Boltzmann Constant - Definition, Formula, Value, FAQs

The Boltzmann constant is a proportionality constant that is associated with the relative average kinetic energy of the particles present inside the gas with the gas’s Thermodynamics temperature. It happens in the description of the gas constant in Planck's ordinance of Black body radiation and the formula of Boltzmann entropy and in the kelvin scale. It was the Austrian researcher Ludwig Boltzmann, who came up with this constant formula, and thus was given the name Boltzmann Constant, consisting of dimensions similar to the entropy which is energy divided by the temperature.

The Boltzmann constant is one of the most well-received seven defining constants which has applications in different mergers to describe the SI base units of the seven defining constants.

History of the Invention of the Boltzmann Constant

The name Boltzmann constant was given the name after the Austrian researcher, Ludwig Boltzmann in the 19th century. Even though it was Ludwig first in the year 1877, who associated probability and entropy, in the black body radiation derivation initially in the 19th century, however, the association was not at all once indicated with a particular constant until Planck, invented the k and presented the most accurate value for it in the ordinance of the black body radiation derivation in the 19th century initially. Prior to the 19th century, calculations regarding the factors of Boltzmann were not put in writing applying the molecule per energy and the Boltzmann constant, instead applying a gas constant form, R and energy in macroscopy and quantities in macroscopy of the compound.

In the year 2017, the most precise calculations of the Boltzmann constant were acquired by gas thermometry which is acoustic, which represents the velocity of the volume of the monatomic gas in a chamber which is triaxial ellipsoid with applying resonances of acoustic and microwave. This time-consuming exertion was accepted with various experiments by many laboratories. It is one of the most well-received cornerstones of the year 2019 SI base unit redefinition. Based upon the calculations, the CODATA endorsed 1.380 649 × 10^-23 J.K^-1 to be the concluding Boltzmann constant value to be applied for the international system of the units.

The Formula of Boltzmann Constant

On account of the equation of Ideal Gas Equation

PV=nRT

Where P stands for the pressure in the units of pascals

V stands for the volume in the units of meter cube

N stands for the number of moles of the gas present or given

R stands for the proportionality constant or the gas constant

T stands for the temperature in the unit of kelvin.

R=N/NA

Where NA represents Avogadro’s number and N is the number of molecules present

The numerical value of Avogadro’s number is 6.022×10²³ per mole molecules

The numerical value of R is 8.314 Joule/mole K.

Since the pressure, the number of moles present and the volume are all quantities in macroscopy, now we are considering the gas molecules which would be represented only in microscopic quantities.

After rearranging the equation, we get

PV=N× K b× T

Where again P stands for pressure in the unit of Pascals

V stands for the volume in the unit of meter cubes

N stands for the number of moles of gas present or given in consideration

And T stands for the temperature.

Since the values N×K b and n× R are equivalent

Equating both sides of the equation we get

n× R= N× Kb

Kb = n/ N×R

The Boltzmann constant formula: K b = 1 / Na ×R

The above equation gives the relationship between the Boltzmann constant and R.

Upon calculation of the above equation, we get the value of the Boltzmann constant,

Kb = 1.3806452 × 10^-23 Joule per kelvin

Relation Between Boltzmann Constant and Gas Constant

Gasses at low pressures and high temperatures at which they liquefy can be satisfied by a simple relation,

$$
P V=K T \text {. }
$$
Where $\mathrm{T}=$ temperature of the gas.
$\mathrm{V}=$ volume of gas.
$\mathrm{P}=$ pressure.
And K is the constant which changes with respect to the volume of the gas.
$\mathrm{K}=\mathrm{nk}$, where the Boltzmann constant k is directly proportional to the number of moles.
It can also be written as,

$$
\frac{P_1 V_1}{N_1 T_1} \frac{P_2 V_2}{N_2 V_2}=k_B
$$
If $P, V$ and $T$ are the same then $N$ is also the same for all the gasses and is known as Avogadro's hypothesis, given by the value $6.022 \times 10^{23}$ at standard temperature and pressure.

The perfect gas equation can also be written as

$$
P V=\mu R T
$$
Where $\mu$ is the number of moles and $R=N_A k_B$, which is a relation between the Boltzmann constant and the Gas Constant.

SI units of Boltzmann constant are joule per kelvin

The equation written above concludes that the energy of the molecules of the gas is directly proportional to the absolute temperature.

Boltzmann constant in e V

Boltzmann constant is applied to represent the factor of Boltzmann, the idea of entropy in the description of the idea of this constant as we are discussing the stochastic of the gas molecules upon the gas which is warmed.

Boltzmann constant in electron volt/Boltzmann constant in ev

is equivalent to 0.000086173324 electron volts per kelvin.

The Boltzmann constant value in erg per kelvin

The Boltzmann constant value in K b in cgs units is given by 1.3806542×10^-16 erg per kelvin

Boltzmann Constant Applications

Boltzmann constant is an interconnected link between microscopic and macroscopic branches of physics

For a classical system of temperature of equilibrium denoted by E, the average energy per degree of freedom is k ×E/ 2

The total thermal energy of the gas molecules present is given by 3× D ×E/2 = m×z²/2

Where z² stands for the average of the squared speed of the molecules of the gas present, and E stands for the absolute temperature in the unit of kelvin.

Thermodynamics of Clausius

Boltzmann proved that the numerical quantity is equivalent to the 2/3^rd of the Clausius entropy of thermodynamics of a molecule of ideal gas.

Boltzmann called the gamma as the permutability measure.

Constant of Planck

Dividing the constant of Planck p by the 4.8 ×10^-34

Kb = p/ 4.8×10^-11 = 6.62610^-11

We get the K b value as 1.38041×10^-23

We came to know that the irregularities of the entropy or the particles which directly interconnected to the molecule temperature inside the gas, which means the higher the temperature the more will be the entropy.

Summary

The Boltzmann constant was named after Ludwig Boltzmann. As the average kinetic energy of gas particles is connected to the temperature, it is a connection between microscopic and macroscopic physics. Defined as $1.380649 \times 10^{-23} \mathrm{~J} / \mathrm{K}$, it forms part of many equations in thermodynamics and is an important tool for understanding gas behaviour, entropy, and black-body radiation.