Boltzmann Constant – Definition, Formula, and Applications.

The Boltzmann constant is a critical factor in proportionality. The physical constant that connects a gas particle’s average kinetic energy to its temperature is known as the Boltzmann constant in thermodynamics. The symbol for it is kB or k. 

The Boltzmann constant value can be measured using J/K or m2Kgs-2K-1. This is mainly observed in the entropy Boltzmann constant formula and the Law of black body radiation by Planck. 

Value of Boltzmann Constant

The Boltzmann constant value can be derived by dividing the gas constant R by NA, or Avogadro’s number. Below is the value of k or the Boltzmann constant value:

Boltzmann constant, kB= 1.3806452 × 10^-23J/K

Value of k

The Boltzmann constant value in eV is 8.6173303 × 10^-5 eV/K. It can be expressed in terms of different units. The Boltzmann constant value or the value of k in different units can is shown in the table given below:

Value of k	Units
8.6173303 × 10-5	eV.K-1
1.3806452 × 10-23	m2.Kg.s-2.K-1
3.2976230(30)×10-24	cal.K-1
1.38064852 × 10-16	erg.K-1
0.69503476(63)	cm-1.K-1
2.0836612(12)×1010	Hz.K-1
−228.5991678(40)	dB.WK-1.Hz-1
4.10	pN.nm
0.0083144621(75)	kJ.mol-1K-1
1.0	Atomic unit (u)

What is the Boltzmann Constant?

Max Planck introduced the Boltzmann constant, while was named after a scientist named Ludwig Boltzmann. It can be obtained by taking the ratio of two other forms of physical constants, namely the Avogadro number and the gas constant.

The formula of the Boltzmann Constant  

Boltzmann constant is better understood by studying the behavior of gases. It is mathematically expressed as:

k = R/NA

Here, k stands for Boltzmann’s constant.

R stands for the gas constant, and

NA stands for the Avogadro number

Applications of Boltzmann Constant

The Boltzmann Constant has a diverse range of applications in Physics, some of which are given below:

• The Boltzmann Constant is employed for the expression of the equipartition of the energy of an atom in classical statistical mechanics.
• It also expresses the Boltzmann factor.
• The Boltzmann constant plays an important role in the definition of entropy from the statistical point of view.
• It is employed for the expression of thermal voltage in the field of semiconductor physics.

History of the Boltzmann’s Constant

The Boltzmann constant has been named after an Austrian scientist named Ludwig Boltzmann, who discovered it in the 19th century. Although Boltzmann first linked probability and entropy in 1877, the relationship never had been expressed with a specific constant until the time Planck first gave k and introduced a more precise and accurate value for it. 

The Gas Constant and energy for the macroscopic quantities of a substance were employed in place of the Boltzmann Constant and energies per molecule in the calculations involving Boltzmann factors prior to the 19th century. 

Boltzmann Constant and Ideal Gas Equation

The product of Volume V and Pressure p for an ideal gas is directly proportional to the product of the absolute temperature and the amount of the substance in moles.

𝑝𝑉=𝑛𝑅𝑇,

Here R stands for the molar gas constant whose value equals 8.31446261815324 J⋅K⁻¹⋅mol⁻¹

The Boltzmann Constant transforms the ideal gas law into an alternative form as the gas constant per molecule. 

k = R/NA:𝑝𝑉=𝑁𝑘𝑇 

Here, N represents the number of molecules of the gas. 

Boltzmann Constant in Chemical Kinetics

Eyring Equation

The Eyring equation is an equation of chemical kinetics describing the chemical reaction rate as a function of temperature. Below is the Eyring Equation:

𝑘= κ𝑘𝐵𝑇ℎ𝑒Δ𝐺‡𝑅

Here, T stands for the absolute temperature

ΔG represents the Gibbs energy of activation

k stands for the transmission coefficient 

kB stands for the Boltzmann Constant

R represents the Gas Constant

h stands for the Planck Constant

Arrhenius Equation

Arrhenius’ equation gave a formula representing the temperature dependence of the rate of reactions. It is given by:

 𝑘=𝐴𝑒−𝐸𝑎𝑘𝐵𝑇, 

Here, 𝐸𝑎 stands for the activation energy of the reaction

k represents the rate constant

T stands for the absolute temperature of the reaction, and

Stands for the pre-exponential factor, which is constant for every chemical reaction.

kB stands for Boltzmann Constant. 

The Boltzmann Constant in Statistical Mechanics

A Degree of Freedom

The average thermal energy that each microscopic degree of freedom in a thermodynamic system carries at absolute temperature is equal to 1/2kT

Here, k stands for the Boltzmann constant

Kinetic Theory of Gases

The kinetic theory of gases gives the average pressure p of an ideal gas as 

𝑝=1/2𝑁𝑉𝑚𝑣²

We know from the ideal gas equation that 𝑝𝑉=𝑁𝑘𝑇

Therefore,1/2𝑚𝑣²=32𝑘𝑇.

Partition Function

The system in equilibrium at temperature T generally occupies a state i with probability Pi and energy E weighted by the corresponding Boltzmann factor:

𝑃𝑖∝𝑒𝑥𝑝(−𝐸𝑘𝑇)𝑍

Here, Z stands for the partition function.

Statistical Entropy

In the statistical mechanics of an isolated system at thermodynamic equilibrium, the entropy S can be defined as W’s natural algorithm.

𝑆=𝑘ln𝑊.

Method for Calculating the Boltzmann Constant

There are various methods that can be used to calculate the Boltzmann Constant. The Boltzmann constant at a particular temperature is calculated using the value of k using the Clausius–Mossotti relation.

The estimation of the Boltzmann Constant is best done with the help of acoustic thermometry, which employs the fact that the speed of sound in the gaseous state is directly proportional to its temperature. Another important technique used to determine how a gas reacts to electric field changes is dielectric steady gas thermometry (DCGT) 

Calculating the dielectric constant is the final step. As the dielectric constant is influenced by temperature, we calculate the Boltzmann Constant this way. The Boltzmann Constant estimation is also achieved by using some optional approaches like Johnson Noise Thermometry.

Significance of the Boltzmann Distribution 

Boltzmann Constant is a very important concept in Physics. The likelihood of a system being in a particular state as a function of its energy is predicted by the Boltzmann distribution. On the other hand, the speed or energies of a particle in a perfect gas is predicted by the Maxwell-Boltzmann Constant.

The usual active energy of a framework in equilibrium is connected to its temperature at every degree of opportunity by the Boltzmann Constant. For example, the Boltzmann Constant gives a relation between the normal dynamic energy of gas particles with their temperature.

Important Points

• Boltzmann Constant establishes the connection between the kinetic energy contained in each molecule of a gas with its absolute temperature. It is represented by the symbol kB or k. The name of the constant is taken after the Austrian scientist Ludwig Boltzmann who significantly contributed to the creation of statistical mechanics in the 19th century.
• The equipartition of the energy of an atom in classical statistical mechanics is also described by the Boltzmann Constant, which is also a symbol for the Boltzmann factor. The statistical definition of entropy is also significantly impacted by it.
• The Boltzmann Constant is used in the expression of thermal voltage in semiconductor physics.
• This constant establishes a relationship between temperature and wavelength. One micrometer is equal to 14387.777 K. It also establishes a relationship between temperature and voltage where one volt equals 11604.518 K. In eVm, the numerical value of hc represents the ratio of these two temperatures 14387.777 K /11604.518 K = 1.239842 K.

A Sample Question on Boltzmann Constant

Calculate the value of β in the relationship p =α/β e(-αz/kbθ), where z stands for the distance, p stands for the pressure, and θ represents the temperature.

Ans. [M¹L^-1T^-2] is the dimension of P.

z = [L¹]

Kb = [M¹L²T^-2K^-1]

θ = [K¹]

e-αz/kbθ is a dimensionless quantity

α = θKb/z = [M¹L²T^-2K^-1][K¹]/[L¹] = [M¹L¹T^-2K⁰]

As, p = α/β or β = α/p

So, the dimension of β is derived as [M¹L¹T^-2K⁰]/[M¹L^-1T^-2]= [M⁰L²T⁰K⁰].

Conclusion

Boltzmann A fundamental idea in physics is the constant. It is crucial to understand its foundational ideas as a result. We trust that this article has improved your comprehension of the subject. 

Try solving more problems like the one given above using the Boltzmann constant formula to gain practical knowledge about it.

Frequently Asked Questions.

Q1) Give the significance of the Boltzmann constant unit.

 The Boltzmann constant unit relates the temperature with the energy.

Q2) Give the dimension of the Boltzmann constant unit.

 Dimensionally, the Boltzmann Constant is represented as [M¹ L² T^-2 K^-1]

Q3) Where do we use the Stefan Boltzmann constant?

The Stefan Boltzmann constant describes the distribution of energy of an atom. As it has a significant impact on the statistical definition of entropy, it is employed in the expression of thermal voltage in semiconductor Physics.

Q4) What is the Boltzmann constant in simple terms?

The Stefan Boltzmann constant is a physical constant that relates the average kinetic energy of particles in a gas with its temperature.