Kinetic Gas Equation
Let us consider a cubical container has only
one type of gas molecules.
Cubical vessel
have side length=l.
So the total
volume (V) of the vessel is= l3
Total inside
surface area of the vessel is = 6l2
The mass of
each gas molecules is= m
The total
number of gas molecules inside the vessel is= n
So total mas of
gas is= mn
Now according
to kinetic theory of gases, these molecules are moving randomly in different
directions with different velocities= v1, v2, v3,
v4, v5………..vn. Now from physics we know that
a vector can be resolve into three component, for any velocity say v1
can be resolve into three component vx, vy, vz.
along the three axis X, Y, Z respectively. now we know that:
consider the vx
velocity component along the X-direction, we have two opposite faces A & B.
The momentum of the molecule before colliding the face B is mvx.
since the collisions are perfectly elastic so the molecules will return with
the same velocity but in opposite to the initial direction so new momentum of
the molecule is -mvx.
Therefore the
change in momentum is = mvx-(-mvx) = 2mvx
Now for the
next collision, the molecule has to travel from the wall B to wall A, collide
there and return back from wall A to wall b, so for next collision the molecule
should have to travel twice the AB=2l
distance. But the molecule has vx velocity along the X-axis i.e. the
molecule travels vx distance in 1 sec time.
Number of
collisions for 2l distance is= 1
Therefore
the number of collisions in 1 sec for vx distance travel is
So
change in momentum in 1 sec on the B wall is
But in
X-direction there is an another face A, so the total change in momentum along
the X-direction
due to two
faces A & B is
Similarly for
both Y & Z-direction the change in momentum in 1 sec are
Therefore
for 1 molecule the total change in momentum in 1 sec is
But from
newton’s 2nd law of motion we know that change of momentum in 1 sec
is the force, so the force (F) by the molecule 1 is
Similarly the 2nd
molecule gives force
…………………………………………………………………………………
Similarly the nth
molecule gives force
Therefore the
total force (F) is
Where u is
called the root mean square velocity or rms velocity
Now, we know tha the pressure (P) is
But we know
that the total inside surface area of the vessel is = 6l2
So in this case the pressure exerted on the six wall’s
of the container is
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