Page 20 - Bird R.B. Transport phenomena

§0.3 The Conservation Laws: An Example 5

Fig. 0.3-1 A collision
between homonuclear
diatomic molecules,
such as N and O .
2
2
Molecule A is made up
Molecule A before collision /
Molecule В before collision of two atoms Л1 and
A2. Molecule В is made
up of two atoms B\
and B2.

Molecule В after collision
Molecule A after collision

temperatures lower than 50 K, the kinetic theory of gases can be developed quite satis-
factorily by use of classical mechanics.
Several relations must hold between quantities before and after a collision. Both be-
fore and after the collision the molecules are presumed to be sufficiently far apart that
the two molecules cannot "feel" the intermolecular force between them; beyond a dis-
tance of about 5 molecular diameters the intermolecular force is known to be negligible.
Quantities after the collision are indicated with primes.
(a) According to the law of conservation of mass, the total mass of the molecules enter-
ing and leaving the collision must be equal:

m B (0.3-1)
Here m A and m B are the masses of molecules A and B. Since there are no chemical reac-
tions, the masses of the individual species will also be conserved, so that

m = m = m (0.3-2)
A A B
(b) According to the law of conservation of momentum the sum of the momenta of all
the atoms before the collision must equal that after the collision, so that

m A\*A\ m B\*B\ + B2*B2 = m> A\*A\ m> B2*B2 (0.3-3)
m
in which r A1 is the position vector for atom 1 of molecule A, and r M is its velocity. We
now write t M = r A 4- K M so that r M is written as the sum of the position vector for the

Atom Л2

Center of mass
of molecule A

Arbitrary origin Fig. 0.3-2 Position vectors for the atoms
fixed in space A\ and AT. in molecule A.