Internal energy
The internal energy for one mole of ideal gas is:
,
And the capacities are both constant:
;
The experimental temperature's dependence of heat capacity does not in accordance with the theory.
Quantum theory has solved this problem.
,
- is a rotational quantum number;
,
- is an oscillation's quantum number;
- is a
Plank's constant.
Example for
hydrogen
.
at
The rotational and oscillation's degrees of freedom are "freeze out" for this temperature.
Barometrical formula
From the second Newton's law in equilibrium:
; 
The solution of this equation is:
where 
- is a pressure at z
height;
- is a pressure at height
.
Gas of Van der Waals
At height pressures the equation for ideal gas is not valid:
At
the expression
is twice more than
.
At normal
conditions (NC:
,
)
the proper volume of molecules is more less then the volume of vessel
with gas:
At the proper volume of molecules is match with the volume of vessel
!
The forces of molecule interactions. Attraction and repulsion.
The dependence of potential energy on distance.
Van der Waals' equation:
For moles of gas the equation is:
The
correction
in
pressure is due to the attraction forces between the molecules.
The
correction
in
pressure is due to the repulsion forces between the molecules at low
distances.
Statistical physics. Probability. Distribution functions
Statistical regularities. Random events.
The probability of random event:
- is a
probability of the "i"
event.
In practice usually:
The normalization requirement:
The probability of complex events.
The addition of probabilities.
The probability for two exclusive events is equal to sum of two probabilities for each event.
The multiplication of probabilities.
The probability for two independent events is equal to product of two probabilities for each event.
The average values for random functions:
The distribution functions.
- is a
probability density, frequency function.
- it is the probability for x
to be within the interval
.
The
probability for x
to be within the finite interval
is
the integral:
- it is the square under the curve
The normalization requirement:
Where
- is the range of all possible values for x.
The average values for continual random functions:
Maxwell's distribution
J.K.Maxwell 1859.
The space for velocities of molecules.
For projection of the velocity:
Because of independent of motions:
Maxwell's distribution:
The normalization requirement:
The average energy for one degree of translational motion is:
,
so
Maxwell's distribution for modulus of velocity
The volume of spherical layer:
and
The normalization requirement:
The characteristic speeds
The most probable speed:
This speed
corresponds to max
of
.
Average speed:
Root-mean-square speed:
; 
The dependence of distribution on temperature.
The reduced distribution
Let us introduce new dimensionless quantity:
The distribution for energy of the molecules
- it is a new variable.
Verify that
and
