4.3.2.2 Consideration of Polyatomic Polyatomic Gas Molecules
and External Forces
I will now generalize the formulas obtained so far, by first extending them to
so-called polyatomic gas molecules and then including external forces and thereby
finally beginning to extend the discussion to any solid and liquid [bold emphasize
by C.B.]. In order not to consider too many examples, I will in each case deal with
the most important case, where, aside from the equation for kinetic energy, there is
no other constraint.
The first generalization can be applied to our formulas without difficulty. So far,
we assumed each molecule was an elastic sphere or a material point, so that its
position in space was entirely defined by three variables (e.g., three orthogonal
coordinates). We know that this is not the case with real gas molecules. We shall
therefore assume that three coordinates are insufficient to completely specify the
position of all parts of a molecule in space; rather r variables will be necessary
p 1 , p 2 , p 3 . . . p r ,
the so-called generalized coordinates. Three of them, p 1 , p 2 , p 3 , are the orthogonal
coordinates of the center of mass of the molecule, the others can be either the
coordinates of the individual atoms relative to the center of mass, the angular
direction, or whatever specifies the location of every part of the molecule. We will
also remove the restriction that only one type of gas molecule is present. We assume
instead, there exists a second type whose every molecule has the generalized
coordinates
´
p 1 , ´
p 2 , ´
p 3 , . . . . . . . . . ::´ p r
For the third type the generalized coordinates are
p} 1 , p} 2 , p} 3 . . . p} r} ,
If there are v + 1 types of molecules, the generalized coordinates of the final type
are
p
v
ð Þ
1 , p
v
ð Þ
2 , p
v
ð Þ
3 . . . p
v
ð Þ
r v
ð Þ :
The first three coordinates are always the orthogonal coordinates of the center of
mass. Of course, the necessary assumption is that many molecules of each type are
present. Let l be the total kinetic energy of the first type of gas; χ is its potential
energy
7 (so that χ + l is constant if internal forces only are acting). Furthermore
7 The quantity χ called “Kraftfunktion” or “Ergal” by Boltzmann is translated as potential energy.
4.3 Evolution of Thermodynamic State Index (Φ)
161
and External Forces
I will now generalize the formulas obtained so far, by first extending them to
so-called polyatomic gas molecules and then including external forces and thereby
finally beginning to extend the discussion to any solid and liquid [bold emphasize
by C.B.]. In order not to consider too many examples, I will in each case deal with
the most important case, where, aside from the equation for kinetic energy, there is
no other constraint.
The first generalization can be applied to our formulas without difficulty. So far,
we assumed each molecule was an elastic sphere or a material point, so that its
position in space was entirely defined by three variables (e.g., three orthogonal
coordinates). We know that this is not the case with real gas molecules. We shall
therefore assume that three coordinates are insufficient to completely specify the
position of all parts of a molecule in space; rather r variables will be necessary
p 1 , p 2 , p 3 . . . p r ,
the so-called generalized coordinates. Three of them, p 1 , p 2 , p 3 , are the orthogonal
coordinates of the center of mass of the molecule, the others can be either the
coordinates of the individual atoms relative to the center of mass, the angular
direction, or whatever specifies the location of every part of the molecule. We will
also remove the restriction that only one type of gas molecule is present. We assume
instead, there exists a second type whose every molecule has the generalized
coordinates
´
p 1 , ´
p 2 , ´
p 3 , . . . . . . . . . ::´ p r
For the third type the generalized coordinates are
p} 1 , p} 2 , p} 3 . . . p} r} ,
If there are v + 1 types of molecules, the generalized coordinates of the final type
are
p
v
ð Þ
1 , p
v
ð Þ
2 , p
v
ð Þ
3 . . . p
v
ð Þ
r v
ð Þ :
The first three coordinates are always the orthogonal coordinates of the center of
mass. Of course, the necessary assumption is that many molecules of each type are
present. Let l be the total kinetic energy of the first type of gas; χ is its potential
energy
7 (so that χ + l is constant if internal forces only are acting). Furthermore
7 The quantity χ called “Kraftfunktion” or “Ergal” by Boltzmann is translated as potential energy.
4.3 Evolution of Thermodynamic State Index (Φ)
161