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Boltzmann constant: the relationship between energy and temperature

Introduction to the Boltzmann constant

The Boltzmann constant written ordinarily as k or denoted with the letter kB was named after an Austrian physicist as well as a philosopher whose works were towards the development of mechanics in the field of statistics that explains and also predicts how the physical properties of matter can be determined using the properties of atoms

Therefore, Boltzmann constant is a physical constant bringing energy together at the individual particle level together with temperature. This Boltzmann constant is written in formula as followings:

  • k= R/NA, where R is the gas constant while NA represents the Avogadro constant

The Boltzmann constant has dimension energy that is divided by temperature (ToC) that is similar to entropy but with the value of 1.38064852(79) ×10−23 J/K in SI units. The letter k of Boltzmann constant acts as a bridge between macro- and microscopic physics. Macroscopically, the ideal gas law states that the products of pressure (p), as well as volume (V) is proportional to the product of the amount of a substance that can be measured in moles at complete temperature (T) for an ideal gas i.e.:

  • pV = nRT, where the letter R represents the gas constant (8.3144621(75) J⋅K−1⋅mol−1)
  • Introducing the Boltzmann constant changes this ideal gas law into an alternative form which is: pV = NkT, where letter N stands for the amount of gas molecules. For n = 1 mol, N is equal to the amount of particles in a mole

In Boltzmann constant, the ideal gas equation is also obeyed by molecular gases. This is because the molecules have extra internal degrees of freedom, as well as the 3o of freedom for movement of the molecule as a whole. In Boltzmann constant, diatomic gases have a total of 6o of freedom per molecule that are associated with atomic motion. At lower temperatures ToC, not all these degrees of freedom may fully engage in the gas heat capacity, due to quantum mechanical limits on the excited states that are available at the relevant thermal energy per molecule

Given a thermodynamic system at a temperature T that is absolute, the average thermal energy carried by each microscopic degree of freedom in the system is on the order of magnitude of 12kT, i.e., about 2.07×10−21 J, or 0.013 eV, at room temperature

In normal statistical mechanics, this average is speculated to hold exactly for homogeneous ideal gases. Monatomic ideal gases possess 30 of freedom per atom, that also equate to the 3-spatial directions, which gives a thermal energy of 32 kT per atom in Boltzmann constant. Thermal energy can be used to estimate the root-mean-square speed of atoms, which turns out to be proportionally inverse to the square root of the atomic mass. The root mean square speeds found at room temperature accurately reflect this, ranging between 1370 meters per second for helium, down to 240 meters per second for xenon. Kinetic theory for an ideal gas as gives the average pressure p as follows:

  • p = 13NV mv2
  • Generally, systems in equilibrium at a temperature T have the probability Pi of occupying a state with an energy that is weighted by a corresponding Boltzmann constant as follows:
  • Pi α exp (-E KT )Z, where letter Z represents the partition function while KT which is the energy like quantity that is of more importance in the equation

Likewise, in statistical mechanics, the entropy represented by letter S of any separate system at thermodynamic equilibrium can be defined as the natural logarithm of W, i.e. the amount of distinct microscopic states made available to the individual system given the macroscopic constraints. Therefore, entropy can be explained as such below:

  • S = k ln W

The proportionality constant k serves to make the statistical mechanical entropy equal to the thermodynamic entropy of Clausius: ∆S= dQ T. One could choose a rescaled dimensionless entropy instead of the above in microscopic terms, i.e.: ∆S′ =- ln W, ∆S′ = dQ KT. This is more of a natural form and it is also a rescaled entropy that corresponds exactly to Shannon’s subsequent information entropy. The characteristic energy kT is thus that energy needed to increase the rescaled entropy by one nat

Role and history of Boltzmann constant in semiconductor physics

In semiconductors, the Shockley diode equation which is the relationship that exists between the flow of an electric current and the electrostatic potential across the junction p–n is dependent on a characteristic voltage known as the thermal voltage, represented the symbol VT. This thermal voltage also depends on absolute temperature T as follows:

  • VT = kTq

Where letter q represents magnitude of the electrical charge on an electron with the value of 1.602176565(35)×10−19 C while the constant k is the Boltzmann constant with the value of 1.38064852(79)×10−23 J/K. In electron volts, the Boltzmann constant is 8.6173324(78)×10−5 eV/K which makes it very easy to calculate at room temperature approximately ≈ 300 K. Therefore, the value of the thermal voltage will approximately be equal to 25.85 millivolts ≈ 26 mV. The thermal voltage is also of importance in plasmas and electrolyte solutions for Boltzmann constant and in both cases it gives the measurement of the amount of spatial distribution of electrons or ions is affected by the boundary held at a fixed voltage

Even though Boltzmann was the first physician to link entropy and probability in the year 1877, but its relation was never shown with a particular Boltzmann constant until another scientist by the name Max Planck first introduced k, then gave an exact value as 1.346×10−23 J/K in one of his derivations in the years between 1900 till 1901 called the law of blackbody radiation. Before the year 1900, equations involving Boltzmann factors were not written in form of energies of each molecule but Boltzmann constant rather use a form of gas constant R, and macroscopic energies for macroscopic quantities of the substance was adopted

In 1920, Max Planck wrote in his Nobel prize while giving his lecture: “The constant is sometimes called Boltzmann constant, but Boltzmann himself did not present it as constant that can be explained by the fact that Boltzmann, as shown from his infrequent comments never gave thought about the likelihood of trying to carry out an exact measurement of the constant”

There was disagreement in this period as to whether atoms as well as molecules were real or whether they were just tools used for solving different problems. There was no agreement whatsoever that chemical molecules, as measured by atomic weights were the same with the physical molecules, as measured by kinetic theory. Planck’s lecture further said: “Nothing can best describe the positive as well as hectic pace of progress that has been made over the past 20 years other than the fact that since that time, several number of ways have been discovered and can be used to measure the mass of any molecule with similar accuracy

Finally, in 2013 the UK National Physical Laboratory tried to find the speed of sound of a monatomic gas with use of a microwave and acoustic resonance measurements in a triaxial ellipsoid chamber to find a more correct as well as an accurate value for the constant as a part of the review of the international unit system. The new value is now: 1.38065156(98) ×10−23 J⋅K−1 and this should now be accepted by the system of international units following the review of it

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