Boltzmann’s Constant

What is Boltzmann’s Constant?

It is a physical constant which provides the computation of the amount of energy to the random thermal motions of the particles making up the substance. A great Austrian physicist Ludwig Boltzmann said: If the temperature of the gas molecules is high, then the average kinetic energy of molecules is large.

 

Temperature (R)      ∝    Kinetic Energy (H.E.)

Boltzmann’s Constant

We know that everything in this world is made of atoms and molecules.


Any vessel filled with water, cover it with a lid, and put it on fire, now before it starts boiling, just remove the lid, put your hand over the vessel of the steam, and it feels hot to your hand. Why do we feel the hotness?


This steam is made of atoms and molecules that are randomly moving inside the vessel, these particles with high entropy are striking your hand with high kinetic energy.

You would feel like your hand is getting bombarded by these tiny particles.


Boltzmann said that the heat energy which is leading these particles to resonate randomly is actually the Kinetic Energy (H.E.) of each H2O (water) molecule. The more is the temperature, more will be the entropy of these particles inside the vessel, the greater will be the transmission of energy, then greater would be the impact on your hand which means as fastly these particles go, the hotter you would feel.


We conclude that particles resonate at very high speed, they transfer kinetic energy to your hand, when kept inside the vessel your hand absorbs more energy, because the molecules move around your skin, causing your hands to get burnt. This was the average molecular explanation of the temperature.

Boltzmann’s Constant Formula

Consider an ideal gas equation: 

      PV = nRT  (Universal gas law)

       Where P is the Pressure in Pascals

         V is the Volume in Meter cubes

         n  = no of moles of the gas

         R = The gas constant

         T = Temperature in Kelvin

  

  R = N /  Na   ( Na = Avogadro’s number)


Where  N  is the number of molecules   and  Na  is the Avagadro's number or Avogadro Constant


    Value of  Na=  6.022 x 10 ^ 23 molecules per mole


    R =  8.314 Joule/ Mol - Kelvin


Since  A (Pressure), B (Volume ), n (no of moles) are all macroscopic quantities, but here we are talking about molecules present in the gas so we would consider only microscopic quantities.


Now, the equation, we drive here would be:


PV   =  N x  Kb x T


  Where P is the Pressure in pascals

  V is the volume in meter cubes or m ^ - 3

   N = No of molecules of the gas

   T = Temperature


Since the value of n x R  will be equal to N x Kb


Now equating both:

  n  x R  =    N x Kb

   Kb =  n / N x R

Boltzmann’s Constant Formula

 

Kb =  1 /  Na x R



=  (8.314 Joule / Mol - Kelvin) /  (6.022 x 10 ^ 23 molecules per mole)


On calculating the above equation, we get:

Value of Boltzmann’s Constant


Boltzmann’s Constant Kb = 1.3806452 x 10 ^ - 23 J/K 



S.I. unit of Kb = Joule per kelvin

This equation states that the energy in the gas molecule is directly proportional

to the absolute temperature.

Boltzmann’s Constant in eV

This constant is used to express the Boltzmann factor, the concept of entropy in explaining the concept of this constant as we are talking about the randomness of the molecules of the gas upon the gas being heated.


Boltzmann’s constant (Kb) in electron Volt (eV) is equal to:  0.000086173324 eV/ kelvin

Value of Boltzmann’s constant in erg/kelvin

 The value of Boltzmann’s constant Kb  in CGS given by:


1.3806542 x 10 ^ - 16 erg / Kelvin


Dimensional formula of Boltzmann’s constant Kb: [ M L ^ 2 T  ^ - 2  Ө ^ -1]

Applications of Boltzmann’s constant

Boltzmann’s Constant (Kb) is a basic constant of physics occurring in the statistical formulation of both classical and quantum physics.


Kb is a Bridge between Macroscopic and Microscopic Physics.

For a Classical system at equilibrium at temperature (E), the average energy per Degree of freedom is 

k x E / 2.


In the elementary illustration of the gas comprising D non-interacting atoms, each atom has  three transitional degrees of freedom (X, Y, & Z directions)

The aggregate thermal energy of the gas will be given by 3 x D x E / 2.


Here, 3 x D  x  E / 2  =  m x z ^ 2 /2 


 Where z  ^ 2  is the average of the squared velocity of the gas molecules, and E is the absolute temperature in Kelvin.

Clausius’s Thermodynamics

Boltzmann showed that the statistical mechanical quantity (γ) is equal to the 2/ 3 rd of Clausius thermodynamic entropy (R) of an ideal gas molecule.

Boltzmann called  “ γ  “  as the Permutability measure.

Planck’s Constant

Dividing Planck’s constant ‘p’ by 4.8 x 10 ^ - 11 meters, we get the value of Boltzmann’s constant.


Where,  c x q =  4.8 x 10 ^ - 11

 and,  c is the speed of light, and q is the charge of an electron.


We know that  p = 6.626 x 10 ^ - 34


Kb  =  p / 4.8 x 10 ^ - 11


  = 6.626 x 10 ^ - 34 / 4.8 x 10 ^ -11


We get the value:


Kb = 1.38041 x 10 ^ - 23

Summary

We concluded that the randomness of particles or the Entropy (H) is directly related to the temperature of the molecules inside the gas, which means the more the temperature, the higher will be the entropy. In the three states of matter, the order of randomness (Entropy):


Gaseous > Liquid > Solid


The gaseous state has got high entropy among the three states of matter.


FAQ (Frequently Asked Questions)

1. Derive the dimensional formula for Boltzmann’s Constant.

We know that  P X  V = Kb x N x T

We also know for n  =  1 mole, we get Na =  no of particles in one mole (Avogadro’s number) which has no unit.


We can see that Boltzmann's Constant (Kb) has the same dimensional formula as of Entropy (S).


2. Describe how Boltzmann’s constant (Cb) has significance in semiconductor physics? How is thermal voltage related to the Boltzmann’s constant?

In a p -n junction diode, the thermal voltage (U) is related to the absolute temperature (E) with this equation:


U = Kb x E /  n,  n is the charge on an electron which is equal to 1.6 x 10 ^ - 19 C.