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A short theory
A hydrogen atom has one electron which "rotates" in a nuclear field. An electric force on Coulomb attraction acts between the electron and the nucleus. The potential energy of an electron in a nuclear field is
where e is the charge of an electron and r is the distance between the nucleus and electron. Such an atom constitutes a peculiar kind of potential well and is illustrated in fig.11.1.
The electron inside the atom has a negative potential energy since the minimum value of potential energy tends to infinity when r → 0 and the maximum value is equal to zero. Fig. 11.2 shows the energy levels obtained from the solution of the Schrodinger equation
An important feature of the solution is the drawing together of the levels as the quantum number n increases. The scales of values, which are proportional to energy are given in the units adopted in spectroscopy: volts and reciprocal centimetres. The energy level formula may by written in the form
For historical reasons, it is customary to write this formula in the for
is the Rydberg’s constant.
The atomic electron may be located at any one of n levels. The energy of a free hydrogen atom on which no force acts is at the lowest energy level
The energy ε = cRh is called the ionisation energy. If the energy imparted to hydrogen atom is less than cRh, a transition of the atom occurs to one of the n levels. Such an atom is said to be in an exited state.
An atom stays in an excited state for a small fraction of a second and then passes to a lower level with the emission off a photon in accordance with the equation
hνmn = εm - εn = cRh (1/n2 - 1/m2). (11.7)
By calculating for a given n the ν frequencies corresponding to the numbers m = n+1, n+2, ..., we obtain a series of frequencies of lines in the hydrogen spectrum. The series corresponding to n = 2 is known as the Balmer series.
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