Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

Physics: Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

The document Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics is a part of the Physics Course Kinetic Theory & Thermodynamics.
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Kinetic theory is a basic mathematical modeling that bridges mechanics and

thermodynamics.

Postulates of kinetic theory of gases

  • Matter is made up of identical molecules.
  • Thermal energy can be identified with molecular motion.

Methodology

Using laws of classical mechanics and statistical averaging one can find total energy of Ideal gases. We can use this theory to study viscosity, conduction, diffusion, effusion and Brownian motion.

Basic Assumption of Kinetic Theory

1. Any infinitely small volume of a gas contains a large number of molecules. The order of number of molecule is 1023

2. A gas is made up of identical molecules, which behaves as rigid, perfectly elastic, hard sphere but considered as point mass .

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

The diameter of molecule is order of 10-10m at STP

3. The molecules continuously moves in random directions. All direction of motion are  equally probable.

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

4. The size of the molecules is much less than the average distance between them. The average distance between them is order of 10-9m and diameter is of order of 10-10m.

5. The molecule of a gas exerts no force on each other except when they collide. These forces are due to only exchange of momentum during collision

6. The collision between molecules and with walls are perfectly elastic.

7. The molecular velocities are assumed to be uniformly distributed in all directions, which means velocity is isotropic symmetric and there is not any preferential direction.

8. The molecules move with all speeds ranging from 0 to ∞.

9. The time of collision is much less than the time between collisions.

Basic Statistical Formula to be used for Kinetic Theory:

If there is a sample random variable X = Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

And total number of variable is N , then some statistical phenomena is defined as Mean Value:Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

If group of

n1 number of x1 

n2 number of x2

...

nnumber of xn

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

Root mean square value Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

ΔX = Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics is identified as standard deviation

Example 1: Consider a system of particles and its velocities in one-dimension given in the table.

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

Find:

(a) Average velocity

(b) Root Mean Square velocity

(c) Most probable velocity

(a) Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics 3.36m/ s

(b) Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics = 25 9697

(c) vp = 3 m/s most number of particles are having velocity 3 m/s

Pressure Exerted by a Gas

Suppose there are N molecules per cubic meter each of mass m and it is assumed that ni number of molecule have velocity vi .

Mathematically,Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics and Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics, where vix,, viy and viz are x, y and z components of velocity of gases respectively

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

Figure 1, shows a typical gas molecule, of mass m and velocity Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics, that is about to collide with the shaded wall, where the time Δt between collisions is the time the molecule take to travel to the opposite wall and back again at speed vix.

From assumptions of kinetic theory of gases, Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

Suppose, molecules are kept in the cubic container of parameter L .

A molecule moving in the x - direction will have momentum mvix normal to face of the cube before collision.

Hence, the momentum Δpix delivered to the wall by the molecule during collision is

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

Force acting on the wall by molecule isKinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

Pressure exerted on the walls of container by molecules, Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

So, the pressure in the x - direction exerted by all molecules is,

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

Average value of v2 is given by,

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

For three dimensional system, Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics and

For isotropic system, Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

So, Px can be written as

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

PV = Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

where V is volume of the container and (v2) is average value of square of velocity.

Gas Law for Ideal Gases

Boyle’s Law

At constant temperature (T), the pressure (P) of a given mass of a gas  is inversely proportional to its volume (V) i.e. Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

Charle’s Law at constant pressure and Gay Lussac’s law at constant volume

At constant pressure (P) the volume of a given mass of a gas is proportional to its temperature (T), V ∝ T.

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

At constant volume (V) the Pressure of a given mass of a gas is proportional it temperature (T), P ∝ T.

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

Avogadro’s Law

At the same temperature and pressure, equal volume of all gases contain equal number of molecules (N)

N1 = N2

Graham’s Law of Diffusion

When two gases at the same pressure and temperature are allowed to diffuse into each

other, the rate of diffusion (r) of each gas is inversely proportional to square root of density of gas (ρ)

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

Dalton’s Law of Partial Pressure

The sum of pressure exerted (P) by each gas occupying the same volume as that of the mixture (P1,P2,P3,....) where P = P+ P+ P3+...

Ideal Gas Equation

Consider a sample of an Ideal gas at pressure P , volume V and temperature T, the gas follows the equation

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

where, n is the number of molecules and R is proportionality constant known as gas constant

R = 8.314 J/mol/K

Boltzmann constant K is ratio between R to Avogadro number Ni.e. 

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

Kinetic Interpretation of Temperature

According to the assumption of kinetic theory of gases, there is only translation motion of the molecule and there is not any potential acting between them, so

Average Energy (E) of gases are equivalent to average translational energy of a molecule i.e. Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

Pressure, PV = Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

= Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics where PV = nRT Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

= Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics put value of PV = Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics , where kB is Boltzmann constant kB = 1.38 x 1023 J/K

So, average kinetic energy is given by (E) = Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics , where  T is absolute temperature

Internal energy of gases with degree of freedom

Number of  independent coordinates required to specify its position correctly 

f = 3N - k

Where N is number of particles and k is Total number of constraints. For a system in

equilibrium at temperature T , the total energy is equally partitioned among the different degrees of freedom and Energy associated with each degree of freedom is 1/2kBT. If there is f degree of freedom then internal energy of system molecule is U = Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

If U is internal energy of Ideal gas then specific Heat at constant volume is defined as Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

Example 2: Mono atomic gas in three dimensional volume:

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

f = 3x1- 0 = 3⇒ f = 3

U = Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics


Example 3: Diatomic molecule at low temperature:

At low temperature Diatomic molecule can be under stood by a model in which two points mass are connected by a mass less rod .so distance between both of particle is constant so k = 1

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

So, degree of freedom is f = 3 x 2 -1 = 5

U = Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics


Example 4: Diatomic molecule at high temperature:

At high temperature Diatomic molecule can be under stood by a model in which two point mass are connected  by a mass less spring. Due to spring one extra mode of degree of freedom(one due to potential energy) will excited hence

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

So degree of freedom is f = Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics = 7

U = Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics


Example 5: Find the maximum attainable temperature of ideal gas in each process given by, Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics, where and P0 and α are positive constants, and V is the volume of one mole of gas.

P = P0 - αV2                                         (i)

Number of moles of gas = 1

We know that,Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

Putting in (i), Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

For T maximum, Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

Putting in equation (ii), one will get Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics


Example 6: Two thermally insulated vessel 1 and 2 are filled with air. They are connected by a short tube with a valve. The volume of vessels and the pressure and temperate of air in them are Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics respectively. Calculate the air temperate and pressure established after opening of valve, if air follows Ideal gas equation.

For vessel (1), Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

For vessel (2),Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

After opening the valve, let pressure, volume and temperature are P, V and T respectively.

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

Hence, system is isolated and thus,

Energy of system (1) + energy of system (2) = energy of composite system

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics


Example 7: A horizontal cylinder closed from one end is rotated with a constant angular velocity ω about a vertical axis passing through the open end of the cylinder. The outside air pressure is equal to p0, the temperature equal to T and the molar mass of air equal to M. Find the air pressure as a function of the distance r from the rotation axis. The molar mass is assumed to be independent of r .

Force equation of dr element,

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics, if A is cross section area then

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

Also we know that , PV = nRT

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

P = Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics


Example 8: Prove that, PA = Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics and Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physicsin two dimension.

A molecule moving in the x direction will have momentum Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physicsnormal to face of the cube before collision.

Hence, the momentum Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics delivered to the wall by the molecule during collision is,

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

Force acting on the wall by molecule is Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

Pressure exerted on the walls of container by molecule is, Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

So, the pressure in the x - direction exerted by all molecules is, Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - PhysicsAverage value of v2 is given by, Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

For two dimensional system Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

So,Px can be written as Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

Thus, Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics


Example 9: It is possible to treat electromagnetic radiation in a container whose walls are mirror, as a gas of particle (photons) with a constant speed c and whose energy is related to their momentum p , which is directed parallel to their velocity by, E = pc . Show that if container is full of radiation, the equation of state is PV = Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

Pressure Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

For photon v = c and velocity is parallel to momentum, so

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

Mean Free Path (λ)

The Mean free path is the average distance travelled by molecule between two successive

collisions. If Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physicsdenotes successive free length path traversed in time t and N is total number of ion suffered in this period then mean free path λ given by Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics which is equivalent to total distance travelled divided by total number of collision.

Consider two molecules of masses m1 and m2 diameters d1 and d2

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

After collision they can be considered as one body system. Reduced mass, Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics and centre to centre distance, Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

Let the number of densities of these gases = n1 and n2

We construct a cylinder of cross section area πd2, length Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics, which defines 

Average speed characterizing maxwellian molecules when molecule of first type moves it collides will all the molecules of second type in the cylinder.

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

Total number of such collisions is given by Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

= Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

If all molecule will identical d1 = d2 = d , m1 = m2 = m and n1 = n2 = n

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

Distribution of Free Path

We wish to estimate the numbers that have not made a collision at some later time. Let the number of molecules surviving a collision in traveling a distance x be N = N(x). If N0 is number of molecule at x = 0 then number of molecule N = N(x) which will suffer no collision between distance x to 0 is N(x) = N0Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics with mean path λ as parameter of gas.

The equation N(x) = N0Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics is identified as survival equation. For Maxwellian distribution into account the survival equation can be represented as

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

The survival equation is shown in figure.

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

Example 10: The fraction of molecules that are undeflected is 1/2, then find the distance that undeflected particles will travel .

N = Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics


Example 11: In a sample of 1026 gas molecules, each molecule is moving with the same speed. Calculate the fraction of molecules that will continue to travel undeflected after traversing of λ,2λ,5λ and 10λ.

The values of f (x) are tabulated below for different below for distance traversed by the gas molecules and you can readily verify these using Eq. Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

To illustrate for x= λ we can write Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

We will note that after traversing a distance of λ and 2λ , nearly 37% and 13.5% molecules respectively survive collisions.  However, after covering a distance of 10λ , almost all molecules will have undergone a collision.

Basic of Transport phenomenon

When a gas is endowed with mass motion, random molecular motion can lead to transport of energy, momentum or mass, depending on the physical conditions, which lead to the phenomena of thermal conduction, viscosity and diffusion, respectively.

Coefficient of viscosity (η) : The coefficient of viscosity for a gas having velocity gradient in a direction normal to the direction of mass motion is given by Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics where m is the molecular mass and v' is the average speed of a molecule. At a given temperature, η is independent of pressure.

Thermal conductivity (K) : The thermal conductivity, K is given by Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics, where f denotes the number of degrees of freedom.

K and η are connected by the relation Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics, where M is molecular weight of gas.

Coefficient of diffusion (D) : The coefficient of diffusion D is given by

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

Example 12: Calculate the coefficient of viscosity, thermal conductivity and diffusion coefficient for air. Given ρ = 1.29kgm-3 , v' = 460 ms-1 and λ = 6.4x10-8m. Take molecular weight of air as 29kg kmol-1 and R = 8.31kJ K-1 kmol-1 .

We know that the coefficient of viscosity η is given byKinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

That the coefficient of thermal conductivity K is given by K = Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

Here Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics is specific heat capacity per unit mass, and M = 29kg kmol-1

For a diatomic gas,Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics= 20.775 x 103 JK-1kmol-1. hence 

K = Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

The coefficient of diffusion D is given as D = Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

= 9.8x10-6 m2 s-1

Phase space and Gibbs postulates and Density of state

Phase curve any between positions coordinate along horizontal axis and conjugate momentum along vertical axis for given value of energy is known as phase curve.

Gibbs take account of uncertain principle then Gibbs make postulates

The minimumKinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics for one dimension

Minimum Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physicsfor two dimension

MinimumKinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics for two dimension

Density of State

g(E)dE is number of level (number of quantum state) in the range of E to E + dE then

g (E) is known as Density of state

Density of State in three dimension

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

where V is volume of container

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

which is equation of sphere with coordinate px, py, pz.

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

So,Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics is volume of sphere with radius Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics is equivalent to Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

where ∅(E) volume of spherical shell between radius Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

Using Taylor expansion thenKinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

g(E)dE in three dimension g(E)dE = Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics where V is volume of three dimensional space.

Density of State in Two Dimension

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

where A is volume of container Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

which is equation of circle with coordinate px , py

So, Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics is area of circle with radius Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

where ∅(E) volume of spherical shell between radius Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

Using Taylor expansion, then Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics is equivalent to Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

g(E)dE in two dimension g(E)dE = Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics, where A is area of the two dimensional space.

Density of State in One Dimension

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

where A is volume of container Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics

g(E)dE in one dimension g(E)dE = Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics where L is area of the one dimensional space

The document Kinetic Theory of Gases Notes | Study Kinetic Theory & Thermodynamics - Physics is a part of the Physics Course Kinetic Theory & Thermodynamics.
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