The **Boltzmann Constant**, generally represented by k_{B} or k, is a proportionality factor which compares the **average relative kinetic energy of particles **in a gas with the **thermodynamic temperature** (T) of the gas.

Index

**Value of ****Boltzmann Constant**

**Boltzmann Constant**

The Boltzmann’s constant is measured in energy per degree of temperature.

The value of Boltzmann’s constant is **1.380649 10 ^{-23} joule per kelvin (K)**, or

**1.380649 10**.

^{-16}erg per kelvinBoltzmann constants’ value can be expressed in various units. The table is given below consists of the value of k along with different units

Value Of k | Units |

1.3806452 × 10^{-23} | m^{2}.Kg.s^{-2}.K^{-1} |

8.6173303 × 10^{-5} | eV.K^{-1} |

1.38064852 × 10^{-16} | erg.K^{-1} |

2.0836612(12)×10^{10} | Hz.K^{-1} |

3.2976230(30)×10^{-24} | cal.K^{-1} |

0.69503476 | cm^{-1}.K^{-1} |

−228.5991678 | dB.WK^{-1}.Hz^{-1} |

4.10 | pN.nm |

0.0083144621 | kJ.mol^{-1}K^{-1} |

1.0 | Atomic unit (u) |

**History**

The Boltzmann’s constant is named after **Ludwig Boltzmann**, an Austrian who discovered it in the 19th century.

Although Boltzmann first linked entropy and probability in 1877, the relation was never expressed with a specific constant until Max Planck first introduced ** k**, and gave a more precise value for it.

Before 1900, calculations involving Boltzmann’s factors were written using a version of the gas constant R and macroscopic energies for macroscopic quantities of the substance, rather than the energies per molecule and the Boltzmann’s constant.

**Boltzmann Constant and Ideal Gas Equation**

The product of pressure p and volume V for an ideal gas is proportional to the product of substance n (in moles) and absolute temperature T:

\(pV = nRT\),

where \(R\) is the molar gas constant (8.31446261815324 J⋅K^{−1}⋅mol^{−1})

The ideal gas law is transformed into an alternative form by the Boltzmann constant as the gas constant per molecule, k = R/NA:

\(pV = NkT\) where \(N\) is the number of molecules of gas.

**Boltzmann Constant in Chemical Kinetics**

**Arrhenius Equation**

The Arrhenius equation is a formula for the temperature dependence of reaction rates and given by,

\(k = Ae^{\frac{-E_a}{k_BT}}\),

Where,

\(E_a\) is the activation energy for the reaction

\(k\) is the rate constant

\(T\) is the absolute temperature

\(A\) is the pre-exponential factor, a constant for each chemical reaction

\(k_B\) is the Boltzmann’s constant.

**Eyring Equation**

The Eyring equation is a chemical kinetics equation that describes the rate of a chemical reaction as a function of temperature.

\(k = \frac{κ k_B T}{h}e^{\frac{\Delta G^\ddagger{}}{RT}}\)

Where,

\(T\) is the absolute temperature

\(\Delta G^\ddagger{}\) is the Gibbs energy of activation*κ* is the transmission coefficient

\(k_B\) is the Boltzmann’s constant.

\(h\) is the planck’s constant

\(R\) is the gas constant

\(k\) is the rate constant

**Boltzmann Constant in Statistical Mechanics**

**Degree of Freedom**

For a given thermodynamic system, at absolute temperature *T*, the average thermal energy carried by each microscopic degree of freedom in the system is **1/2 kT ** where k is the Boltzmann’s constant

**Kinetic Theory of Gases**

Kinetic theory of gases gives the average pressure *p* for an ideal gas as \(p = \frac{1}{3} \frac{N}{V}m \bar{v^2}\).

From ideal gas equation we know that \(pV = NkT\).

So we get,

\(\frac{1}{2} m\bar{v^2} = \frac{3}{2}kT\).

**Partition Function**

Generally, systems in equilibrium at temperature *T* have probability *P _{i}* of occupying a state

*i*with energy

*E*weighted by the corresponding Boltzmann factor:

\(P_i \propto \frac{exp\left(-\frac{E}{kT}\right)}{Z}\), where Z is partial function.

**Statistical Entropy**

In statistical mechanics, the entropy *S* of an isolated system at thermodynamic equilibrium is defined as the natural logarithm of *W*.

\(S = k \ln W\).

**FAQs**

**What is the significance of Boltzmann constant?**Temperature and energy are related by the Boltzmann constant (kB).

**What is the dimension of Boltzmann constant?**The Boltzmann constant is dimensionally represented as **[M ^{1} L^{2} T^{-2} K^{-1}]**.

**Where is Boltzmann constant used?**The Boltzmann Constant is a term that describes how an atom’s energy is distributed. It’s a symbol for the Boltzmann factor. It has a major impact on the statistical definition of entropy. It is used to express thermal voltage in semiconductor physics.