Page 95 - Modern physical chemistry

5

Entropy and the

Second Law

5.1 Accessibility of States
A MACROSCOPIC SYSTEM OF GIVEN COMPOSITION is made up of a very large number
of molecules, atoms, radicals, and/or ions. The internal energy of the system is associ-
ated with random chaotic movements of these particles. Throughout a uniform region,
the Kelvin temperature is proportional to the average kinetic energy in a classically excited
degree of freedom.
Raising the temperature of a uniform section of a system requires adding energy to
that section; lowering the temperature requires removing energy therefrom. These
processes may be effected by work done on and/or heat transferred to the section.
A continuous manifold of states of a uniform system is generated from a given state
by reversible adiabatic processes. In the space of the independent variables, this mani-
fold forms a surface.
From a given point on such a surface, the given system may be moved at constant
volume to lower temperatures or to higher temperatures. If only work of compression
is allowed and the volume is kept constant, then to take the system to a lower tempera-
ture requires removal of heat. Similarly, to reach the point from the high temperature
side at constant volume requires removal of heat. But through any given state of a uniform
system, the adiabatic surface exists. Furthermore, the complete set of such surfaces
exhausts the space of the independent variables. We are thus led to the Principle of
Caratheodory:
Near any given state of a uniform system, there are states that cannot be reached by
adiabatic processes. But if adiabatic transition from state 2 to state 1 is impossible, adi-
abatic transition from state 1 to state 2 can be carried out.
With the Caratheodory principle, one can order all states of a given uniform system along
a line. Each point s on the line corresponds to a reversible adiabatic surface. A given s then
separates the states that cannot be reached adiabatically starting on the corresponding
surface from the states that cannot move to the surface adiabatically. See figure 5.1.
We have thus correlated all states of the system with points on the straight line. Let
us assign the same entropy S to the states correlated with point s. These states are linked
by adiabatic reversible processes. A larger S is assigned states to the right, those states
for which removal of heat is needed to reach the surface identified with s. So if such a
state is to be reached reversibly from this surface, heat would have to be added.
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