can have five values, +2, +1, 0, -1, -2. It means that d-sub-shell has five orbitals.

For l = 3 (,i -.e2.,,f sub-shell), m1 can have seven values, +3, -3. It means that f-sub-shell has seven orbitals.

For l = 2 (i.e., d-sub-shell), m1 can have five values, +2, + 1, 0, -1, -2. It means that d sub-she U has five orbitals.

For l = 3 (i.e., /-sub-shell), m1 can have seven values, + 3, +2, +1, 0, -1, -2, -3. It means that f-sub-shell has seven orbitals.

The number of orbitals in various types of sub-shells are given below in tabular form.

Sub-shell                                 s           p          d          f           g

Value of l                                0          1          2          3          4

No. of orbitals (2l + 1)            1          3          5          7          9

The relationship between the principal quantum number (n), angular momentum quantum number (Z) and magnetic quantum number (m1) is summed up in Table 5.4.

A boundary surface which encloses the regions of maximum probability (say 90%) best describes the shape of the orbital. It is not possible to draw a boundary surface diagram which encloses the region of 100% probability because probability density has always some value, howsoever small it may be, at any finite distance from the nucleus. The boundary surface diagrams for s-orbitals are spherical in shape. The size of the s-orbital, however, increases with

increase in value of n. If we observe the probability density curve for 2s orbital we find that the probability density function for 2s orbital decreases to zero, increases to maximum value and finally approaches zero with increase in distance from the nucleus. The region where the probability density function reduces to zero is called nodal surface or node. In any ns orbital there are (n – 1) nodes. Thus, 2s orbital has only one node.

The boundary surface diagrams of the three 2p orbitals are not spherical. Each p-orbital consists of two lobes which are separated by a region of zero probability called node. The three 2p orbitals lie along x-, y- and z-axis respectively and are designated as 2p x’ 2p Y and 2p ,.

Like s-orbitals, p-orbitals also increase in size with the increase in value of n.

The boundary surface diagrams of ls, 2px, 2py and 2pz orbitals are shown in Fig. 5.24.

Fig. 5.24. The boundary surface diagrams of 1 s and 2p orbitals.

The five d-orbitals are designated as d xy , d yz , d xz , dx 2 -y 2 and d z2 The boundary surface diagrams of the five 3d orbitals z are shown in Fig. 5.25.

Each d-orbital  has two nodal planes or angular nodes. For example, d xy orbital has two nodal planes passing through the origin and bisecting the principal axes.

Fig. 5.25. Boundary surface diagrams of the five 3d orbitals.

An alternative way to describe the probability is in terms of negative charge cloud, the density of charge cloud being proportional to w2 or probability. In this method, the charge cloud is represented in term of small dots. To understand this method of representation, let us imagine the electron as a very small dot. Let us further suppose that we take a very large number of photographs of the electron in a hydrogen atom at very short intervals of time on the same film. If the film were developed, the picture obtained would be similar to that shown in Fig. 5.26 (a). In terms of electron cloud representation the probability of finding the electron in a particular region of space is directly proportional to the density of such dots in that region.

The charge cloud pictures of ls, 2s and 3s orbitals have been given in Fig. 5.26.

Fig. 5.26. (a), Charge cloud picture of 1s orbital

(b) Charge cloud picture of 2s orbital

(c) Charge cloud picture of 3s orbital

For p-orbitals there are three possible orientations of electron cloud. These three orientations or orbitals of a p-sub-shell are designated as P x, P y and P z or P +t‘ p-1 and Po respectively. P x‘ P y and P z orbitals are oriented along x-axis, y-axis and z-axis respectively. Each p-orbital has two lobes which are separated by a plane of zero probability called nodal plane. Each p-orbital. is, thus, dumb-bell shaped. The spatial distributions of 2p orbitals are shown in Fig. 5.27.

Fig. 5.27. 2p orbitals.

In the absence of an external electric or magnetic field, the three p-orbitals of particular energy level have same energy. Such orbitals which have equal· energy are called degenerate orbitals. In the presence of external magnetic or electric field this degeneracy is lost because  the three p- orbitals are oriented differently with respect is applied field.

SOLVED EXAMPLES

Example 5.10  (a) What sub-shells a