Of Physics, Narajole Raj College Prof

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C6T ( Thermal Physics) , Topic :- Kinetic Theory of Gases(Part-2): Circulated by-Prof. Surajit Dhara, Dept. Of Physics, Narajole Raj College

Prof. Surajit Dhara SACT,

Dept. Of Physics, Narajole Raj College

C6T ( Thermal Physics) , Topic :- Kinetic Theory of Gases(Part-2)

 Maxwell’s Law of Distribution of Molecular Speeds:

Let us represent at all the molecules in a velocity digram.

Suppose, OX,OY and OZ be the axes of a rectangular coordinate system with O as origin and u,v and w be the components of any velocity vector c along the axes,

. : 𝑐 = 𝑢 + 𝑣 + 𝑤 … … … (1) Differentiating equation (1) we have,

𝑢𝑑𝑢 + 𝑣𝑑𝑣 + 𝑤𝑑𝑤 = 0 … … … . (2)

Equation (2) is a condition equation which shows that 𝑑𝑢, 𝑑𝑣 and 𝑑𝑤 are not independent of each other.

Now, the probability that a molecule , randomly selected , has a velocity between 𝑢 and 𝑢 + 𝑑𝑢 is 𝑓(𝑢)𝑑𝑢. Similarly, the probabilities that a molecule has velocity components lying between 𝑣 and 𝑣 + 𝑑𝑣 and 𝑤 and 𝑤 + 𝑑𝑤 are 𝑓(𝑣)𝑑𝑣 and 𝑓(𝑤)𝑑𝑤 respectively.

.: The probability that a molecule has velocity components between 𝑢 and 𝑢 + 𝑑𝑢, 𝑣 and 𝑣 + 𝑑𝑣 and 𝑤 and 𝑤 + 𝑑𝑤 is the product of the individual probabilities, i.e.

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𝑓(𝑢)𝑓(𝑣)𝑓(𝑤)𝑑𝑢𝑑𝑣𝑑𝑤

If N be the total number of molecules in the gas ensemble and 𝑑𝑁 represents the number having the specified range of velocities then,

𝑑𝑁 = 𝑁𝑓(𝑢)𝑓(𝑣)𝑓(𝑤)𝑑𝑢𝑑𝑣𝑑𝑤……..(3)

⟹ 𝑑𝑁

𝑑𝑢𝑑𝑣𝑑𝑤= 𝑁𝑓(𝑢)𝑓(𝑣)𝑓(𝑤)

⇒ 𝜌 = 𝑁𝑓(𝑢)𝑓(𝑣)𝑓(𝑤) ……….(4) Where, 𝜌 =𝑑𝑁

𝑑𝑢𝑑𝑣𝑑𝑤 , represents the number of molecules of the specified type per unit volume of the velocity space. Now as the gas is in a steady state, the molecular density in any direction is constant, i.e.

𝑑𝜌 = 0

. : 𝑑[𝑁𝑓(𝑢)𝑓(𝑣)𝑓(𝑤)] = 0

⇒ 𝑑[𝑓(𝑢)𝑓(𝑣)𝑓(𝑤)] = 0

⇒ 𝑓 (𝑢)𝑓(𝑣)𝑓(𝑤)𝑑𝑢 + 𝑓(𝑢)𝑓 (𝑣)𝑓(𝑤)𝑑𝑣 + 𝑓(𝑢)𝑓(𝑣)𝑓 (𝑤)𝑑𝑤 = 0……..(5) Dividing throughout by 𝑓(𝑢)𝑓(𝑣)𝑓(𝑤) we have ,

( )

( ) 𝑑𝑢 + ^{( )}

( ) 𝑑𝑣 + ^{( )}

( ) 𝑑𝑤 = 0 …………(6)

We have already come through the point that du, dv and dw are not independent. So, we can’t equate coefficients of the differentials to zero. But , we can in effect make them so by the theorem of undermined multipliers of Lagrange. Multiplying equation(2) by 𝛼 and adding the product with equation (6), we get,

𝑓 (𝑢

𝑓(𝑢)+ 𝛼𝑢 𝑑𝑢 + 𝑓 (𝑣)

𝑓(𝑣) + 𝛼𝑣 𝑑𝑣 + 𝑓 (𝑤)

𝑓(𝑤) + 𝛼𝑤 𝑑𝑤 = 0 … … (7)

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Now, du , dv and dw are in effect independent. Soo, we may write

(

( )+ 𝛼𝑢 = 0 ^{( )}

( ) + 𝛼𝑣 = 0 …….(8)

𝑓 (𝑤)

𝑓(𝑤) + 𝛼𝑤 = 0

Integrating each of the equation of set (8) , we get

𝑓(𝑢) = 𝐴𝑒 ; 𝑓(𝑣) = 𝐴𝑒 ; 𝑓(𝑤) = 𝐴𝑒 .: From equation (3) we have

𝑑𝑁 = 𝑁𝐴 𝑒 ⁽ ⁾𝑑𝑢𝑑𝑣𝑑𝑤 ……..(9)

⇒ = 𝜌 = 𝑁𝐴 𝑒 ⁽ ⁾ = 𝑁𝐴 𝑒 ……(10)

Where b = and 𝑐 = 𝑢 + 𝑣 + 𝑤 by equation (1)

Equation (10) implies that the molecular density is a function of c only. So, the distribution of molecules is spherically symmetrical; the density of representative points is the same everywhere on the sphere of radius c in velocity space. This type distribution is called isotropic distribution. The plot of 𝜌 against is shown below which shows that 𝜌 falls off exponentially with c which is always positive.

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 Evaluation Of A : We have earlier that the total number of molecules in the gas assembly is N.

𝜌𝑑𝑢𝑑𝑣𝑑𝑤 = 𝑁

⇒ 𝑁𝐴 𝑒 ⁽ ⁾𝑑𝑢𝑑𝑣𝑑𝑤 = 𝑁

⇒ 𝑒 ⁽ ⁾𝑑𝑢𝑑𝑣𝑑𝑤 = 1

𝐴 [.: ∫ 𝑒 = √ ]

⇒ ( 𝜋

𝑏) = 1 𝐴

𝐴 = √ ……….(11)

 Evaluation of b: The pressure of the gas p is given by the following equation –

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𝑝 = ∑ 𝑛 𝑢

Where , 𝑛 =number of molecules per c.c having velocity in the range u and u+du

= 𝑛𝑓(𝑢)𝑑𝑢 ; n being the number of molecules per c.c.

. : 𝑝 = 2𝑚𝑛 𝑓(𝑢)𝑢 𝑑𝑢

= 2𝑚𝑛𝐴 𝑒 𝑢 𝑑𝑢

= 2𝑚𝑛𝐴 ×

.: 𝑒 𝑢 𝑑𝑢 = 1 4

𝜋 𝑏

= 𝑚𝑛

2 × 𝑏

𝜋× 𝜋 𝑏

= 𝑚𝑛 2𝑏

⇒ 𝑏 = = = [.: 𝑝 = 𝑛𝑘𝑇]

𝒃 = ^𝒎

𝟐𝒌𝑻 ; k=Boltzmann’s Constant

Expression of A , from equation (11), comes out as

𝑨 = ^𝒎

𝟐𝝅𝒌𝑻 …(13)

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Finally, the molecular density comes out as

𝝆 = 𝑵( ^𝒎

𝟐𝝅𝒌𝑻)^𝟑^𝟐𝒆 ^𝒎𝒄𝟐^𝟐𝒌𝑻 ..(14)

The left-hand side of equation (14) represents the number of specified molecules per unit volume. So, the number molecules (𝑑𝑁 ) having velocity in the range c and c + dc is 𝜌 times the volume between the two spheres (Spherical shell) of radii c and c +dc.

𝑑𝑁 = 𝜌. 4𝜋𝑐 𝑑𝑐

⇒ 𝑑𝑁 = 4𝜋𝑁( ) 𝑒 𝑐 𝑑𝑐 ……..(15)

This is the celebrated Maxwell’s law of distribution of molecular speeds.

.: Fraction of the total number of molecules having velocity in the range c and c+dc is

𝑑𝑁

𝑁 = 4𝜋 𝑚

2𝜋𝑘𝑇 𝑒 𝑐 𝑑𝑐 = 𝐹𝑑𝑐

Where , 𝐹 = 4𝜋 𝑒 𝑐 ……(16)

.: A plot of F againt c gives the distribution curve of molecular velocity, and it is shown in the figure below.

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From Maxwell’s law 𝑑𝑁 = 0, when c =0 and when 𝑐 = ∞,which is depicted by the nature of the curve. It means that the number of molecules having velocities tending to zero and also tending to infinity is quite small. Between these two extremes there is a value of c for which F is maximum i.e., maximum number of molecules will posses this velocity which is called the most probable velocity.