A system of physical chemistry - Index of

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A system of physical chemistry - Index of

1 6 A SYSTEM OF PHYSICAL CHEMISTRY

and independent of the conditions under which the system is examined.

A more general form of the above expression is—

S = -^ log W + constant.

We have now to find out the significance of the universal constant k.

To do this we make use of a statistical expression arrived at by

Boltzmann for the entropy of a perfect monatomic gas. A monatomic

gas is one in which the total or internal energy U is due entirely to the

kinetic energy of the molecules. Boltzmann's expression is—

S = 3/2/^N log U + /^N log z; + K

where U and k are defined above, N is the number of molecules in the

system, v the volume of the system, and K is a constant independent of

the energy and volume but involving the number and mass of the

molecules. It has already been shown in Volume II. that the following

purely thermodynamical relation holds good :—

^ _ J yu

Differentiating Boltzmann's expression for the entropy of a perfect

monatomic gas we obtain :—

Hence U = s/s^NT.

If N be taken as denoting the number of molecules in one gram-molecule,

then U denotes the total energy of one gram-molecule of monatomic

gas.

But we have already seen (Vol. I.) that in the case of a perfect gas,

the total kinetic energy of all the molecules forming one gram-molecule

is 3/2RT, where R is the gas constant per gram-molecule. Further

in the case of a monatomic gas the internal energy is entirely kinetic.

Hence for one gram- molecule of a monatomic gas : U = 3/2 RT.

It follows therefore that— N/^ = R,

or k is the gas constant per single molecule.

Further, in the case of a perfect gas, 3/2R = C„ where Ct, is the

gram-molecular heat at constant volume. Hence the equation of

Boltzmann for the monatomic gas becomes—

S = C„ log T -f R log z; -1- K^

where S now denotes the entropy of one gram- molecule. This expression

is in complete agreement with that already deduced in Volume II.

on thermodynamical grounds, viz. :—

S = C„ log T -f R log z; -t- Si

if we identify S^ with K^. This constant represents the value of the

entropy under certain conditions. On purely thermodynamical grounds

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