Boltzmann constant (k), named after Ludwig Boltzmann a 19th-century Austrian physicist, is a fundamental constant relating the average kinetic energy of the gas particles and the temperature of the gas represented by k or kB. Boltzmann constant has a defined value of 1.380649 × 10−23 joule per kelvin (K) or 1.380649 × 10−16 erg per kelvin which is mostly observed in Boltzmann’s entropy formula and Planck’s law of Black body radiation.

The Boltzmann constant, denoted by the symbol k is defined as the proportionality constant that illustrates the relationship between a gas’s thermodynamic temperature and its average relative kinetic energy. The Boltzmann constant has dimensions of energy per degree of temperature and is measured using Joule/Kelvin or m2Kgs-2K-1. The molar gas constant or Ideal gas constant R is defined as the product of Avogadro’s number and the Boltzmann constant.

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

The formula of the Boltzmann Constant

Boltzmann and Plank brought the behavior of gases one step closer to understanding by establishing constants and the Boltzmann constant’s value can be stated mathematically as-

K=RNA

where,

K= Boltzmann’s constant

R= universal gas constant and

NA = Avogadro’s number.

Consider an ideal gas equation i.e.,

PV = n*R*T (Universal gas law)

where,

P = Pressure in Pascal,

n = no of moles of the gas

V= Volume in Metre cubes

R = The gas constant (8.314 Joule/ Mol – Kelvin)

T = Temperature in Kelvin

R=N/Na

Here, Na= Avogadro’s number (6.022×1023 molecules per mole),

N is the number of molecules

Since Pressure, Volume, and no of moles are all macroscopic quantities, here we are talking about molecules present in the gas, so we would consider only microscopic quantities.

Now, the equation would be:

PV = N x Kb x T

The value of n x R will be equal to N x Kb

Now on equating both, we get

n x R = N x Kb

Kb= n/N×R

Thus, Boltzmann’s constant formula is given as:

(8.314 Joule / Mol – Kelvin) / (6.022×1023 molecules per mole)

On calculating the above equation, the value of Boltzmann’s Constant can be calculated. The above equation states that the energy in the molecule of gas is directly proportional to the absolute temperature.

Value of Boltzmann Constant

In 2017, the most exact measurements of the Boltzmann constant were made using acoustic gas thermometry, which estimates the speed of sound of a monatomic gas (mono= one) in a triaxial ellipsoid container using acoustic resonances and microwaves. This decade-long effort of measurement of the Boltzmann constant was carried out by several laboratories using varied approaches and is one of the components of the 2019 SI base unit redefinition. Based on these findings, CODATA (Committee on Data for Science and Technology) suggested 1.380649 x 1023 JK1 as the final fixed value of the Boltzmann constant for the International System of Units.

The value of Boltzmann’s constant K_b in SI is 1.380649 × 10⁻²³ joule per kelvin and in CGS is 1.3806542×10⁻¹⁶ erg / Kelvin

Ideal Gas Constant and the Boltzmann Constant

we can write the ideal gas equation as PV= nRT= nN_akT.

Because the number of molecules in the sample, N, is N=nN_a

Na= Avogadro’s number

we have

PV=NkT

Applications of Boltzmann Constant

In classical physical mechanics, the Boltzmann Constant is used to represent the proportionality of a molecule’s energy. The Boltzmann Constant is employed to convey heated voltage i.e., the thermal voltage in semiconductor material research. Boltzmann showed that the statistical mechanical quantity is equal to the 2/ 3rd of Clausius thermodynamic entropy i.e., R of an ideal gas molecule and hence plays a major role in the statistical definition of entropy. Boltzmann factor can be expressed by using this constant.

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