drive gas equation by kinetic equation
Answers
What Is Kinetic Theory Of Gases?
The kinetic theory of gases describes gas as a large number of submicroscopic particles such as atoms and molecules, all of which are in random and constant motion. The randomness arises from the particles collisions with each other and with the walls of the container.
Derivation Of Kinetic Gas Equation
Derivation Of Kinetic Gas Equation
Consider a cubical container of length ‘l ’ filled with gas molecules each having mass ‘m’ and let N be the total number of gas molecules in the container. Due to the influence of temperature, the gas molecules move in random directions with a velocity ‘v.’
The pressure of the gas molecules is the force exerted by the gas molecule per unit area of the wall of the container and is given by the equation
P=FA
Let us consider a gas molecule moving in the x-direction towards face A. The molecule hits the wall with a velocity Vx and rebounds back with the same velocity Vx, and will experience a change of momentum which is equal to Δp=−2mVx.
For a total of N number of gas molecules in the container, all such change in momentum is given by
Δp=−2NmVx
The force is given by the equation
F=Δpt
Therefore,
F=−2NmVxt
Gas molecules will hit the wall A and will travel back across the box, collide with the opposite face and hit face A again after a time t which is given by the equation
t=2lVx
Substituting the value of t in the force equation, we get the force on the molecules as
F=−2NmVx2lVx
Fmolecules=−2NmVx2lVx=−NmV2xl
Therefore, the force exerted on the wall is Fwall=NmV2xl.
Now, the pressure P is given by the equation
P=ForceonthewallArea=NmV2xll2=NmV2xl3
Hence, PV=NmV2x (1)
Since Vx, Vy and VZ are independent speed in three directions and if we consider the gas molecules in bulk, then
V2x=V2y=V2z
Hence,
V2=3V2x
Substituting the above condition in eq (1), we get
PV=NmV23
Therefore,
PV=13mNV2
This equation above is known as the kinetic theory equation.
The velocity V in the kinetic gas equation is known as the root-mean-square velocity and is given by the equation
Vrms=V21+V22+V23……..+V2n√N
We use this equation to calculate the root-mean-square velocity of gas molecules at any given temperature and pressure
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