1. Derive different gas laws from kinetic equation of the gas.
Answers
Answer:
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Explanation:
Kinetic theory of gases relates the macroscopic property of the gas, like – Temperature, Pressure, Volume to the microscopic property of the gas, like – speed, momentum, position. In this model, the atoms and molecules are continually in random motion, constantly colliding one another and the walls of the container within which the gas is enclosed. It is this motion that results in physical properties such as heat and pressure. In this article, let us delve deeper into the kinetic theory of gases.
Table of Contents
What is Kinetic Theory of Gases?
Kinetic theory Assumptions
Kinetic Theory of Gases Derivation
Average Kinetic Energy of a Gas Molecule
Kinetic Theory of Gases Solved Numericals
Kinetic theory of gases Significance
What is Kinetic Theory of Gases?
The gases are made up of a large number of molecules and they are flying in a random direction with a certain speed. By knowing their speed or position, one can figure out the macroscopic properties. In other words, by knowing the value of velocity or internal energy of gas molecules, one should be able to figure out the temperature or pressure.
Assumptions of Kinetic theory
The Kinetic theory of gases makes some basic assumptions. They are as follows:
Kinetic Theory Assumptions
The molecules do not interact with each other.
The collision of molecules with themselves or wall will be an elastic collision.
The momentum is conserved.
Kinetic energy will be conserved.
How is the Kinetic Theory of Gases Derived?
Consider a cubic box of length l filled with the gas molecule of mass m, moving along the x-axis with velocity vx Therefore its momentum is mvx.
The gas molecules collide the walls. At wall 1, it collides and the gains momentum mvx.
Similarly, the molecules collide wall 2, reversing the momentum i.e., -mvx.
Thus, the change in the momentum is given by
Δp = mvx-(-mvx) = 2mvx—–(1)
After the collision, the molecule travels a distance of 2l before colliding again with wall 1.
Thus, the time taken is given by
Time=Distancetravelledvelocity=2lvx—-(2)
These continuous collisions carry a force, given by
Force=ChangeinmomentumChangeinTime=ΔpΔt——(3)
Thus, substituting the values from equation(1) and (2), we get
Force=2mvx2lvx
Force(F)=mv2xl ——(4)
The continuous collisions also exert pressure on the wall given by-
Pressure=ForceArea=FA
Pressure(P)=(mv2xl)l2
P=mv2xl3——(5)
We know that the gas is made of N number of molecules and move in all possible directions. Thus, the pressure exerted on wall 1 by the collision of N number of gas molecules is given by-
P=ml3(v2x1+v2x2+v2x3+…..+v2xN)
⇒P=ml3(Nv¯2x)
⇒P=mNv¯2xV——(6)
where,
v¯2x=v2x1+v2x2+v2x3+…..+v2xN is the average velocity (or velocity component) of all gas molecules colliding with wall 1 along x-direction.
And l3 = V
On extending the above equation to three dimensions, we get-
v2=v2x+v2y+v2z
v¯2=v¯2x+v¯2y+v¯2z
For a large number of gas molecules:
v¯2x=v¯2y=v¯2z
or
v¯2=3v¯2x
⇒v¯2x=13v¯2-—-(7)
Substituting equation (7) in equation (6), we get
P=Nm(13v¯2)V
On rearranging,
⇒PV=13Nmv¯2 ——(8)
What is the Average Kinetic Energy of a Gas Molecule?
We know that PV=nRT —(9)
Thus equating equation (8) and (9) we get-
nRT=13Nmv2
Multiplying and Dividing by 2 we get-
nRT=23N(12mv2) —–(10)
Rearranging equation (10) we get-
12mv2=32nRTN =32RT(Nn) —-(11)
Here, N is the total number of gas molecules in a given region of space
n is the number of moles of gas present in a given region of space.
Nn=NA , Avogadro’s number. Avagadro’s number in this context is the number of molecules present in the one mole of gas. NA = 6.022140857 × 1023.
Substituting NA in equation (11),
(11)⇒12mv2=32RTNA —–(12)
Thus, Average Kinetic Energy of a gas molecule is given by-
⇒K.E=32kT
Here, K.E=12mv2 and k=RNA a Boltzmann constant
For monoatomic molecules, the total internal energy is given by-
Etotal=32PV [∵N(12mv2)=N(K.E)=Etotal=32PV] Etotal=32NkT Etotal=32nRT
Solved Numericals on Kinetic Theory of Gases
Watch the video below and learn to solve numericals based on kinetic theory of gases
74,127
What is the significance of the Kinetic theory of gases?
By knowing the Temperature we can directly figure out the average Kinetic energy of a gas molecule. No matter what gas are you considering. Unless and until it is an ideal gas.
Knowing the macroscopic parameters of gases like Pressure, Volume, Temperature etc., one can accurately calculate the microscopic parameters like Momentum, Velocity, Internal energy, Kinetic energy, Thermal energy etc. and Vise-Versa.
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