Derive Gas equation PvP =1/3mnu square
Answers
Answer:
Kinetic Theory of Gases has put forward a series of assumptions in order to explain what has been observed experimentally in gases.
Although, their number may vary, the core message is the same. They are as follows:
The molecules of a particular gas are identical and in random motion.
The collisions between the molecules and the between them and the walls of the container are perfectly elastic.
The volume of a molecule is negligible, compared with the volume of the container.
There are no intermolecular attractions between the molecules.
The time taken for a collision between two molecules is negligible compared with the time taken for the same between a molecule and the wall.
Based on these assumptions, a formula can be derived that connects the pressure, volume, the number of molecules, individual mass and of course, the mean velocity.
If an individual molecule collides with a wall, as shown in the animation, its momentum gets doubled.A gas molecule can move in any direction at a given time - in the x-direction, y-direction or z-direction.
Let's consider the motion in the x-direction, as shown in the animation; let the velocity be Ux
If the length of the cube, mass of the molecule and velocity are l, m and v respectively,
Momentum in the x-direction = mUx
Momentum in the -x-direction = -mUx
Change in momentum = 2mUx
Total time taken - from one end to the other and vice versa - = 2l / Ux
Rate of change in momentum = 2mu/(2l / Ux)
= mUx2/l
According to Newton's Second Law, the rate of change of momentum is the force exerted by the molecule on the wall.
Therefore, Force = [mUx / l]
Since pressure, P = force / area
Pressure on the wall, P = [mUx2/l] / l2
= mUx2 / l3
= mUx2 / V, where V is the volume of the container, THE cube.
If there are N molecules in the container,
P = m[U1x2 + U2x2 + U3x2 + ... + Unx2 ] / V
If the velocities are equal,
P = m[NUx2 ] / V P = NmUx2 / V