A system of physical chemistry - Index of


A system of physical chemistry - Index of


The total potential energy of the system consisting of the electrons

and the nucleus is—


P = - -(E - esn)

Sn = -2cosec — •

^ S = I

. (5)

1 The above expressions require probably a little explanation. Let us consider a

positively charged nucleus, charge E surrounded by n electrons, each of charge e,

the electrons being arranged at equal distances on the circumference of a circle of

radius a. We shall number the electrons i, 2, 3,

etc., up to «. The problem is to find the expression

for the potential energy of this entire system, the

potential energy being measured by the work

vv'hich it is necessary to do upon the system in

order to remove all the electrons to an infinite distance

from the nucleus and from each other. First

of all consider the nucleus with respect to electron

I. There is a force of attraction between the electron

and the nucleus amounting to e-^Eja^. The

potential of the electron with respect to the nucleus

being - e^EJa, the minus sign being introduced

because, in pulling the electron away, we oppose

the natural direction in which the electron tends to

move {viz. towards the nucleus). For electron 2

we have a similar term, so that for the « electrons

we have the potential energy of the electrons with

respect to the nucleus given us by the expression


neEla. We have now to consider the effect of the

electrons upon each other. The force is now one

of It

repulsion. therefore aids the process of pulling

apart. The expression for the potential energy of

each electron with respect to every other appears

therefore with a positive sign. Consider electron i, first of all with respect to

electron 2. The force of repulsion is

e-jr^^, o where r^, 2

is the chord connecting

the two electrons. The potential of electron i with respect to 2 is therefore e'^jr-^,^.

The potential of electron i with respect to electron 3 is given by e-jf^, 3 and so on

up to the ;ith electron. The total potential energy due to the \n - i) electrons

acting upon electron i is thus given by a sum of (n - i) terms :—

Now from the figure it is seen that—

rj, 2 = 2a sin 7r/« where 27r/M is the angle between successive electrons,

yj, 3 = 2a sin 27r/« ,, „ „ ,, „

^'i, 4 = 2a sin 3ir/« ,, „ „ „ >.

r^, M = 2a sin {n- i) Tr/« „ ,, ,, „

Hence the sum of the potential energy terms of electron i with respect to all the

others is—


2a [cosec 7r/« + cosec 2ir/» + cosec 3ir/?j + . . . + cosec (n - ijtt/hJ.

For electron 2 we can write down an analogous set of terms and so on up to the

wth electron. The sum of all such terms is therefore n times the series given. That

is, for all electrons mutually acting upon one another we have—•

2a [cosec Tr/n + cosec 27r/» + cosec 37r/» + . . • + cosec (« - i)ir/M].

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