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Boltzmann Constant

18th May 2019 @ 6 min read

Physical Chemistry

The Boltzmann constant is a very important constant in physics and chemistry. The constant relates the average kinetic energy of molecules of a gas with thermodynamic temperature. The Boltzmann constant is denoted as kB or k. The dimension of the Boltzmann constant is energy per thermodynamic temperature. The SI unit is J K1, which is the same as of entropy. The value of the Boltzmann constant is 1.380 649 × 1023 J K1.

Values of Boltzmann Constant

The value of Boltzmann's constant in different units is presented in the table below.

The value of the Boltzmann constant in different units
ValueUnit
1.380 649 × 10−23J K−1
1.380 649 × 10−16erg K−1
1.018 314(44) × 10−23ft lbf K−1
3.299 830 3(06) × 10−24cal K1
1.833 239 0(59) × 10−24cal R1
8.617 330 3(50) × 10−5eV K−1
2.083 661 2(12) × 1010Hz K−1
3.166 811 4(29) × 10−6EH K−1, H is the hartree.

History of Boltzmann constant

The Boltzmann constant is named after Austrian physicist Ludwig Boltzmann. He made significant contribution to the field of statistical mechanics. In 1877 Boltzmann formulated the relation between entropy and probability; this relation is connected by the Boltzmann constant. That time, the constant was not christened after Boltzmann. It was Max Planck who first entitled the constant in his work on black body radiation in the 1900s.

Today, the Boltzmann constant is found in various expressions. Some of them are discussed below.

Boltzmann constant and Gas constant

Both constants are related by Avogadro's constant. The gas constant (R) divided by Avogadro's constant (NA) gives the Boltzmann constant.

Boltzmann constant, kB=R∕NA

Boltzmann constant and Ideal gas equation

The relation between the ideal gas equation and the Boltzmann constant is as follows:

Ideal gas equation, PV=NkBT

where:
P is the pressure of an ideal gas,
V is the volume occupied by an ideal gas,
N is the molecules in an ideal gas; it is defined as N = n × NA,
and T is the temperature of an ideal gas.

Boltzmann constant in Chemical kinetics

Relationship with Arrhenius equation

Arrhenius equation is a very important equation in chemical kinetics. It associates the rate constant of a chemical reaction with temperature. The rate constant is a function of temperature and activation energy.

Arrhenius equation

where:
k is the rate constant of a reaction,
A is the Arrhenius constant,
T is the temperature in the kelvin,
and Ea is the activation energy.

According to the collision theory, A is defined as:

Arrhenius constant according to collision theory

where:
Z is the collision frequency in m−3 s−1,
nA and nB are the collision frequency of gas A and gas B respectively,
ρ is the steric factor,
μAB is the reduced mass, Reduced mass,μAB=mAmB∕(mA+mB),
and σAB is the reaction cross section area in m2.

Relationship with Eyring equation

The Eyring equation is another important equation in chemical kinetics. The equation is named Mexican-born American chemist Henry Eyring. The Eyring equation is similar to the Arrhenius equation. It correlates the rate constant of a reaction with Gibb's energy of activation ∆G and temperature in the kelvin.

Eyring equation

where:
k is the rate constant,
κ is the transmission coefficient,
h is the Planck constant.

Boltzmann constant in Statistical mechanics

Relationship with degree of freedom

Let N be the number of degree of freedom. The average internal energy of a system with N degree of freedom is given as:

Average internal energy, E=N/2kBT

A monatomic gas like helium, neon, xenon has three degrees of freedom (3 spatial direction). For monatomic gas, the average internal energy is

Average internal energy for monatomic molecules, E=3∕2kBT

Relationship with Kinetic theory of gases

From the kinetic theory of gases, the pressure of a gas (P) is

Pressure by the kinetic theory of gases, P=1∕3 N∕Vmv2

where v2 is a velocity vector in the three-dimensional space.

Substituting PV = NkBT in the above equation,

Therefore, the kinetic energy associated with monatomic ideal gases is expressed in the equation below.

Kinetic energy, KE=1∕2mv2=3∕2kBT

Relationship with Partition function

Pi is the probability of a system occupying state i at equilibrium temperature T is inversely proportional to partition function Z.

Partition function

Relationship with statistical entropy

Statistical entropy (S) at thermodynamic equilibrium is defined as the Boltzmann constant times the natural logarithm of the distinct microscopic states (W) at fixed total energy.

Statistical entropy, S=kBlnW

Boltzmann published this equation in 1877.

Boltzmann constant in Shockley diode equation

The Boltzmann constant is also used in calculating thermal voltage in the Shockley diode equation. As per the Shockley diode equation, the diode current (I) is given as:

Current as per Shockley diode equation

where:
Is is the reverse bias saturation current,
VD is the voltage across diode,
n is identity faction,
VT is the thermal voltage.

The thermal voltage at temperature (T) is calculated from the below expression.

Thermal voltage at temperature (T)

Here, q is the charge on an electron, 1.602 176 620 08(98) × 10−19 C.

At 27 ℃, its value is 25.856 mV.

Boltzmann constant with Black body radiation

In Planck equation

The Boltzmann constant is useful in estimating the spectral radiance Bν of a body at temperature (T).

spectral radiance by Planck equation

where:
h is the Planck constant,
ν is the frequency of radiation,
c is the speed of light in a given medium.

With Stefan-Boltzmann constant

The Stefan-Boltzmann constant (σ) is as follows:

Stefan-Boltzmann constant, 5.670×10−8 J∕s∕m2∕K4

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Imran
14th Jun 2020
Amazing illustrations, great to see how you explain things, so easy to understand the concept. I personally think this article deserves 100 plus claps.


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