We had earlier studied that `psi_r^ 2` gives the radial probability density of finding an electron at a point. It refers to the radial probability of finding an electron in a unit volume in an atom at a radial distance of r from the nucleus . Hence, total radial probability in a spherical shell of thickness `dr` at a radial distance of r from the nucleus (which will have a volume of `4pir^2dr`) is given by `4pi r^2psi_r^ 2dr`. Sometimes, `psi_r^ 2` is often represented as `R^2`

As stated above, the radial probability density at a radial distance r is `R^2(r)`. Therefore radial probability of finding the election in a volume `dv` will be `R^2(r) dv`.

The radial probability is the probability of finding the election in a radial shell between spheres of radii `r` and `r + dr`, where `dr` is small radial distance.

Volume of spherical shell of thickness

`dr = 4/3 pi(r + dr )^2 - 4/3 pir^2`

`= 4/3 pi [ r^3 +(dr)^3 + 3rdr(r +dr) -r^3]`

`=4/3pi [ (dr)^2 + 3r^2 dr + 3r(dr)^2]`

Since dr is very small, so the terms can be neglected.

`=> dv = 4/3 pi xx 3r^2dr = 4 pir^2dr`

Radial probability of finding an `e^-` in a shell of thickness `dr` at a distance ` 'r'`

`= R^2(r) xx 4 pi r^2 dr ; = 4 pir^2R^2(r) dr`

� Radial Probability Density curves are between `psi_r^2 vs r` whereas Radial Probability Distribution curves which would be more useful are between `4 pi r^2 psi r^2` vs `r`.

(i) For the same value of `n`, the distance of max probability, `r_text(max)` of various orbitals is inversely dependent upon the value of `l`

`l uparrow , r_(text(max)) downarrow`

`(r_(text(max)))_(3s) > (r_(text(max)))_(3p) > (r_(text(max)))_(3d)`

(ii) Penetration power of an orbital is a measure of its closeness to the nucleus. Due to the additional maximas in `3 s` curve, electron in `3 s` spends some of its time near the nucleus making it to be more penetrating than `3 p` which in turn more penetrating than `3 d`.

`undersettext(Decreasing order of penetration powder) overset(3s > 3p > 3d)->`


`text(Important points to note about the)` `4 p r^2R^2` `text(vs. r plots:)`

i) Radial probability is ALWAYS SMALL near the nucleus (`4 p r^2` small near the nucleus).

ii) The maximum in the `4 p r^2R^2` `1 s` vs. `r` plot occurs at `0.53` `A^0` - just the radius of the `n=1` orbit of the Bohr model.

iii) On average a `2s` electron spends its most time at a greater distance from the nucleus than the `1 s` electron which is consistent with the observation that (r_(1 s) )_text(max) < ( r_(2s))_text(max)

iv) The position of the principal (i.e. largest) maximum depends on `n` and `I`. For fixed `l`, as `n` increases the position of the principal maximum moves to larger `r` values. (For fixed `n`, the position of the principal maximum moves to shorter `r` values as `l uparrow`)


i.e., `(r_(2p)_(max) < (r_(2s))_(max) ; (r_(3d))_(max) < (r_(3p))_(max) < (r_(3s))_(max) : r_(2p)_(max) < ( r_(3p))_(max)`

Angular probability density `f^2(theta , phi)` determines the shape of orbitals and its orientation in space

`f(theta ,phi)` for s-orbitals does not depend upon angles `theta & phi `, for all other orbitals `f(theta , phi)` will be a function in terms of `theta & phi `. The angular probability density curves are same as the shape and orientation of the orbital in space.

Total no. of angular nodes for any orbital = `l`

For `s` orbital there will be no angular node.

See fig.1.

For `P_x` orbital, ( = l = no. of angular nodal = 1 (yz plane)

For `P_y` orbital, nodal plane (xz plane)

For `P_z` orbital, nodal plane (xy plane)

For `d_(xy)` orbital, ' = `2` therefore 2 nodal planes, nodal planes: xz & yz planes

See fig.2.

For `d_(yz)` orbital, xy & xz planes are nodal planes.

See fig.3.

For `d_(zx)` orbital xy & yz planes are nodal planes .

See fig.4.

� no. of radial nodes `= n - l - 1`

no. of angular nodes `= l`

total no. of nodes `= n - l - 1 + l'`

`= (n - 1)`

No. of peaks in `psi_r^2` vs r curve is `(n - l)`