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Dimensions of Boltzmann Constant

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Last updated date: 14th Jul 2024
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Introduction

The Boltzmann constant which is generally represented by either k or kB is a proportionality factor that helps compare the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. The Boltzmann constant is measured by using energy per degree of temperature units. The value after calculation of the Boltzmann constant rounds up to 1.380649 X 10-23 joule per kelvin. This Boltzmann constant can be expressed in various ways with a change in the units as well. It is now possible to know everything about the Dimensions of Boltzmann Constant – Explanation, Formula, and Important FAQs via Vedantu and prepare for their IIT JEE Main exams.


Dimensions of Boltzmann Constant – Explanation

We know that derived units can be obtained from the fundamental units of mass, length, and time. Let’s represent mass with M, length with L, and time with T.  Then we can say that the dimensional formula for a physical quantity is an expression that tells us the following:

  1. The fundamental (basic) units on which the quantity depends, and

  2. The nature of dependence.


For example, \[[M^0L^1T^{-1}]\], is the dimensional formula for velocity. It shows that velocity depends upon L and T. Furthermore, the unit varies directly as a unit of length and inversely as the unit of time.


So, [v] = \[[M^0L^1T^{-1}]\] is the dimensional equation for velocity.


Dimensional Constants

We understood the concept of finding the dimensional formula of a physical quantity. So there are other quantities that we are going to use, that is dimensional constant.


So, dimensional constants are those quantities whose values are constant, and they possess dimensions.

 

For example, Gravitational constant, Boltzmann constant, Planck’s constant, etc.


Here, we will find out the dimension of a Boltzmann constant. 


Let’s get started.


The Dimension of Boltzmann Constant

We know that the dimensional formula for Boltzmann constant (Kb

=\[\frac{Dimensional \:formula \:for\: energy\: (E)}{Dimension \:of\: temperature \:in \:Kelvin}\]

=  \[\frac{[ML^2T^{-2}]}{[K]}\] = \[[ML^2T^{-2}K^{-1}]\]

 

What is Boltzmann’s Constant?

Do You Know What the Boltzmann Constant is?

Let’s say, you keep a water-filled container on fire, and cover it with a lid. After some time, when you remove the lid, you can see that there are several vibrations inside the container. 


So, Why is There a Vibration inside the Container?

Well, the molecules inside the water gain three kinetic energies, i.e., translational kinetic energy, rotational kinetic energy, and vibrational kinetic energy.


So, when the temperature increases, their kinetic energy also increases. As the volume increases, the molecules reach the edge of the container and start creating pressure on the walls. 


Since the molecules are at high temperatures, when you put your hand inside the container, the molecules on your hand absorb the energy transferred by these gas molecules, spread around your hand, and damage your skin.


On increasing the temperature, the kinetic energy of a molecule increases. This means that the kinetic energy of each molecule is directly proportional to the temperature.

                                T ∝ KE

We know that PV = n * RT is the ideal gas law, where 

P is the pressure, measured in Pascals

V is the volume, measured in a cubic meter

n is the number of moles,

R =  universal Gas Constant, and

T = the temperature in Kelvin

We know that n = N/NA = total number of molecules in the gas/Avagadro’s number

Let’s rewrite the ideal gas law:

PV = N__T

Here, we are considering the number of molecules (because n is a tiny value), so we will use a different constant and that constant is the Boltzmann constant.

So, the new equation becomes, PV = NKbT

Here, Kb = Boltzmann's constant

nR = NKb

⇒ Kb = (n/N)R = (1/NA)R= \[\frac{1}{6.022 \times 10^{23}\:molecules \:per \:mole}\] = 1.38 x 10-23J/K

Do You Know What the Dimensions of Boltzmann’s Constant are?


Dimension Formula of Boltzmann Constant

We know that Boltzmann constant = Energy x temperature

The formula for energy = the product of mass, acceleration, and displacement

= \[[M^{1}]x[M^0LT^{-2}][L^1]\] = \[[M^1L^2T^{-2}]\]...(1), and

The dimensional formula for the temperature is \[[M^0L^0T^{0}K^{1}]\]…(2)

On substituting the equation (2) and (3) in equation (1), we get

Boltzmann’s constant, Kb = \[[Energy]\times[Temperature]^{-1}\]

= \[\frac{[M^1L^2T^{-2}]}{[M^0L^0T^{0}k^1]}\] = \[[M^1L^2T^{-2}K^{-1}]\]

∴ The Boltzmann’s constant is dimensionally expressed as \[[M^1L^2T^{-2}K^{-1}]\].

Deriving it by using the formula, 1/NA R…(a)

We know that NA is Avogadro’s number, which equals the number of atoms divided by the number of atoms in one gram atom per mole. So, it is a constant value.

Then, its dimensional formula is \[[M^0L^0T^{0}]\], and

R is the universal gas constant, whose value is 8.314 J/K-mol, and its dimensional formula is \[[M^1L^2T^{-2}K^{-1}]\].

So, substituting the formula of R and NA in equation (a), we get

=  \[\frac{[M^1L^2T^{-2}K{-1}]}{[M^0L^0T^{0}]}\] = \[[M^1L^2T^{-2}K^{-1}]\]

Hence, the dimensional formula for Boltzmann’s constant is \[[M^1L^2T^{-2}K^{-1}]\].

 

Use of Boltzmann Constant in Chemical Kinetics

A Boltzmann constant can be used in two types of equations:

1. Arrhenius Equation:

K = Ae\[\frac{-E_{a}}{K_{B}T}\]

This equation is a formula for the temperature dependence of reaction rates which is provided as follows:

Where,

  • Ea is the activation energy for the reaction

  • K is the rate constant

  • T is the absolute temperature

  • A is the preexponential factor that will remain constant for a chemical reaction

  • KB is the Boltzmann constant.


2. Eyring Equation:

K = \[\frac{k_{B}T}{h}\] e\[\frac{\Delta G}{RT}\]

The Eyring equation is a chemical kinetics equation that describes the rate of chemical reaction as a function of the temperature.

Where,

  • T is the absolute temperature

  • ∆G is the Gibbs free energy of activation

  • KB is the Boltzmann constant

  • H is Planck’s constant

  • R is the gas constant

  • K is the rate constant

FAQs on Dimensions of Boltzmann Constant

1. What is Kb in Boltzmann's Formula?

Kb in Boltzmann's formula describes the relationship between entropy (randomness of the gas molecules) and how the number of ways the atoms or molecules of a thermodynamic system can be arranged.

2. How is Boltzmann Constant Calculated?

In thermodynamics, Boltzmann constant is the physical constant that relates the average kinetic energy of the gas molecules with the temperature of the molecules. It is represented by k or Kb. The value of the Boltzmann constant is measured using J/K or Kgm2s-2K-1.

3. In a relation, p =α/β  e-αz/kbӨ where p is the pressure, z is the distance, Ө is the temperature, then the value of β is?

The dimension of P = [M1L-1T-2]

z  = [L1]

Kb = [M1L2T-2K-1]

Ө = [K1]

e-αz/kbӨ is dimensionless

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

Since p = α/β or β = α/p

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

4. What is Boltzmann constant in eV?

The value of Boltzmann constant in eV is  8.6173 x 10-5eV/K.

5. What are the applications of the Boltzmann constant?

The Boltzmann constant has found uses in various fields of Physics and can be provided as follows:

  • The Boltzmann constant in classical statistical mechanics is widely used in expressing the equipartition of the energy of an atom.

  • It plays a critical role in the statistical definition of entropy.

  • It is also used in the expression of the Boltzmann factor.

  • In semiconductor physics, the Boltzmann constant is used in order to express thermal voltage.

  • It helps in finding out the kinetic theory of gases.

6. Is Boltzmann constant always a constant?

The Boltzmann constant is not a real physical constant the same way that the fine structure constant is a fundamental constant. Boltzmann constant is essentially just a ratio of the average energy of gas with its temperature. The values that have been listed have an uncertainty however that’s not a fundamental uncertainty in the constant itself. It is found out that the uncertainty is in the definition of the Kelvin which is involved with the Boltzmann constant. This constant is just a choice to measure the temperature in Kelvins. If the temperature is measured in energy then the Boltzmann constant will be dimensionless and will stay exactly equal to 1.

7. What is the history of the Boltzmann constant?

The Boltzmann constant is named after the 19th century Austrian discovered Ludwig Boltzmann. Although Boltzmann had linked the relation between entropy and probability it was found that there was no proper expression for it until Max Planck who provided the value k which had a more precise value to it which was introduced in the year 1900 during the derivation of black body radiation. Before the 1900s the equations that and use the Boltzmann constant were not written by using the energies per molecule and the Boltzmann constant instead it was written using a form of gas constant R and the macroscopic energies for larger quantities of the substance. It was in 2017 that the most accurate measures of Boltzmann constant were obtained by acoustic gas thermometry.

8. What is the role of the Boltzmann constant in statistical definitions of entropy?

In statistical mechanics, the entropy S of an isolated system at thermodynamic equilibrium can be said to be the natural logarithm of W which indicates the number of distinct macroscopic states available in the system given the macroscopic constraints. This formula can be represented as follows:

S = kB log W


This equation relates the microstates of the system with the macroscopic state of the system. This relation is the central idea of statistical mechanics. The constant of proportionality k served to make the statistical mechanical entropy equal to the classical thermodynamic entropy of Clausius:

ΔS = \[\int\frac{dQ}{T}\]

9. Why is it important to learn more about the Dimensions of Boltzmann Constant – Explanation, Formula, and Important FAQs?

Apart from its wide uses, Boltzmann Constant is one of the most important factors in Physics and helps in solving a lot of problems. By learning more about Boltzmann Constant – Explanation, Formula, and Important FAQs students get to know how to apply them in problems they face in real life. This topic is also important exam-wise. It is also an important topic on which there is numerical put during the JEE Main exams. Students who would like to know more about the questions that are added in the IIT JEE Main based on this topic can now check the Vedantu IIT JEE Main sample papers for practice. You can also attempt some mock tests that are free of charge available on Vedantu that will help you prepare well.