Relative sizes of hydrogenic orbitals and the probability criterion

4 Relative sizes of hydrogenic orbitals and the probability criterion 

For hydrogenic orbitals, the average distance of the electron from the nucleus, r, can be evaluated exactly from the equation 

The r values for orbitals Is through 4f and their size ratios, calculated relative to the Is value as a standard, are given in the second and third columns of Table 2.1.5. 

For any given orbital, the probability of finding the electron in the volume elementdr = r2 dr sin 0 dddcp [eq. (2.1.7)] in the vicinity oflhc point (r, 0,4 ) is 

If we integrate this expression over all possible values of 0 and p, we get 

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