6
1 Nucleation Theory
1.1.3 The Principle of Detailed Balance
In the framework of thermodynamics up until now, there has been no concept of
time so far. That two phases (1) and (2) can coexist in thermodynamic equilibrium
when μ 1 (P, T ) = μ 2 (P, T ) does not mean complete absence of transformation of
molecules between the two phases. Rather, there are equal numbers of molecules
transforming in one direction as the other. The principle of detailed balance states
that each elementary process should be balanced by its reverse process in equilibrium.
Consider a situation in which a single-component liquid is introduced to a closed
system that is maintained at a constant temperature. The amount of liquid is plentiful
and assumes that the space above the liquid in the closed system at time = 0 is a
vacuum. How quickly will the liquid evaporate?
After a long time, after the system has attained an equilibrium, the space above
the liquid will be totally saturated with the vapor. Since the system is maintained
at a constant temperature, any cooling due to the latent heat of evaporation will be
promptly replenished by the heat reservoir, and the vapor will attain an equilibrium
vapor pressure of the liquid at that temperature.
The principle of detailed balance states that, after the system has achieved an
equilibrium, the elementary rate of evaporation of a molecule is equal to the elementary rate of condensation of the molecule. For the system as a whole, the rate
of evaporation is constant with time after time = 0 because the amount of the
liquid is plentiful. The rate of condensation, in contrast, is zero at the beginning
(time = 0) because the upper space has started from a vacuum. Even though the
elementary rate of evaporation of a molecule is equal to the elementary rate of
condensation of the molecule, since there is no molecule in the vacuum to condense
from at time = 0, the total rate of condensation is zero. The system-wide rate of
condensation then gradually increases with time as the upper space is progressively
populated with the vapor molecules. Eventually, the system-wide rate of condensation equals the system-wide rate of evaporation at which point an equilibrium is
established.
The great value of the principle of detailed balance comes from the realization
that the elementary rate of evaporation of a molecule can be estimated from the
elementary rate of condensation of a molecule in equilibrium and vice versa. For
example, consider two liquids, A and B. Liquid A has a much greater equilibrium
vapor pressure than Liquid B at a certain temperature. Which liquid takes longer
to attain equilibrium if each liquid is introduced to a pre-evacuated chamber of the
same temperature at the same time?
That the equilibrium vapor pressure of A is higher than B means that both the rate
of evaporation and the rate of condensation of the molecule of A are greater than
those of the molecule B. Therefore, Liquid A will attain equilibrium first.
1 Nucleation Theory
1.1.3 The Principle of Detailed Balance
In the framework of thermodynamics up until now, there has been no concept of
time so far. That two phases (1) and (2) can coexist in thermodynamic equilibrium
when μ 1 (P, T ) = μ 2 (P, T ) does not mean complete absence of transformation of
molecules between the two phases. Rather, there are equal numbers of molecules
transforming in one direction as the other. The principle of detailed balance states
that each elementary process should be balanced by its reverse process in equilibrium.
Consider a situation in which a single-component liquid is introduced to a closed
system that is maintained at a constant temperature. The amount of liquid is plentiful
and assumes that the space above the liquid in the closed system at time = 0 is a
vacuum. How quickly will the liquid evaporate?
After a long time, after the system has attained an equilibrium, the space above
the liquid will be totally saturated with the vapor. Since the system is maintained
at a constant temperature, any cooling due to the latent heat of evaporation will be
promptly replenished by the heat reservoir, and the vapor will attain an equilibrium
vapor pressure of the liquid at that temperature.
The principle of detailed balance states that, after the system has achieved an
equilibrium, the elementary rate of evaporation of a molecule is equal to the elementary rate of condensation of the molecule. For the system as a whole, the rate
of evaporation is constant with time after time = 0 because the amount of the
liquid is plentiful. The rate of condensation, in contrast, is zero at the beginning
(time = 0) because the upper space has started from a vacuum. Even though the
elementary rate of evaporation of a molecule is equal to the elementary rate of
condensation of the molecule, since there is no molecule in the vacuum to condense
from at time = 0, the total rate of condensation is zero. The system-wide rate of
condensation then gradually increases with time as the upper space is progressively
populated with the vapor molecules. Eventually, the system-wide rate of condensation equals the system-wide rate of evaporation at which point an equilibrium is
established.
The great value of the principle of detailed balance comes from the realization
that the elementary rate of evaporation of a molecule can be estimated from the
elementary rate of condensation of a molecule in equilibrium and vice versa. For
example, consider two liquids, A and B. Liquid A has a much greater equilibrium
vapor pressure than Liquid B at a certain temperature. Which liquid takes longer
to attain equilibrium if each liquid is introduced to a pre-evacuated chamber of the
same temperature at the same time?
That the equilibrium vapor pressure of A is higher than B means that both the rate
of evaporation and the rate of condensation of the molecule of A are greater than
those of the molecule B. Therefore, Liquid A will attain equilibrium first.