Boltzmann Constant - Definition, Formula, Value, Applications


Introduction

Boltzmann’s constant is denoted by letter $\mathrm{k_B}$. Certain physical quantities in physics remain constant with time and are universal constants. These are called fundamental physical constants. For example - speed of light (c), Planck's constant $\mathrm{(\hslash)}$, Boltzmann constant $\mathrm{(k_B)}$ are such quantities. Boltzmann constant first time arises in physics in the context of statistical physics. However, its importance is not limited to that field only, today it is used in chemistry, thermodynamics, etc. Boltzmann’s constant was named after the famous physicist Ludwig Boltzmann.

He was an Austrian physicist. His greatest contribution was the development of statistical mechanics. He also worked in other fields such as the kinetic theory of gases and thermodynamics. Boltzmann constant works as a proportionality factor between the kinetic energy of molecules and their temperature.

It is interesting to note that his famous equation which connects the entropy with probability is written in his grave in Vienna.

Value of Boltzmann Constant

Boltzmann constant’s value is $\mathrm{1.38×10^{−23} JK^{−1}}$ in SI units.

Its Dimension is given by

$$\mathrm{ [k_B]=[ML^2 T^{−2} K^{−1}]}$$

For different purposes, it is expressed in different units. Here are some examples

  • If energy is expressed in $\mathrm{eV, k_B=8.61×10^{−5} eVK^{−1}}$

  • In CGS units: $\mathrm{k_B=1.38×10^{−16} ergK^{−1}}$

  • In atomic calculations, for Hartree energy $\mathrm{ E_H :\:\:k_B=3.16×10^{−6} E_H K^{−1}}$

  • In thermal noise calculations: $\mathrm{k_B=-228.5991672 dB(W/K/Hz)}$

What is Boltzmann Constant

Boltzmann derived a relation between thermodynamic probability and entropy.

$$\mathrm{S=f(\Omega)}$$

He worked out the dependence of this function $\mathrm{f(\Omega)}$ on entropy.

Finally, he reached at

$$\mathrm{f(\Omega)=kln\Omega+C}$$

Here the constant k is the same for all systems. But Boltzmann himself was not able to tell the meaning and nature of the constants k and c.

Max Planck used the fact that at T=0K, we get zero entropy. Hence

$$\mathrm{S=0=kln\Omega+c}$$

This means at T=0K,Ω=1 and c should be taken zero. It leads to the famous equation -

$$\mathrm{S=k_B\:ln\Omega}$$

Where $\mathrm{k_B}$ is identified as Boltzmann constant. This relation is a very important result in statistical mechanics and is referred to as the Boltzmann relation.

Significance of Boltzmann Constant Formula

Boltzmann constant plays a very significant role in physics. Here are some examples

SI unit of Temperature

Boltzmann constant is used to define Kelvin, the SI unit of temperature. 1K is defined as the temperature which makes the value of $\mathrm{k_B= 1.38× 10^{−23}\:J/K}$

Boltzmann Factor

In statistical mechanics, the Boltzmann factor gives us the probability of occupancy of some energy state at temperature T. In this formula, the Boltzmann factor comes inside of the exponential.

$$\mathrm{P\:\propto\:exp(\frac{-E}{k_B T})}$$

Equipartition Theorem

The Equipartition theorem tells the relation between the temperature of the system and its average energy. It says that every degree of freedom will contribute energy $\mathrm{\frac{1}{2}\:k_B\:T}$ to the system. Here we can see the prominence of Boltzmann's constant.

A bridge between Macroscopic and Microscopic Quantities

Boltzmann’s constant works as the ridge between macroscopic and microscopic quantities. As we saw in the case of Boltzmann's relation

$$\mathrm{S =k_B lnΩ}$$

It works as a proportionality factor between macroscopic quantity entropy and microscopic quantity thermodynamic probability.

Planck’s Law

Boltzmann constant is also present in Planck’s law of black body radiation.

$$\mathrm{f_
u (T) = \frac{2
u^{2} h
u}{c^2(exp\lbrace\frac{hν}{k_B T}\rbrace -1)}}$$

Here ν = frequency

$\mathrm{f_
u}$= spectral radiance

Applications

Boltzmann's constant can be used in several other places in physics. Some examples are the following

Can Form Planck Unit

Using four fundamental constants - speed of light, Planck’s constant, Boltzmann constant, and gravitational constant we can form Planck’s temperature.

$$\mathrm{T_p=\sqrt{\frac{hc^5}{Gk^2B}}}$$

Gas Equation Conversion

The ideal gas equation can be written as

$$\mathrm{PV = nRT,}$$

Using Boltzmann’s constant, we can write it in terms of no. of molecule in gas. Since gas constant is defined as $\mathrm{R=k_B\:N_A}$, where $\mathrm{N_A}$ is Avogadro number.

Hence

$$\mathrm{ PV=nk_B T }$$

Thermal Voltage

We also use it in the diode equation to define thermal voltage.

$$\mathrm{I_D=I_s\lbrace exp(\frac{eV_D}{nk_B T})\rbrace-1}$$

Here $\mathrm{\frac{e}{k_B T}}$ is called thermal voltage and its value at room temperature is 25.9mV

Thermodynamic Beta

Boltzmann constant and product of temperature defines a quantity called beta.

$$\mathrm{\beta=\frac{1}{k_B T}}$$

In statistical mechanics, it is considered a more fundamental quantity than the usual thermodynamics temperature.

Thermal Noise

In electronic devices thermal noise power is given by the formula

$$\mathrm{P=k_B T\Delta f}$$

Here and in other related formulas like RMS current and voltage, the Boltzmann’s constant comes as proportionality is constant.

Conclusion

Boltzmann constant plays an important role in physics. It was named after the famous physicist Ludwig Boltzmann; however, he had not calculated its value. It relates the average energy of molecules to the temperature. Its unit is the same as entropy. It is useful in defining the gas constant, thermal voltage, inverse temperature, etc.

FAQs

Q1. Can entropy decrease? Give the reasoning from the Boltzmann formula.

Ans. The value of entropy is given by

$$\mathrm{S = k_B lnΩ}$$

We know that the Boltzmann constant is a positive number and the value of the logarithm can’t be negative. Hence

$$\mathrm{S\ge 0\:always}$$

Q2. What is the average energy of a system having 3 degrees of freedom?

Ans. According to the equipartition theorem: one degree contributes $\mathrm{\frac{1}{2}k_B T}$ of energy. Hence energy due to three degrees of freedom will be $\mathrm{\frac{3}{2}k_B T}$

Q3. How is Boltzmann's constant measured?

Ans. Boltzmann constant has been measured in many ways so far. The agreeable value was released in 2018. Here are the two most reliable techniques for the measurement of the Boltzmann constant.

  • Acoustic Thermometry: In this method, physicists use the fact that the speed of sound is related to temperature,

  • Dielectric Constant gas thermometry: In this technique, scientists use the relation between dielectric constant and temperature,

Q4. Calculate the Thermal voltage at $\mathrm{T=20^{\circ}\:C}$?

Ans. Thermal voltage is given by $\mathrm{\frac{e}{k_B T}}$

Here $\mathrm{T=20^0 C=293K,k_B=1.38×10^{−23} JK^{−1}}$

Hence Thermal voltage $\mathrm{=\frac{1.6×10^{−19}}{293 × 1.38 ×10^{−23}}=25.2mV}$

Q5. Give the examples where Boltzmann’s constant is used in chemistry?

Ans. In chemistry, the relation between temperature and the rate of a chemical reaction is given by the Arrhenius equation.

$$\mathrm{k = A\:exp(-\frac{E_A}{k_B T})}$$

Where k = rate constant, $\mathrm{E_A}$= Activation energy

Updated on: 18-Apr-2023

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