Dimensional Formula of Boltzmann Constant

Boltzmann constant is one of the essential contents of thermodynamics. This constant establishes a relationship between kinetic energy and temperature. The constant has a widespread application in the field of thermodynamics and thermochemistry. Boltzmann constant is represented by k or kb. According to the latest definition of basic constants

The value of it is defined as 1.380×10−23 J/K.

Kinetic Energy and Temperature

The relationship between kinetic energy and temperature is that the higher the temperature, the faster the particles move. The only motion possible for atoms in a simple monatomic gas, such as helium or neon, travels from one location to another in a straight line until they collide with another atom or molecule. So, a gas’s average kinetic energy and temperature are directly proportional. The relative kinetic energy of the two colliding atoms can and often does change as a result of these collisions: if one slows down, the other accelerates.  

Dimensional Formula of Boltzmann Constant

As mentioned in the above definition, it is defined as

Boltzmann Constant(k) = energy/temperature.

So, dimensions of Boltzmann Constant = dimension of energy/dimension of temperature.

Dimension of energy = dimension of work done = dimension of force x dimension of displacement = [MLT-2] x [L]=  [ML2T-2].

So, the dimension of Boltzmann Constant =  [ML2T-2]/[K] = [ML2T-2K-1].

Hence, the dimension of  Boltzmann Constant is  [ML2T-2K-1].

Meaning of Boltzmann Constant/Definition

Boltzmann constant is one of the very constants in thermodynamics. In several definitions, including kelvin, black-body radiation, gas-constant etc., it is defined as energy per unit temperature change. Boltzmann constant k or kb = R/NA where R=universal gas constant and NA  is Avogadro’s number.  

Unit of Boltzmann Constant: the SI base unit is J/K.

Kinetic Energy

The kinetic energy of an atom or molecule is directly proportional to this type of motion, known as translational motion. 

So,

KE = 12mv2 = 32 kT, 

where v = the average velocity of the population’s molecules.

m = mass.

k = the Boltzmann constant.

T = temperature.

Atoms in any given gaseous sample collide multiple times in unit time. Yet, these collisions do not affect the total energy of the system.

There is kinetic energy in every atom and molecule, but not temperature. This is a crucial distinction to make. Individual molecules do not have a temperature; they have kinetic energy. Populations of molecules have a temperature linked to their average velocity. Temperature as a characteristic of a system, rather than its components, is an essential concept. While a system’s temperature is unique, the kinetic energy of the individual molecules that make up the system might be varied. Even though the temperature of the system remains constant, an individual molecule’s kinetic energy might change rapidly due to collisions between molecules. Individual kinetic energy will be crucial when it comes to chemical bonds.

The mathematical expression of the ideal gas law is given by PV = nRT

Where, P – absolute pressure.

             n – amount of substance.

             T – absolute temperature.

             V – volume.

             R – ideal gas constant (product of the Avogadro constant and Boltzmann constant), = 8.31 J/mol K.

Ideal gas law is widely used, and whenever any of the two variables is given in P,V and T, the third can be easily found.

Dimensional Formula

In terms of dimensions, a dimensional formula is an equation that expresses the relationship between fundamental and derived units (equation). The letters L, M, and T represent the three basic dimensions of length, mass, and time in mechanics.

All physical quantities can be stated in terms of the fundamental (base) units of length, mass, and time, multiplied by some factor (exponent).

The dimension of the amount in that base is the exponent of a base quantity that enters into the expression.

The units of fundamental quantities are expressed as follows to determine the dimensions of physical quantities:

• ‘L’ stands for length, 
• ‘M’ for mass, and 
• ‘T’ for time.

Example: The area is equal to the sum of two lengths. As a result, [A] = [L2]. That is, an area has two dimensions of the length and zero dimensions of mass and time. In the same way, the volume is the sum of three lengths. As a result, [V] = [L3]. The volume dimension has three dimensions of length ande zero dimensions of mass and time.

Conclusion

In this chapter of the Dimension formula for Boltzmann Constant, we learned the basics of Boltzmann Constant and its dimensional analysis. For an ideal gas, when temperature increases, the randomness also increases. The Boltzmann constant k represents the proportionality.