Boltzmann Constant Formula: Definition And Applications

Boltzmann constant (kB) is a constant named after Ludwig Boltzmann, which relates the average kinetic energy of particles in a gas to the temperature of the gas.

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Boltzmann constant formula

The physical constant relates the average kinetic energy of gas particles to the temperature of the gas, which is represented by k or kB in thermodynamics. As observed in Boltzmann’s entropy formula and Planck’s law of black body radiation, the Boltzmann constant is measured as J/K or m2Kgs-2K-1.

Boltzmann Constant Definition

The energy of a gas molecule is proportional to the absolute temperature in most cases. When a gas is heated, the molecules in it move faster. The kinetic energy per molecule increases as the temperature increases. As a result, either a higher pressure or an increased volume will result if the gas is contained within a constant volume space.

The Boltzmann constant, k, is of physical significance as it measures the energy (heat) assigned to the random thermal motions of a material’s particles. 

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For example, the average energy per degree of freedom in a classical system in equilibrium at temperature T is kT/2. Taking one atom as an illustration, with three translational degrees of freedom (i.e. able to move in either the x, y or z-direction), and considering a gas made up of N non-interacting atoms; then its total thermal energy can be derived as 3NkT/2.

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Value Of Boltzmann Constant

Gas constant R is divided by Avogadro’s number NA to obtain Boltzmann constant values. The value of k is

Boltzmann constant, kB= 1.3806452 × 10-23J/K

Boltzmann’s Constant

Everything in this world is made up of atoms and molecules. Put a vessel full of water on fire, lid it with a lid, and remove the lid before it boils. When we put our hand over the steaming vessel, it feels hot. Why? The steam comprises randomly moving particles that strike your hand with high kinetic energy. According to Boltzmann’s idea, this kinetic energy comprises the (water) molecules’ kinetic energy. The higher the temperature, the more entropy these particles have, and hence, the greater is the transfer of energy to your hand, which makes you feel hotter faster. In conclusion, when large amounts of particles hit our skin at a very high speed, they transmit kinetic energy and cause our hands to get burnt due to more molecules circling our skin. This was an average molecular explanation for how temperature works.

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Boltzmann’s Constant Formula

Consider an ideal gas equation: 

   PV = nRT  (Universal gas law)       

Where P is the Pressure in Pascals         

V is the Volume in Metre cubes         

n  = no of moles of the gas         

R = The gas constant         

T = Temperature in Kelvin

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Applications Of Boltzmann’s Constant

  • To express the energy partition of an atom in classical statistical mechanics, the Boltzmann Constant is used. 
  • In semiconductor physics, it is used to express thermal voltage. The statistical definition of entropy is used to express the Boltzmann factor.
  • It calculates the quantity of heat emitted by the dark body. 
  • It converts temperature (K) into intensity (Wm-2), essentially power per unit area.

Facts About Boltzmann’s Constant

  • The Boltzmann Constant, named after Austrian physicist Ludwig Boltzmann and proposed by Max Planck, is a key tool in classical statistical mechanics for expressing the equipartition of energy in an atom. 
  • Furthermore, it also demonstrates the relationship between energy and temperature and is used to study heat. Its dimensions are [M1L2T-2K-1], with a value of 1.3806452 x 10-23J/Kand 8.6173303 x 10-5 eV/K in eV. 
  • This constant plays a major role in semiconductor physics’ statistical definition of entropy and thermal voltage.

Boltzmann Constant Formula FAQs

What is the Boltzmann constant (k)?

The Boltzmann constant (k) is a fundamental physical constant representing the relationship between temperature and the average kinetic energy of particles in a system. It is approximately equal to 1.380649 × 10^-23 joules per kelvin (J/K).

How is the Boltzmann constant related to the gas constant (R)?

The gas constant (R) is the Boltzmann constant (k) multiplied by Avogadro's number (NA), i.e., R = k * NA. This relation connects the microscopic behavior of individual particles to macroscopic gas properties.

What is the significance of the Boltzmann constant in statistical mechanics?

The Boltzmann constant plays a crucial role in statistical mechanics by linking the microscopic behavior of particles to macroscopic thermodynamic quantities, allowing for the study of thermal properties and entropy.

Can the Boltzmann constant be used in different units?

Yes, the Boltzmann constant can be expressed in various units, such as electronvolts per kelvin (eV/K) or calories per mole-kelvin (cal/mol·K), depending on the context and the system of units being used.

How does the Boltzmann constant relate to the Boltzmann distribution?

The Boltzmann constant is a fundamental parameter in the Boltzmann distribution, which describes the probability distribution of particle energies in a system at a given temperature. It quantifies the relationship between energy and temperature in statistical physics.

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