Derive the relation P= 1/3 mnv2
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Answer:
Derivation of pV = 1/3Nmc2
The Kinetic Theory of Gases and the Ideal Gas Equation
ASSUME:
Ideal gases are composed of:
- Numerous
- elastic molecules
- of Negligible Size compared to Bulk Container
- whose Thermal Motion is 'Random'
Consider a rectangular box length l, area of ends A, with a single molecule travelling left and then right the length of the box because of a collision with the end wall.
The time between collisions with the left wall is the distance of travel between wall A and the other wall divided by the speed of the particle in the x-direction, u.
t = 2 L - (equation 1)
u
According to Newton, force is the rate of change of the momentum
F = D (m u)
Dt
The momentum change upon collision is the momentum after the collision minus the momentum before the collision To hit the left wall the initial velocity must have been -u, so:
change in momentum, D (m u) = m u - (-m u) = 2m u
The average force on the left end wall is the rate of change of momentum
F = 2m u
Dt
Combine equation (1) into the above equation and we get:
F = 2m u
(2 L /u)
The 2's cancel and the formula reduces to:
F = m u2
L
The pressure, p, exerted by this single molecule constrained to move in one horizontal direction (one dimension) is the average force per unit area
p = F = m u2 = m u2
A L . A V
where V = A . L is the volume of the rectangular box.
Now consider N gas molecules in the box
p = Nm u2
V
But they could be moving moving with velocities in ALL directions - not just horizontally. They could be moving in the:
x direction (ux)
y direction (uy)
z direction (uz)
Using the rule for adding vectors at right angles to each other - we have to use Pythagoras to add the three velocities. (Square all the velocities and add them)
"mean square speed" of the gas molecules: c2 = ux2 + uy2 + uz2
but on average only a third of all molecules will be moving in any given direction,
so ux2 = uy2 = uz2
and so c2 = 3 ux2 OR ux2 = 1/3 c2
If the molecules are free to move in three dimensions, they will hit walls in one of the three dimensions one third as often. The pressure then of a gas sample of N molecules in 3-D is
p = 1 Nmc2
3 V
pV = 1 Nmc2
3
where p = gas pressure
V = gas volume
N = number of molecules
m = mass of each molecule
c2 = mean square speed of the gas molecules