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 which tells us the following:

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

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]. Further, 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.

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 with.

We know that the dimensional formula for Boltzmann constant (Kb)

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

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

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.

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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 temperature, so 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 damages 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 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= \[\frac{1}{6.022 x 10^{23}molecules per mole}\] = 1.38 x 10^{-23}J/K

Do you know what the dimensions of Boltzmann’s constant are?

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 = \[[Energy]x[Temperature]^{-1}\]

Now, dividing equation (1) by equation (2), we get our desired formula:

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

∴ 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 the 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 \[[M^1L^2T^{-2}K^{-1}]\].

FAQ (Frequently Asked Questions)

Q1: What is K_{b} in Boltzmann's Formula?

Ans: K_{b} 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.

Q2: How is Boltzmann Constant Calculated?

Ans: 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 K_{b}. The value of Boltzmann constant is measured using J/K or Kgm^{2}s^{-2}K^{-1}.

Q3: 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?

Ans: The dimension of P = [M^{1}L^{-1}T^{-2}]

z = [L^{1}]

Kb = [M^{1}L^{2}T^{-2}K^{-1}]

Ө = [K^{1}]

e^{-αz/kbӨ} is dimensionless

[α] = ӨK_{b}/z = [M^{1}L^{2}T^{-2}K^{-1}][K^{1}]/[L^{1}]= = [M^{1}L^{1}T^{-2}K^{0}]

Since p = α/β or β = α/p

So, we get the dimension of β as [M^{1}L^{1}T^{-2}K^{0}]/[M^{1}L^{-1}T^{-2}]= [M^{0}L^{2}T^{0}K^{0}].

Q4: What is Boltzmann constant in eV?

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