# Shape of P-Orbital

**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 p

_{0}, 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 p_{z}, p_{x}and p_{y}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.Ïˆ

^{2}_{2,1,0 }= RÎ˜^{2}_{1,0 }Î¦Ïˆ

_{,1,+1 }= R_{,1}Î˜_{,1}Î¦Ïˆ

_{,1,-1 }= R_{1}^{Î˜}_{,1}Î¦For the three 2p orbitals, the plots of

*R*_{2,1}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 R _{2},_{1} Versus r and (b) Plot of, R_{,1} Versus r. Here a_{o} 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 Î˜ and Î¦. Before describing the plot of Ïˆ

*2, it is essential to consider these directional characteristics.***Directional Characteristics of**

*Ïˆ*

_{2, 1, 0 }The function Î¦

_{0}is found to be independent of angle Ï•. Thus, the directional characteristics of Î˜

_{ 1,0 }Î¦

_{0}is found to be the same as that of Î˜

_{ 1, 0}. The plots of Î˜

_{ 1, 0}versus Î¸ and Î˜

^{2}

_{1,0}versus Î¸ are shown as follows:

The plot of Î˜

_{1, 0}versus Î¸ looks like two circles, each passes through origin and are symmetrical placed about the*z*-axis. It is because of these reasons, the orbital Ïˆ_{ 2, 1, 0}is known as 2_{pz }orbital. The plot of Î˜^{2}_{1,0}versus Î¸ 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

*Ïˆ*_{2,1,+1}*Ïˆ***and**_{2,1,-1}**Orbitals:**Without going into detail, it may be stated that 2p_{+1}(= Ïˆ_{ 2, 1, +1}) and 2p_{-1 }(= Ïˆ_{ 2, 1, -1}) cannot be plotted as such as they involve an imaginary quantity in the Î¦ functions. This difficulty is, however, removed by taking linear combinations (Î¦_{+1}+ Î¦_{-1}) and i (Î¦_{+1}- Î¦_{-1}). The resultant orbitals are designated as follows:2p

2p

*=*_{x}*R*_{2,1}Î˜_{1,1}{Î¦_{+1}+ Î¦_{-1}}2p

*=*_{y}*R*_{2,1 }Î˜_{1,1}{i(Î¦_{ +1 }-Î¦_{-1})}The plots of 2p

*and 2p*_{x}*orbitals are of the type 2*_{y}_{pz }except that the directions are now pointing in the*x*- and*y*-axis, respectively. These are shown as follows: