Table of contents  
What are Electron Configurations?  
Magnetic moment (m):  
Radial wave function (R)  
Radial Probability density (R2)  
Radial probability functions 4πr2R2  
Angular wave Function QF. 
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Electronic configuration can be expressed in the following ways :
(i) Orbital notation method : nl^{x}
Li → 3 → 1s^{2} 2s^{1}
(ii) Orbital diagram method :
Li → 3 →
(iii) Condensed form :
[He] 2s^{1}
To write the electronic configuration of ions, first, write the electronic configuration of the neutral atom and then add (for a negative charge) or remove (for a positive charge) electrons in the outer shell according to the nature and magnitude of charge present on the ion. e.g.
Al → [Ne]3s^{2} 3p^{1},
,
Similarly, in the case of transition elements, electrons are removed from n^{th} shell for e.g. 4^{th} shell in case of 3 d series.
Cl : [Ne] 3s^{2}.3p^{5}
Cl^{} : [Ne] 3s^{2}.3p^{6}
Expected configuration Cr  24 > [Ar] 4s^{2} 3d^{4}
Observed configuration
Expected configuration
Observed configuration
Above exceptional configuration can be explained on the basis of following factors.
(i) Symmetrical Electronic Configuration :
It is wellknown fact that symmetry leads to stability, orbitals of a subshell having halffilled or fullfilled configuration (i.e. symmetrical distribution of electrons) are relatively more stable. This effect is dominating in d and f subshell.
d^{5}, d^{10}, f^{7}, f^{14} configurations are relatively more stable.
(ii) Exchange Energy :
It is assumed that electrons in degenerate orbitals, do not remain confined to a particular orbital rather it keeps on exchanging its position with electron having same spin and same energy (electrons present in orbitals having same energy). Energy is released in this process known as exchange energy which imparts stability to the atom. More the number of exchanges, more will be the energy released, more the energy released stability will be more.
→ 3 2 1 = 6
where n represents no. of electron having same spin.
The relative value of exchange energy and transfer energy (4s →3d) decides the final configuration. For example, if magnitude of Dep. energy Ex. energy > transfer energy as in cases of Cu and Cr then configuration is:
respectively.
In case of carbon magnitude of transfer energy > dep. energy exch. energy and configuration of carbon is:
It is the measure of the magnetic nature of substance
where n is no. of unpaired electrons.
When m = 0, then there is no unpaired electron in the species and it is known as diamagnetic species.
These species are repelled by magnetic field.
When , for paramagnetic species. These species have unpaired electrons and are weakly attracted by the magnetic field.
Ques. Calculate magnetic moment of these species
(i) Cr
(i) Cr : No. of unpaired electrons = 6.
= BM.
Generally, species having unpaired electrons are coloured or impart colour to the flame. Species having unpaired electrons can be easily excited by the wavelengths corresponding to visible lights and hence they emit radiations having characteristic colour.
In all cases R approaches zero as r approaches infinity. One finds that there is a node in the 2s radial function. In general, it has been found that nsorbitals have (n1) nodes, nporbitals have (n2) nodes etc.
The importance of these plots lies in the fact that they give information about how the radial wave function changes with distance r and about the presence of nodes where the change of sign of R occurs.
Fig: Variation of R with r.
The square of the radial wave function R^{2} for an orbital gives the radial density. The radial density gives the probability density of finding the electron at a point along a particular radius line. To get such a variation, the simplest procedure is to plot R^{2} against r. These plots give useful information about probability density or relative electron density at a point as a function of radius. It may be noted that while for sorbitals the maximum electron density is at the nucleus. all other orbitals have zero electron density at the nucleus.
Fig: Plot of R^{2 }with r.
The radial density R^{2} for an orbital, as discussed above, gives the probability density of finding the electron at a point at a distance r from the nucleus. Since the atoms have spherical symmetry, it is more useful to discuss the probability of finding the electron in a spherical shell between the spheres of radius (r dr) and r. The volume of the shell is equal to 4/3π(r dr)^{3}4/3πr^{3} = 4πr^{2}dr.
This probability which is independent of direction is called radial probability and is equal to 4πr^{2}drR^{2}.
Radial probability function (4πr^{2}drR^{2}) gives the probability of finding the electron at a distance r from the nucleus regardless of direction.
The radial probability distribution curves obtained by plotting radial probability functions versus distance r from the nucleus for 1s, 2s and 2p orbitals are shown in fig.
Hydrogen 1s Radial Probability:
Hydrogen 3s Radial Probability
Hydrogen 2p Radial Probability
Hydrogen 3p Radial Probability
Hydrogen 3d Radial Probability
The angular wave function ' Q F ' depends only on the quantum number l and m_{l} and is independent of the principal quantum number n for a given type of orbital. It therefore means that all s orbitals will have same angular wave function.
(A) Angular Wave Function, ' Q F '
For an sorbital, the angular part is independent or angle and is, therefore, of constant value. Hence this graph is circular or, more properly, in three dimensions i.e., spherical. For the p_{z} orbital, we get two tangent spheres. The p_{x} and p_{y} orbitals are identical in shape but are oriented along the x and y axes respectively. The angular, wave function plots for d and forbitals are four lobed and sixlobed respectively.
It is necessary to keep in mind that in the angular wave function plots, the distance from the center is proportional to the numerical values of ' Q F ' in that direction and is not the distance from the center of the nucleus.
(B) Angular Wave Function,  Q F ^{2}
The angular probability density plots can be obtained by squaring the angular function plots shown in fig. On squaring, different orbitals change in different ways. For an s orbital, the squaring causes no change in shape since the function everywhere is the same: thus another sphere is obtained.
Fig: Angular wave function.
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