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Shape of Orbitals

Concept of Orbitals
A large number of orbitals are possible in an atom. An orbital of smaller size means there is more chance of finding the electron near the nucleus. Similarly the shape and orientation mean that there is more probability of finding the electron along certain directions than along others.

Shape of Orbitals

The symbols s, p, d and f find their origin from the words sharp, principal, diffuse and fundamental, respectively, which have been used to identify the spectral lines in the atomic spectra of different atoms.
The total permitted values of m for a given value of l gives the number of orbitals of one type within a subshell. 
For example, there is one orbital of s-type if the value of l is zero, three p-orbitals if the value of l is one, five d-orbitals if the value of l is two and so on.

Shape of 1s Orbital

For s orbitals, the values of quantum numbers l and m are l = 0 and m = 0. For these values, the functions Θ and Φ are independent of angles θ and ϕ , respectively (see tables 1.2 and 1.3).
Each of these two functions is equal to a constant term. Hence, for such orbitals, equation (1.26) is reduced to
ψ2n,0,0  α R2n,0

The plots of R1,0 versus r and R21,0 versus r are shown.

The plots (a)  R1,0 Versus r and (b) R21,0 versus r.
Here ao Represents the Bohr Radius (52.9 pm)

It can be seen from the above that for 1s orbital, the probability of finding the electron is maximum at r = 0 and it decreases exponentially with distance r. As described earlier, the plots of R and R2 are spherical symmetrical around the nucleus, so such plots exist all around the nucleus.
Three-dimensional plots of ψ2 versus r can be shown conveniently by either dot-population picture (or the electron-cloud density pattern) or by boundary surface (or equal probability contour) plots.
In the dot-population picture, the value of relative probability at a given location is shown by the density of dots near that location. The dot-population picture gives the most realistic description of the electron's time average distribution. The dot-population picture corresponding to the plot versus r is shown.
In the equal probability contours, we draw the contours by joining the points of identical probability. For 1s orbital (in fact, for any s orbital), these contours are spherical symmetrical around the nucleus.
If we are contented with a total of 90-95 % probability of finding the electron, we can draw the contour within which there exists a total of 90 % probability of finding the electron. For 1s orbital, 90 % probability contour diagram (or the shape of orbital) is shown. 

The dot - Population Picutre
(or the Electroncloud Density Pattern) for 1s Orbital
Shape of 1s Orbital

Shape of 2s Orbital

As stated above, for 2s orbital we will have
ψ ,0,0   R,0

The plots of R2,0 versus r and R0, versus r are shown.
For a given value of r, the function R may have positive, zero or negative value. Thus, the plot of R also often includes the sign of the value of R . The probability plots are always positive as the square of a positive or negative quantity is always positive.


It can be seen from the above that for 2s orbital, there are two maxima in the R0 versus r plot, one at r = 0 and the other at about r = 210 pm. In between these two maxima, probability becomes zero at about r = 105 pm. This point is known as nodal point. The dot-population picture of 2s orbital is shown in Figure 1.15 whereas 90 % probability contour diagram (or shape of 2s orbital) is shown in Figure 1.16. The size of 2s orbital will be larger than that of 1s orbital as the most of charges in 2s orbital reside farther away from the nucleus as compared to 1s orbital.

Ploof R2,o Versus r and R,0Versus r. Here ao Represents Bohr Radius ( = 52.9 pm)


 The dot - population Picture of 2s Orbital  Shape of 2s Orbital

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