Dimensions of Boltzmann Constant

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