Kinetic Theory of gases and radiations

Kinetic Theory of Gases and Radiations

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Introduction : Kinetic theory of gas was developed by Maxwell, Boltzmann , others in nineteenth century. We are well aware that gases do not have any shape . They take the shape of the container. The molecules of the gases are always in random motion due to thermal energy and exert pressure on the wall of the container. The possess momentum and change in momentum gives impulse on the wall of the container in which the gases are placed. 

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Let try this interesting experiment given below  before learning This lesson.

How to do this experiment: Lift the piston and push down so that gas are filled in the container. You can increase or decrease the temperature of the system just by lowering or raising the knob provided in the bucket below the container. Observe the change in pressure in the pressure bar at top right side. Observe the experiment and make a conclusion related to pressure, velocity, volume and temperature and try to find out how they are related to each other.

 

Or else check this video

Try this Experiment by your own .

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Laws of gases

Boyle's Law : At constant temperature the volume occupied by the gas is inversely proportional to pressure exerted by a fixed mass of a gas.

V α  1/ P ---- ( Temperature being constant)   ---(1)

Charle's law: At constant pressure , the volume is directly proportional to temperature. 

V  α T ----- (Pressure being constant) ---(2)

Gay Lussac's Law: At constant volume, pressure is directly proportional to temperature.  

P α T ----- (Volume being constant) ---(3)

Combining all the three equation

PV α T

OR

PV / T = Constant

PV = RT   ( R is called as gas constant)

if n is the no of moles of gas then ,equation changes to, 

PV = nRT    -------(A)

Equation  (A) gives Ideal gas equation

Note: The no of moles (n) =  Mass of gas (M) / Molar mass (M₀)

n = M / M₀

R is called called gas constant having value 8

314 J mol^-1 kg^-1

R= k_B N_A

k_B =Boltzman Constant ( 1.38 x 10^-23-23 JK^-1)

N_A = Avagadaro's number

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Ideal Gas : A gas that satisfies equation PV = nRT  exactly at all pressure and temperature i known as ideal gas. In ideal gas the intermolecular interaction is negligible as the molecules are far away from each other.

Real gas: A real gas have some intermolecular interactions but if they are far away from each other they are very near of being ideal gas. We ca achieve this at low pressure and high temperature. There is no real gas which is ideal.

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Assumptions of kinetic theory of gases 

1)A large numbers of  molecule combine together to form a gas.

2) This molecules are always in random direction.

3) This molecules are rigid and perfectly elastic and is spherical.

4) The volume of the gas is large as compared to volume of each molecule.

5) The molecule continuously collide with each other and the collision is perfectly elastic.

6) Between two successive collisions, molecule travel in a straight line with constant velocity. This is called free path.

6) The time taken for collisions is small as compared to time taken for collisions.

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Mean free path

The molecule being in random direction continuously collide with each other. The path between two successive collision is called as mean free path. The magnitude of the free path is given by

λ = 1 / [√2π d² (N/V)]

where,

  d is diameter of molecules

N is the no of molecules enclosed   

V is the volume of a gas.

The average distance travelled by a gas molecule between two successive collisions is called as free mean path.

It is measured in angstrom or meter

If  λ₁+λ₂+---------+ λ_n are the free path ,then mean free path is given by equation,

 λ = [λ₁+λ₂+---------+ λ_n] / n

where n is no of collisions.

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Terms related to velocity

Mean Velocity ( Average velocity) (C)

Let we are having N no of molecules of an ideal gas enclosed in a vessel. 

Let  C_{1 ,} C₂ ------ , C_N be their velocities , thus the mean is given by

C=   [C₁²+  C₂² +-----+ C_N²] / N

Mean square velocity (C²)

The square of average velocity of all molecules is known as Mean Square Velocity

Let we are having n no of molecules of an ideal gas enclosed in a vessel. 

Let  C_{1 ,} C₂ ------ , C_N be their velocities , thus the mean square velocity  is given by

C²=   [C₁²+  C₂² +-----+ C_N²] / N

Root mean Square velocity  (√C²)

It is defined as  square root of mean square velocity

Let we are having n no of molecules of an ideal gas enclosed in a vessel. 

Let  C_{1 ,} C₂ ------ , C_n be their velocities , thus the mean square velocity  is given by

C_rms=   √{[C₁²+  C₂² +-----+ C_N²] / N}

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Pressure of a gas

Pressure of a gas

           Cartesian Coordinate 

Consider a fixed amount of gas  enclosed in a cuboid  of side 'a' The gas molecules are in random direction having momentum, which is imparted on the wall of container . Thu pressure is exerted by  gas molecules on the wall of the vessels.

Assuming some terms

N = Total no of molecules

m= mass of each molecule

M = Nm = total mass of a gas

a = length of each side

A = a² = Area of the container

V = l³ = volume of the container

ρ= M/V  = M/ l³  = Density of the gas

Consider  n molecules having velocity C_{1 ,} C₂ ------ , C_N 

Resolving each velocity into three coordinate system

Let  u_1, v_{1 ,} w₁ be the component of velocity C₁ in x  , y , z axis respectively.

Let  u_2, v_{2 ,} w₂ be the component of velocity C₂ in x  , y , z axis respectively.

Let  u_n, v_{n ,} w_n be the component of velocity C_n in x  , y , z axis respectively.

Then 

C₁²=   u₁² ₊ v₁² +------ + w₁²------(A)

C₂²=  u₂² ₊v₂² +------ + w₂²------(B)

^{.              .         .                    .}

       ^{.              .          .                   .}       

C₁²=   u_n²  ₊v_n² +------  + w_n²------(C)

^{Consider a molecules of mass 'm' moving with velocity C}₁ ^{in direction of wall  PQRS .Let its velocity be u}₁ ^{in X direction.}

^{Thus  the initial momentum of the molecules is given by} 

 ^mu₁ ^{-----------(1)}

 ^{After sometime the molecule collide with wall PQRS And reflect back with same velocity in opposite direction.}

^{Now the momentum after collision will be equal as per laws of conservation  of momentum, and it will be given  y,}

^mu₁^{-----------(2)}

^{The change in momentum is given by,}

^- mu₁ ^{- mu}₁ ^{= -2mu}₂

^{Negative sign indicates  lost in momentum ,but in this case momentum is conserved as no external force was acting, thus equation becomes,}

^= 2mu₁-----------(3)

After striking wall PQRS , the molecules reversed back and again strike wall P'Q'R'S' and return back to wall PQRS with same speed, the distance between this two collision is equal to '2a'. Thus time interval is given by 

Time = Distance / Speed

t = 2a / ^u₁ ---------(4)

The force f₁ exerted on PQRS is given by ,

force = change in momentum / time

 f₁ = [ ^2mu₁ / (2a / ^u₁) ]  -----------------( from 3 and 4)

 f₁ = [^mu₁² / a ] ----------(5) 

Equation (5) gives value of force exerted by particle moving in x direction with velocity u_1.

Similarly force exerted by molecule in X direction with velocity u_{1 ,}u_{2 ,}u_{N is given by}

 f₂ = [ ^mu₂² / a ] ----------(6)

 f_N = [  ^mu_N²/ a ]  ---------(7)

Resultant force on PQRS by molecules with velocity u_1, u_2, u_{N in X direction is given by,}

f_x = f₁+f₂+f_N

f_x =  ^mu₁² / a    +   ^mu₂² / a     +    ^mu_N² / a 

f_x =  m/a  [ ^u₁²+ ^u₂² + ^u_N² ] ------------(8)

Similarly, resultant force on by molecules with velocity v_1, v_2, v_N and  w_1, w_2, w_{N in} Y and Z direction respectively is given by,

f_y =  ^mv₁² / a    +   ^mv₂² / a     +    ^mv_N² / a

f_y =  m/a  [ ^v₁²+ ^v₂² + ^v_N² ] ------------(9)

f_z=  ^mw₁² / a    +   ^mw₂² / a     +    ^mw_N² / a

f_z =  m/a  [ ^w₁²+ ^w₂² + ^w_N² ] ------------(10)

Equation (8) (9) (10) gives Resultant force on wall of the cube in X ,Y , Z direction respectively

Now Pressure exerted due to this force on wall PQRS is given by,

Pressure (P)= Force (F)/Area (a²)

Pressure in X direction is given by,

P_x =  f_x /a²      --------( A= a²)  

P_x = [m/a(^u₁²+ ^u₂² + ^u_N² )]/ a² 

P_x = [m/a³ (^u₁²+ ^u₂² + ^u_N² )]----------(11)

P_y =  f_y /a²      --------( A= a²)  

P_y = [m/a(^v₁²+ ^v₂² + ^v_N² )]/ a² 

P_y = [m/a³ (^v₁²+ ^v₂² + ^v_N² )]----------(12)

P_z =  f_x /a²      --------( A= a²)  

P_z = [m/a(^w₁²+ ^w₂² + ^w_N² )]/ a² 

P_z = [m/a³ (^w₁²+ ^w₂² + ^w_N² )]----------(13)

we know pressure exerted by gas is same in all three direction

P = P_x  =P_y = P_z 

3P = P_x +P_y +P_z 

P = (P_x +P_y + P_z ) / 3 ----------(14)

Putting (11) (12) (13) in (14)

P =  m/3a³ (^u₁²+ ^u₂² + ^u_N² )+ m/a³ (^v₁²+ ^v₂² + ^v_N² ) + m/a³ (^w₁²+ ^w₂² + ^w_N² )

P = m/3a³[ (^u₁²+ ^u₂² + ^u_N² )+  (^v₁²+ ^v₂² + ^v_N² ) +  (^w₁²+ ^w₂² + ^w_N² )]

From (A) (B) (C)

P = m/3a³ [C₁²+C₂²+C_n²]

P = m/3V [C₁²+C₂²+C_n²] ----(a³=V) ------(15)

We know, C_rms=   √{[C₁²+  C₂² +-----+ C_N²] / N}

N * C²_rms =C₁²+  C₂² +-----+ C_N²  --------(16)

putting equation (16) in (15)

P = m/3V [N * C²_rms ]

P= 1/3 * M/V  * C²_rms --------(17)

Equation (17) gives Pressure of gas

Next Topic : Kinetic Energy of a gas

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