Boltzmann's distribution
When the gas is in the vessel in the field of gravity there are two tendencies:
the thermal motion of the molecules scattering them on the accessible space;
external field of gravity collect the molecules in the range with minimal potential energy.
So we can see the equilibrium state with concentration of molecules depends on the position in space. The temperature is supposed to be constant.
Where
- is a force applied to one molecule.
For ideal gas:
Finally
there is the Boltzmann's distribution: 
where
- is the concentration of molecules in space where the potential
energy is equal
.
The atmosphere of the Planets
where 
- is a mass of planet;
- is a mass of molecule;
- is a distance from the center of planet.
where R – is the radius of the planet.
One can see
that at
the concentration
:
Therefore the gas of atmosphere is dissipated.
Maxwell's – Boltzmann's distribution
- the portion of molecules with the components of velocities are in
the intervals:
,
,
.
- is the number of molecules in the volume
at
the position
.
Substitution the distributions:
gives us:
where
- is the mechanical energy of the molecule.
Perrine's experiment
Brownian's particles.
We can define the Boltzmann's constant from the experiment:
Two-level system
The second law
The first law of T/D – it is a "book-keeper of enterprise".
It keeps the energy balance between the system and the surroundings.
The second law of T/D – it is a "director of enterprise".
It permits only such process that in accordance with the principle of entropy growing.
The second law by Klausius (1850):
"It is impossible spontaneous heat transfer from cold to hot body."
The second law by Kelvin (1851):
"It is impossible to convert the heat completely to work. It does not existing the perpetual motion machine of the second kind".
Irreversibility of the process in macro systems.
Entropy
Clausius
(1865):
for
the reversible
processes.
.
Instead of heat Q the entropy S is a function of state.
This integral does not depend on the path; it depends only on the initial (1) and final (2) states.
The characteristics of entropy
The entropy is a function of state.
The entropy is an additional function.
The entropy of an insulated system is not decreased.
The third law of T/D (Nernst's theorem):
when
.
Consequence:
,
from which one has:
when
.
Master equation of T/D
For the reversible process:
and
From the first law:
The entropy of an ideal gas.
and
The expansion of a gas in vacuum.
Thermo-insulated
system (
).
During the
expansion the system does not do any work (
).
It means that:
and
.
Let us consider the reversible isothermal process:
So 
Carnot cycle
A heat engine is a device that takes in energy by heat and, operating in a cyclic process, expels a fraction of that energy by means of work. For instance, in a typical process by which a power plant produces electricity, a fuel such as coal is burned and the high-temperature gases produced are used to convert liquid water to steam. This steam is directed at the blades of a turbine, setting it into rotation. The mechanical energy associated with this rotation is used to drive an electric generator. Another device that can be modeled as a heat engine is the internal combustion engine in an automobile. This device uses energy from a burning fuel to perform work on pistons that results in the motion of the automobile.
F
rom
the energy conservation law:
The thermal
efficiency
of a heat engine is defined as the ratio of the net work done by the
engine during one cycle to the energy input at the higher temperature
during the cycle:
The Kelvin–Planck form of the second law of thermodynamics states the following:
I
t
is impossible to construct a heat engine that, operating in a cycle,
produces no effect other than the input of energy by heat from a
reservoir and the performance of an equal amount of work.
The analysis of the cycle:
1-2:
;
;
;
;
;
.
2-3:
;
;
;
;
;
.
3-4:
;
;
;
;
;
.
4-1:
;
;
;
;
;
.
From T-S diagram:
The thermal efficiency of the Carnot cycle depends only on temperatures of hot and cold reservoirs.
The irreversible engine has the thermal efficiency less then the reversible.
therefore
