# Applications of Boltzmann equation

## Introduction

Boltzmann Equation describes the reactions of fluids with temperatures. When a fluid is in motion, the change in the physical amount of thermal energy and momentum is calculated by Boltzmann's equation. This equation is also known as Boltzmann Transport Equation (BTE). Thermal conductivity, viscosity, and electrical conductivity of fluids are also obtained by the Boltzmann equation. In 1872 Ludwig Boltzmann developed an equation that classified the statistical reactions of thermodynamics as an unbalanced state. In modern physics, statistical mechanics is one of the most important chapters of physics. Maxwell-Boltzmann statistics is one of the parts of modern statistical mechanics.

## What is the Boltzmann Equation?

Boltzmann Equation describes the reactions of fluids with temperatures. This equation is also known as Boltzmann Transport Equation (BTE). The equation is a nonlinear integrodifferential equation.

$$\mathrm{\frac{N_b}{N_a}=\left(\frac{g_b}{g_a}\right)e^{− \frac{(E_b-E_a)}{kT}}}$$

Here

Nb and a = number of atom

k=Boltzman constant

T=temperature of gas

## Statement: Boltzmann Law

Boltzmann law is also called Stefan-Boltzmann law. It states that the total amount of thermal energy radiated per second per unit area of a perfectly black body is directly proportional to the fourth power of its absolute temperature

$$\mathrm{E\:\alpha\:T^4}$$

$$\mathrm{E=\sigma T^4}$$

Here,

E=Thermal Energy

T=Absolute Temperature

Here $\mathrm{\sigma}$ is Stefan's Constant

$$\mathrm{\sigma=5.67×10^{−8} Wm^{−2} K^{−4}}$$

Theoretical proof of this Stefan’s result is given by Boltzmann.

## Ludwig Boltzmann

Ludwig Boltzmann was an Austrian physicist born in 1844. He gives the statistical description of the second law of thermodynamics and the development of Statistical mechanics is the best achievement in his life. The current definition of entropy is given by Boltzmann.

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

Here,

$\mathrm{\Omega}$ is the number of microstates which is equal to the energy of the system.

$\mathrm{k_B}$ is known as the Boltzmann constant.

In 1863 he studied at the University of Vienna. He studied mathematics and physics. In 1866 he completed his doctorate. Joseph Stefan is the mentor of Boltzmann. Stefan is the director of the institute of physics. Boltzmann worked with Stefan very closely. In 1890 the University of Munich in Bavaria appointed Boltzmann as Chairman of Theoretical Physics. He died at age 62 (5th September 1906).

## What is Boltzmann Constant?

Max Planck introduced the Boltzmann constant. It is the constant value that connects the kinetic energy of the gas and the temperature of the gas. Boltzmann constant is the ratio between vapor or gas constant and Avogadro number. Both are constant values.

$$\mathrm{k=\frac{R}{N_A}}$$

Here,

k=Boltzman constant

R=gas constant

$\mathrm{N_A}$-Avogadro number

Normally Boltzmann constant is denoted as $\mathrm{k_B}$.

Unit of Boltzmann constant is joule per kelvin or $\mathrm{m^2 kgs^{−2} K^{−1}}$.

value of $\mathrm{k_B=1.3806452×10^{−23} J/K.}$

$$\mathrm{k_B=1.3806452×10^{−23} m^2 kgs^{−2} K^{−1} }$$

Boltzmann constant exhibits the energy of atoms in statistical mechanics in classical Physics. The Boltzmann factor is expressed by the Boltzmann constant. In the current definition of entropy, the Boltzmann constant plays a crucial role. Thermal voltage is expressed by the Boltzmann constant in semiconductor physics.

Boltzmann law is also called Stefan-Boltzmann law. It states that the total amount of thermal energy radiated per second per unit area of a perfectly black body is directly proportional to the fourth power of its absolute temperature

$$\mathrm{E\:\alpha\:T^4}$$

$$\mathrm{E=\sigma T^4}$$

To derive this formula, we can use Planck’s radiation formula,

$$\mathrm{\frac{dP}{dA}\frac{1}{A}=\frac{2\pi hc^2}{\lambda^5(\frac{hc}{e^{\lambda kT}}-1)}}$$

Integrate both sides for λ with the limits,

$$\mathrm{\int_{0}^{\infty}\frac{d(\frac{P}{A})}{d\lambda}=\int_{0}^{\infty} \begin{bmatrix}\frac{2\pi hc^2}{\lambda^5(\frac{hc}{e^{\lambda kT}}-1)}\end{bmatrix}d\lambda}$$

$$\mathrm{\frac{P}{A}=2\pi hc^2\int_{0}^{\infty}\begin{bmatrix}\frac{d\lambda}{\lambda^5(\frac{hc}{e^{\lambda kT}}-1)}\end{bmatrix}---------(1)}$$

Let $\mathrm{\frac{hc}{\lambda kT}=x}$

$$\mathrm{\therefore\:\:\:\:h=\frac{x\lambda kT}{c}}$$

$$\mathrm{c=\frac{x\lambda kT}{h}}$$

Differentiating the x value with respect to λ

$$\mathrm{dx=-\frac{hc}{\lambda^2 kT}d\lambda}$$

$$\mathrm{\therefore\:\:\:\:-\frac{\lambda^2 kT}{hc}dx}$$

Substituting the values of h, c, dλ, and x in equation (1).

$$\mathrm{\frac{P}{A}=2\pi (\frac{x\lambda kT}{c})(\frac{x\lambda kT}{h})^2\int_{0}^{\infty}\frac{-\frac{λ^2 kT}{hc}dx}{e^x-1}}$$

$$\mathrm{\frac{P}{A}=2\pi (\frac{x^3 \lambda^5 k^4 T^4}{h^3 c^2 \lambda^5})\int_{0}^{\infty}\begin{bmatrix}\frac{dx}{e^x-1}\end{bmatrix}}$$

$$\mathrm{\frac{P}{A}=\frac{2\pi(kT)^4}{h^3 c^2}\int_{0}^{\infty}\begin{bmatrix}\frac{x^3}{e^x-1}\end{bmatrix}dx----------(2)}$$

From the integral formula

$$\mathrm{\int_{0}^{\infty}\begin{bmatrix}\frac{x^3}{e^x-1}\end{bmatrix}dx=\frac{\pi^4}{15}}$$

Substitute this formula in equation (2)

$$\mathrm{\frac{P}{A}=\frac{2\pi(kT)^4}{h^3 c^2}\frac{\pi^4}{15}}$$

$$\mathrm{\frac{P}{A}=\left(\frac{2k^4 \pi^5}{15h^3 c^2}\right)T^4}$$

Here $\mathrm{\left(\frac{2k^4 \pi^5}{15h^3 c^2}\right)=\sigma}$

$$\mathrm{\therefore\:\:\:\:\frac{P}{A}=\sigma T^4}$$

(Or)

$$\mathrm{\varepsilon=\sigma T^4\:\:\:\:\:(\varepsilon=\frac{P}{A})}$$

$\mathrm{\varepsilon=\sigma T^4}$ is the Stefan Boltzmann formula.

## Applications of Boltzmann Equation

• Conservative laws for mass, charge, momentum, and energy equations are derived using the Boltzmann equation.

• It is used to reframe classical mechanics into Hamilton mechanics. Different mathematical methods help to reframe this.

• Relativistic quantum systems where the number of particles in a collision is not protected so there is the possibility to define quantum Boltzmann equations.

• To find the galactic dynamics Boltzmann constant is used.

## Boltzmann Equation Examples

Example: In thermodynamic equilibrium, a gas of Hydrogen atom(H) has an equal number of atoms in the first and second states. Calculate the temperature of the gas.

Ans: Boltzmann Equation is

$$\mathrm{\frac{N_b}{N_a}=(\frac{g_b}{g_a})e^{−\frac{\Delta E}{kT}}}$$

From the given instruction number of atoms, $\mathrm{N_1}$ and $\mathrm{N_2}$ are equal. So, the ratio of the number of atoms is equal to 1.

$$\mathrm{\frac{N_2}{N_1}=1}$$

Degeneracy for hydrogen $\mathrm{g_n=2n^2}$

$$\mathrm{n=1\:then\:g_1=2(1)^2=2}$$

$$\mathrm{n=2\:then\:g_2=2(2)^2=8}$$

$$\mathrm{\frac{g_2}{g_1}=\frac{8}{2}=4}$$

Apply these values in the Boltzmann equation,

$$\mathrm{1=4e^{−\frac{\Delta E}{kT}}}$$

$$\mathrm{\frac{1}{4}=e^{−\frac{\Delta E}{kT}}}$$

Taking ln on both sides

$$\mathrm{ln(0.25)=-\frac{\Delta E}{kT}}$$

ΔE for Hydrogen atom is $\mathrm{\Delta E=1.63×10^{−18}\:J}$

$$\mathrm{\therefore\:\:\:\:Temperature\:\:T=-\frac{\Delta E}{k(ln(0.25))}}$$

Applying the values of energy change and Boltzmann constant value we get,

$$\mathrm{T=8.53×10^4\:K}$$

## Conclusion

The Boltzmann equation was given by Ludwig Edward Boltzmann. In 1872. He developed Statistical mechanics and make the best achievement of his life. When a fluid is in motion, the change in the physical amount of thermal energy, and momentum is calculated by Boltzmann's equation. Thermal conductivity, viscosity, and electrical conductivity of fluids are also obtained by the Boltzmann equation. Boltzmann law is also called Stefan-Boltzmann law. Stefan is the director of the institute of physics. Boltzmann worked with Stefan very closely. Conservative laws for mass, charge, momentum, and energy equations are derived using the Boltzmann equation. It is used to reframe classical mechanics into Hamilton mechanics. Different mathematical methods help to reframe this.

## FAQs

Ans. In one mole of a substance how many molecules are available is called the Avogadro number. $\mathrm{6.023×10^{23}}$ is the value of the Avogadro number and it is a constant value for all mediums.

Q2. Define the First Law and the Second Law of Thermodynamics

Ans. The sum of the work done by the system and change in internal energy is equal to the amount of thermal energy provided. This is the first law of thermodynamics.

Uniformity between work and heat is defined in the first law of thermodynamics.

The second law of thermodynamics is the process of the first law of thermodynamics whether happens or not. There are many explanations for the second law of thermodynamics given by many scientists.

Q3. What is a perfect Black Body?

Ans. When heating a black body, if it absorbs all the thermal radiations completely in every wavelength which drops on the body and emits heat radiations of every wavelength is called a perfect black body.

Q4. The temperature of a gas is 1000K. Calculate the radiated energy. Given $\mathrm{\sigma=5.67×10^{−8}\:Wm^{−2} K^{−4}}$.

Ans. Given the temperature

T=1000K

$$\mathrm{\sigma=5.67×10^{−8}\:Wm^{−2} K^{−4}}$$

$$\mathrm{Radiated\:Energy\:E=\sigma T^4}$$

$$\mathrm{E=(5.67×10^{−8})(1×10^3)^4}$$

$$\mathrm{E=5.67×10^4\:Wm^{−2}\:K^{−3}}$$

Q5. State Newton’s law of cooling

Ans. The rate of cooling of a substance is directly proportional to the temperature variation between the surroundings and the substance. This law carries an advantage for a small variation of the temperature. The radiated thermal energy loss depends on the properties of a surface and the area of the uncovered surface.

Updated on: 18-Apr-2023

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