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