The Boltzmann constant (k or k<sub>B</sub>) is a fundamental physical constant in thermodynamics and statistical mechanics. It relates the average relative kinetic energy of particles in a gas with the absolute temperature of the gas.
The Boltzmann constant is named after the Austrian physicist Ludwig Boltzmann, who was a pioneer of statistical mechanics. The equation S = k log W (which uses this constant) is famously engraved on his tombstone in Vienna.
The easiest way to derive the dimensional formula is from the famous kinetic energy formula for a monoatomic gas molecule:
E = (3/2) k T
Where:
Rearranging to solve for k: k = E / T (Ignoring the 3/2 since numbers are dimensionless).
To find the dimensions of k, we divide the dimensions of Energy by the dimensions of Temperature.
Dimensional formula of Energy (E): Energy = Work = Force × Distance = [M¹ L¹ T⁻²] × [L¹] = [M¹ L² T⁻²]
Dimensional formula of Temperature (T): Temperature is a fundamental quantity, represented by [K] (Kelvin) or sometimes [θ].
Putting it together: [k] = [E] / [T] [k] = [M¹ L² T⁻²] / [K¹] [k] = [M¹ L² T⁻² K⁻¹]
The dimensional formula for the Boltzmann constant is [M¹ L² T⁻² K⁻¹].
Based on the formula k = E / T, the SI unit is Energy divided by Temperature.
The accepted numerical value of the Boltzmann constant is: k = 1.380649 × 10⁻²³ J/K
The Boltzmann constant is simply the Universal Gas Constant (R) divided by Avogadro's Number (N<sub>A</sub>). k = R / N<sub>A</sub> While 'R' applies to a whole 'mole' of gas, 'k' applies to just one single molecule of gas.
Yes. The macroscopic definition of entropy is change in heat divided by temperature (ΔQ / T). Since heat is energy, the dimensions of entropy are also [M¹ L² T⁻² K⁻¹].
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