Flick through any physics textbook and it won't be long before you encounter an equation containing a lowercase letter ‘k’. This, as you probably are aware, is the usual notation for Boltzmann's constant, a value which many of us possibly take for granted due to its ubiquity. And yet, you may not know much about the man behind the number.

Ludwig Boltzmann was born in Vienna, Austria, in 1844. His early years were spent being schooled in the suburban family residence, sadly losing his father at the age of 15. He went on to study mathematics and physics at the University of Vienna, an institute to which he would later return as a professor of mathematics, having also spent time teaching at the University of Graz. 

It could certainly be argued that Boltzmann is one of the 'godfathers' of the kinetic theory of gases, as well as statistical mechanics, having spent a good part of his career extending and refining previous work by James Clerk Maxwell: work that led to the Maxwell-Boltzmann equation which describes the distribution of speeds of constituent particles in an ideal gas in terms of molecular mass and temperature.  Whilst an exhaustive examination of that particular equation is beyond the scope of this discussion, it's definitely worth looking at the much simpler formula for the average translational kinetic energy of a molecule of gas in terms of the absolute temperature of said gas:

\({1\over2}m(v_\text{av})^2 = {3\over2}kT \)

The left-hand side of the equation contains molecular mass along with the average squared molecular speed, whilst on the right-hand side we have the absolute temperature alongside ‘k’: the famous Boltzmann constant of proportionality, which appears in so many related formulas.

Boltzmann's constant has a value of 1.3800649×10^-23 J K^-1 and can be expressed as the ratio of the molar gas constant to Avagadro’s number, N_A. However, since the International System of Units (SI) regards k_B (as it is also often notated) as a fundamental constant, not only is the gas constant actually defined in terms of the Boltzmann constant, but the Kelvin scale of temperature is also defined by the value of k.

Ludwig Boltzmann sadly suffered from periods of severe depression throughout his life which were possibly exacerbated by his difficulty in getting certain contemporary philosophers and scientists particularly in the German-speaking world to accept his theories. There was a competing idea prevalent at the time which asserted that atoms and molecules were merely convenient intellectual contrivances, and, as remarkable as it may sound to us, it wasn't until Einstein's quantitative work on Brownian motion that there finally came a universal scientific acceptance that elements and gases are composed of individual particles.

One of Boltzmann's most significant achievements must surely be his unification of classical and statistical thermodynamics particularly with regard to the entropy version of the second law.  His work demonstrated that the statistical irreversibility of entropy (disorder) in macroscopic systems and the impossibility of heat spontaneously flowing from a cooler to a hotter place are both natural consequences of the enormity of possible microscopic states in an ideal gas.

Despite a brief period of relative celebrity, living and lecturing in California, Boltzmann eventually ended his life on a family holiday near Trieste in Italy. This outcome is rendered all the more tragically ironic by the belated regard with which he has come to be held by the scientific world. This is not least in the case of the quantum physicist Max Planck, who formalized the constant k in honour of Boltzmann, and indeed, whose elegant entropy equation \(S = k\log{W}\) (which relates the entropy S of a macroscopic system to the natural logarithm of the number of its inherent microscopic states w) can be seen fittingly gracing Boltzmann's final resting place, in Vienna.

Images

• Ludwig Boltzmann colorized by PhotoColor, CC BY-SA 4.0 <https://creativecommons.org/licenses/by-sa/4.0>, via Wikimedia Commons
• Brownian motion diagram by Andi schmitt, CC BY-SA 4.0 <https://creativecommons.org/licenses/by-sa/4.0>, via Wikimedia Commons
• Ludwig Boltzmann Grave Zentralfriedhof Vienna 2022 photo by Feldkurat Katz, CC BY-SA 4.0 <https://creativecommons.org/licenses/by-sa/4.0>, via Wikimedia Commons