Gas Molecule Collisions
Question:
In proving PV = (1/3)Nmc*2 from kinetic model, where
P is the pressure, V is the total volume of container, N is total
mole of gas, m is mass and c is the speed, we say that momentum
just before impact of a gas particle at the container wall is mu,
and momentum just after is -mu, thus change in momentum is 2mu.
It is assumed that all collisions between gas molecules and wall
are completely elestic.
It seems that in this case, the energy is conserved but
momentum is not. Is this correct? If it is really the case, then
it is completely different from the cases we learnt about
collisions in the part of Mechanics, where we only learnt about
collisions with momentum conserved while energy can be increases,
decreased or the same. So what makes gas particles collide
differently from the objects we studied in Mechanics? Is there
any other 'special' particles collide like gas
particles?
Answer:
The elastic collisions between gas molecules and the walls of
their container are exactly the same as the collisions we dealt
with in mechanics. The total momentum is conserved. You were
looking only at the change in momentum of the molecule. The wall
of the container also suffered a change in momentum, equal to and
opposite that of the molecule. The reason the container does not
get a velocity as a result of the collision with the molecule is
that over on the other side of the container another molecule was
being hit by a nearby molecule at the same time. The chances of
it happening this way are about 100%. Remember how many molecules
there are in a mole.
With each collision if we divide the change in momentum of the
molecule by the duration of the collision (remember impulse) we
get the force felt by the container. The sum of these collision
impulses over all the molecules hitting the container walls is
the total force on the container. Divided by the area of the
container surface it is the pressure. If the collisions are the
result of a sudden release of energy, as in an explosion, the
force on the inside of the container may exceed its strength and
blow it apart. If momentum were not conserved we would not see
this phenomena.