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 heat?

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 \[H_{2}O\] (water) molecule. The more is the temperature, the 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 Metre cubes

         n  = no of moles of the gas

         R = The gas constant

         T = Temperature in Kelvin

 \[R = \frac{N}{N_{a}}\] ( \[N_{a}\] = Avogadro’s number)

Here,  N  is the number of molecules   and  \[N_{a}\] is the Avagadro's number or Avogadro Constant.

Value of  Na= \[6.022 \times 10^{23}\] molecules per mole

R =  8.314 Joule/ Mol - Kelvin

Since  A (Pressure), B (Volume ) and 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  \[K_{b}\] x T

Here, P is the Pressure in Pascals

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

N = No of molecules of the gas

T = Temperature

The value of n x R  will be equal to N x \[K_{b}\].

Now on equating both, we get

n  x R  =    N x Kb

\[K_{b} = \frac{n}{N \times R}\]

Thus, the Boltzmann’s constant formula is given as

=  (8.314 Joule / Mol - Kelvin) /  (\[6.022 \times 10^{23}\] molecules per mole)

On calculating the above equation, we get the value of Boltzmann’s Constant.

S.I. unit of \[K_{b}\]= 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 (K_b) 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 is given by:

\[1.3806542 \times 10^{-16}\] erg / Kelvin

Dimensional formula of Boltzmann’s constant \[K_{b} : ML^{2}T^{-2}\Theta^{-1}\]

Applications of Boltzmann’s Constant

Boltzmann’s Constant (K_b) is a basic constant of physics occurring in the statistical formulation of both classical and quantum physics. The Boltzmann Constant is a term that has a major impact on the statistical definition of entropy. It is used in semiconductor physics to determine the thermal voltage. 

• K_b 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 Z²

Here, Z²  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 metres, 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 electrons.

We know that  p = 6.626 x 10^-34

\[K_{b} = \frac{p}{(4.8 \times 10^{-11})}\]

\[= \frac{(6.626 \times 10 ^{-34})}{(4.8 \times 10^{-11})}\]

We get the value:

Kb = 1.38041 x 10^-23

Summary

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

Gaseous > Liquid > Solid

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

Facts About Boltzmann’s Constant

Given below are some facts about the famous Boltzmann’s constant: 

1. Ludwig Boltzmann is an Australian scientist from whose name we get the name of the Boltzmann’s constant. Max Planck was the person who named this constant after him.

2. Ludwig Boltzmann has played a major role in statistical mechanics and he also gave the explanation of the second law of Thermodynamics. 

3. The significance of Boltzmann's constant is that it represents how energy is distributed in an atom. It is also used to represent thermal voltage in the study of semiconductor physics. 

4. It is an important tool to study heat and it also tells the relationship between energy and temperature.

5. The dimensions of the Boltzmann’s constant are [M¹L²T^-2K^-1] 

6. The value of the Boltzmann’s constant is kB= 1.3806452 x 10^-23J/K 

7. The value of the Boltzmann constant is 8.6173303 x 10^-5 eV/K in eV.