Page 21 - Bird R.B. Transport phenomena

6 Chapter 0 The Subject of Transport Phenomena

center of mass and the position vector of the atom with respect to the center of mass, and
we recognize that R = -RAU w e ^ s o write the same relations for the velocity vectors.
A2
Then we can rewrite Eq. 0.3-3 as
m r + m x = m x + m r (0.3-4)
A A B B A A B B
That is, the conservation statement can be written in terms of the molecular masses and
velocities, and the corresponding atomic quantities have been eliminated. In getting
Eq. 0.3-4 we have used Eq. 0.3-2 and the fact that for homonuclear diatomic molecules
m M = ™A2 = \ m .
A
(c) According to the law of conservation of energy, the energy of the colliding pair of
molecules must be the same before and after the collision. The energy of an isolated mol-
ecule is the sum of the kinetic energies of the two atoms and the interatomic potential en-
ergy, ф , which describes the force of the chemical bond joining the two atoms 1 and 2 of
А
molecule A, and is a function of the interatomic distance \x A2 — г |. Therefore, energy
Л1
conservation leads to
2
bn r + ФА) + (Ьп гГв1 + \™вгГ вг + ф ) =
в
A2 A1
В
+ \m r \ + ф' ) + &п'в\Гв\ + W r \ + ф ) (0.3-5)
A1 A А B2 B в
Note that we use the standard abbreviated notation that f\ = (г • f ). We now write
x Л1 Л1
the velocity of atom 1 of molecule A as the sum of the velocity of the center of mass of A
and the velocity of 1 with respect to the center of mass; that is, г = г + К . Then Eq.
Л1 л Л1
0.3-5 becomes
(\m r 2 A + u ) + (lm r 2 B + u ) = %m r 2 + u ) + (lm r 2 + u ) (0.3-6)
A
A A
B
B
A
B B
A
B
in which u = \ni R + lm R + ф is the sum of the kinetic energies of the atoms, re-
A M Al A2 A2 А
ferred to the center of mass of molecule Л, and the interatomic potential of molecule A.
That is, we split up the energy of each molecule into its kinetic energy with respect to
fixed coordinates, and the internal energy of the molecule (which includes its vibra-
tional, rotational, and potential energies). Equation 0.3-6 makes it clear that the kinetic
energies of the colliding molecules can be converted into internal energy or vice versa.
This idea of an interchange between kinetic and internal energy will arise again when
we discuss the energy relations at the microscopic and macroscopic levels.
(d) Finally, the law of conservation of angular momentum can be applied to a collision
to give
([г Л1 X т г ] + [г Л2 X т г ]) + ([r B1 X m i ] + [r B2 X m i ]) =
Л1
Л2
Л1
m m
Л2
B2 B2
([г Л1 X ш г ] + [г Л2 X т г ]) + ([r B1 X т г ] + [r B2 X т г ]) (0.3-7)
Л2
В1
Л2
В2
В1
Л1 Л1
В2
in which X is used to indicate the cross product of two vectors. Next we introduce the
center-of-mass and relative position vectors and velocity vectors as before and obtain
1
([г л x т г ] + 1 ) + ([r B X m r ] + ) =
B B
B
л
л
Л
([г X т г ] + 1 ) + ([r X т г ] + ) (0.3-8)
1
B
л
л
л
Л
в
в
B
in which 1 = [К Л1 X т К ] + [R A2 x m R ] is the sum of the angular momenta of the
Л1
Л
Л1
A2
A2
atoms referred to an origin of coordinates at the center of mass of the molecule—that is,
the "internal angular momentum." The important point is that there is the possibility for
interchange between the angular momentum of the molecules (with respect to the origin
of coordinates) and their internal angular momentum (with respect to the center of mass
of the molecule). This will be referred to later in connection with the equation of change
for angular momentum.