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Shape of P-Orbital

Since the quantum number m fixes the direction of angular momentum, this number also fixes the direction of the orbital in space. For example, there are three orbitals of p-type (l = 1). 
These correspond to the three values of m (i.e. + 1, 0, - 1). The plots of p0, p+1 + p-1 and p+1 - p-1 indicate that these are dumb-bell shaped and are perpendicular to one another pointing towards z-, x- and y-axis, respectively. It is for this reason; they are also called as pz, px and py orbitals, respectively.

Shape of 2p Orbitals

For p orbitals, l = 1. There are three orbitals of this type. These correspond to the three different values (+1, 0, -1) of magnetic quantum number, m. The expressions of their probability distribution are given below.
ψ 22,1,0 = RΘ 21,0 Φ
ψ ,1,+1 =  R,1Θ ,1Φ
ψ ,1,-1 =  R1Θ ,1Φ
For the three 2p orbitals, the plots of R2,1 versus r and R versus r are the same as the function R depends only on the quantum numbers n and l. These are shown as follows:

 (a) Plot of R2,1 Versus r and    (b) Plot of, R,1 Versus r. Here ao is Bohr Radius (=52.9 pm)

In contrast to the 2s orbital, 2p orbital has minimum probability at the nucleus. It has a maximum value at about r = 104 pm and thereafter it decreases exponentially with the distance.
The 2p orbitals will have directional characteristics. It is due to the angular functions Θ and Φ. Before describing the plot of ψ  2, it is essential to consider these directional characteristics. 
Directional Characteristics of ψ 2, 1, 0  The function Φ0 is found to be independent of angle ϕ. Thus, the directional characteristics of Θ 1,0 Φ0 is found to be the same as that of Θ 1, 0. The plots of Θ 1, 0 versus θ and Θ 21,0 versus θ are shown as follows:

The plot of Θ1, 0 versus θ looks like two circles, each passes through origin and are symmetrical placed about the z-axis. It is because of these reasons, the orbital ψ 2, 1, 0 is known as 2pz orbital. The plot of Θ21,0 versus θ is no longer circular (Figure 1.18b).
The 90 % probability contour diagram which gives shape of the orbital is shown in Figure 1.19. This look like two spheroidal lobes, with the nucleus located between them.
Shapes of ψ 2,1,+1 and ψ 2,1,-1 Orbitals:  Without going into detail, it may be stated that 2p+1 (= ψ 2, 1, +1) and 2p-1 (= ψ 2, 1, -1) cannot be plotted as such as they involve an imaginary quantity in the Φ functions. This difficulty is, however, removed by taking linear combinations (Φ+1 + Φ-1) and i (Φ+1 - Φ-1). The resultant orbitals are designated as follows:
2px = R2,1Θ1,1{Φ+1 + Φ-1}

2py = R2,1 Θ1,1{i(Φ +1 -Φ-1)}
The plots of 2px and 2py orbitals are of the type 2pz except that the directions are now pointing in the x- and y-axis, respectively. These are shown as follows:

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