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This question is prompted by (comments at) another one. There, I was surprised to find that despite traditional claims to the contrary, Boltzmann himself did once write his formula $S=k\log W$:

$\hspace{11em}$

That’s in his book (1898, §61, p. 172), with a pointer to (1896, §8, p. 60) where he says the same thing in words, and emphasizes that $RM$ is the same constant for all gases. In fact, if we note that his $R$ is the specific gas constant (equal to $P\ /\ \rho T$ by the gas law, p. 53) and his $M$ the molecular mass $\rho V\ /\ N$, we see on multiplying that his $RM$ is indeed our $PV\ /\ NT=k$. So all seems well.

But now: there is another common story, which we get to hear again as 2018 is the year the kelvin should be redefined by freezing $k$. E.g. White and Fischer (2015) (emphasis mine):

Although Boltzmann published his famous definition of entropy in 1877, the constant of proportionality in Boltzmann’s definition, was not identified as Boltzmann’s constant until 1900 when Planck published his analysis of blackbody radiation (1900a, 1900b), where he identified the constant as $k$ and named it after Boltzmann.

And that, again, seems not true. Of course Planck does introduce $k$ — in (1900b, p. 241), after $h$, as “a second constant of nature” such that an entropy is $k\log\mathfrak R_0$. But he does not name it after Boltzmann there — nor, unless I missed it, in any of the obvious or oft-quoted places. Not in the follow-up articles (1901a, 1901b). Not in the Boltzmann Festschrift (1904: no mention of $k$). Not in his books (1906, 1910, 1913, 1930). Not in his Nobel lecture (1920):

This constant is frequently termed Boltzmann’s constant, although to the best of my knowledge Boltzmann himself never introduced it

Not in his autobiography (1948) where $k$ is “the so-called absolute gas constant”. And most remarkably — even if meant with self-deprecating irony — not in his recollections (1943):

I (...) was not even taken seriously, in some places. But I did not let such doubts deter me from trusting my constant $k$.

The literature also seems to have settled slower than legend has it — see chronology in the CW answer below. (A name Planck did propose in (1900b, p. 245) is “Boltzmann-Drude constant” for $\alpha=3k/2$, but few besides Abraham (1905, pp. 284, 362) seem to have adopted it — e.g. Perrin (1909) calls $\alpha$ “la constante d’énergie moleculaire”.)

So: If not in 1900, when did $k$ get its name? Was there ever a debate (e.g. after the unveiling of Boltzmann’s famous tombstone)? Was there a concerted decision? Were the above-quoted pages of Boltzmann’s book ever invoked? Or was it no-one’s doing, just resolution by attrition? Finally, if Planck did not name the constant after Boltzmann, how did we end up with the tale that he did?

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3 Answers 3

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Community-wiki timeline of relevant literature:

By year written, sources where $k$ is named after...

histogram

Author Dated Published Denoted Called
Boltzmann 1877 (e) S ein constanter Factor 1
Boltzmann● 1895.09 1896 (e) RM S für alle Gase gleiche Constante RM 2
Boltzmann● 1898.08 1898 (e) RM S 3
Drude 1900.02 1900 2α/3 E — (school using α = 3k/2 instead)
Planck 1900.12 1900b (e) k S eine zweite Naturconstante
Planck 1901.01 1901a (e) k S eine universelle Constante
Planck 1901.01 1901b (e) k S die Strahlungsconstante k
Planck 1901.05 1901c k U eine universelle positive Constante
Jeans 1901.06 1902 λ R the gas constant
Planck 1901d k S die Proportionalitätsconstante k
Einstein 1902.06 1902 E eine universelle Constante
Lorentz 1903.04 1903 k U een universeele constante
Planck 1903.07 1904 — (Festschrift)
Einstein 1904.03 1904 E eine absolute Konstante
Jeans● 1904.11 1904 R E an absolute constant
Abraham● 1905.03 1905 k U die universelle Konstante k
Einstein 1905.03 1905 C S eine universelle Konstante 4
Rayleigh 1905.05 1905 k U a constant
Jeans 1905.06 1905a R R
Jeans 1905.07 1905b k S
Jeans 1905.10 1905c R R the constant of the theory of gases
Langevin 1905a r R constante de l’équation des gaz
Langevin 1905b r R constante de l’équation des gaz
Einstein 1906.03 1906a R/N R
von Laue 1906.04 1906 k U Konstante des Verteilungsgesetzes
Planck● 1906.04 1906 k S universelle Integrationskonstante
Winkelmann● 1906 k S eine Konstante
Ehrenfest 1906.07 1906a k die Boltzmannsche Konstante 5
Einstein 1906.08 1906b k S eine (universelle) Konstante
Ehrenfest 1906.10 1906b — (obituary)
Einstein 1906.12 1907 R/N R 6
von Laue 1907.03 1907a k U Konstante des Planckschen Gesetzes
von Laue 1907.10 1907b k S eine universelle Konstante
Reiche 1908.01 1908 k U eine gewisse allgemeine Konstante
Planck 1908.06 1908 k S
von Laue 1908.09 1908 k S eine universelle Konstante
Kowalski 1908.10 1908 k U Konstante des Planck’schen Gesetzes
Lorentz● 1909.01 1909 k U a universal physical constant
Einstein 1909.01 1909a R/N R
Reinganum 1909.04 1909 k E die Schwingungsenergie.. beträgt.. kT
Planck 1909.04 1910a (e) k S die Konstante k
Wien● 1909.05 1909 k S
Jahn 1909.05 1909 k die Plancksche Konstante 7
Lorentz 1909.06 1909 2α/3 R α — een werkelijk universeele constante
Einstein 1909.09 1909b k S eine universelle Konstante
Larmor 1909.11 1909 k S a universal constant
Kowalski 1909.12 1910a K die universelle Boltzmannsche Konstante
Planck 1910.01 1910b k U eine Konstante
Haas 1910.03 1910 k U 2/3 der Boltzmann-Drude’schen Konstanten
Kowalski 1910.05 1910b k la constante universelle de Boltzmann
Lindemann 1910.06 1910 k U
Debye 1910.10 1910a k S universelle Konstante, welche..
Einstein 1910.10 1910 R/N R
Debye 1910.10 1910b k S
Einstein 1910.11 1911 k U Konst. der Planckschen Strahlungsformel
Planck 1911.02 1911a k S
Planck 1911.04 1911b (g) k S la constante k
Sommerfeld 1911.07 2004 k Planck’s.. universelle Constante k
Ehrenfest 1911.07 1911 k
Planck 1911.07 1911c k S der universelle Faktor k 9
Debye 1911.08 1911 k S constante de la loi de Boltzmann
Ornstein 1911.09 1911 (f) R/N R
Sommerfeld 1911.09 1911 k E
Perrin 1911.11 1912 r R la constante universelle R/N
Langevin 1911.11 1912 k le coefficient de Boltzmann
Debye 1911.12 1912a k S universelle Konstante, mit welcher..
K.Onnes-Keesom● 1911.12 1912 kP Planck’sche Konstante
Lorentz 1912 k E une constante universelle
Planck 1912.01 1912 k S
Einstein 1912.01 1912 ϰ — (ambiguous!)
Bohr 1912.02 1912 k R gas-constant referred to a molecule
Kroò 1912.02 1912 k die universelle Plancksche Konstante
Warburg 1912.02 1912 k Plancksche Strahlungskonstante
Debye 1912.03 1912b k constante universelle de Boltzmann
Tetrode 1912.03 1912 k S
Frank 1912.04 1912 k U
Keesom 1912.04 1912 kP
Debye 1912.07 1912c k Boltzmannsche Konstante
Drude-Boll● 1912 k S une constante universelle
Schrödinger 1912.10 1912 k die Boltzmann-Planck-Konstante 8
Schrödinger 1912.10 1912 k die universelle Boltzmannkonstante
Westphal 1912.10 1912 k S die Konstante der Entropiegleichung
Nernst● 1912.11 1913 (e) (school using β = h/k instead)
Planck● 1912.11 1913 (e) k S universelle Integrationskonstante
Lorentz● 1912.11 1916 k E constante qui détermine l’énergie
Born-von Kármán 1912.11 1913 k U
Hasenöhrl 1912.12 1913 k S
Reiche 1912.12 1913 k U
Jellinek● 1913.01 1913 k S eine universelle Konstante
Einstein-Stern 1913.01 1913 k U
Tetrode 1913.01 1913 k U
Sommerfeld 1913.02 1913 k E
Born-Bolza & al 1913.02 1913 k die Plancksche universelle Konstante
Einstein 1913.03 1913 R/N R
Born 1913 k U
Planck 1913.04 1913 k U
Planck 1913.04 1914 k E absolute Gaskonstante
Debye 1913.04 1914 k die Boltzmannsche Konstante
Smoluchowski 1913.04 1914 H/N R
Sommerfeld 1913.04 1914 k die Boltzmann-Plancksche Konstante
Lorentz 1913.04 1914a k E
Stern 1913.05 1913 k R
Ehrenfest 1913.05 1913 k U
Ratnowsky 1913.05 1913 k die Boltzmann-Plancksche Konstante
Millikan 1913.06 1913 (g) k the Boltzmann entropy constant
Bolza 1913.07 1913 k die Plancksche universelle Konstante
Born-Courant 1913.07 1913 k die Boltzmannsche Konstante
Debye 1913.07 1913a k die Boltzmannsche Konstante
Ortvay 1913.07 1913 k die Boltzmannsche Konstante
Debye 1913.08 1913b k die Boltzmannsche Konstante
Kroò 1913.09 1913 k die Plancksche Konstante
Lorentz 1913.09 1914b R/N R
Einstein 1915 R/N R
von Laue 1913.10 1913 ϰ die Boltzmannsche Konstante
Fokker 1913.10 1913 k S de gasconstante voor één molecuul
von Laue 1913.10 1921 k la constante de Boltzmann
Sommerfeld 1913.10 1921a k U constante du rayonnement
Lorentz 1913.10 1921 k la constante k de M. Planck
Einstein 1913.11 1913 k S
Schrödinger 1913.11 1913 k Boltzmann-konstante
Schrödinger 1913.11 1914 k Boltzmannkonstante
Pockels 1913 k die Boltzmannsche Konstante
Stern 1914.02 1914 k R
Bragg 1914.03 1914 k die Boltzmannsche Konstante
Richardson● 1914.05 1914a k S a universal constant
Griffiths 1914.06 1914 k Boltzmann’s constant
Lorentz 1914.09 1917 k Planck’s well known coefficient
Richardson 1914b R R the gas constant for one molecule
Lindemann 1914.12 1915 R/n R
Czukor 1915.02 1915 k die Boltzmannsche Konstante
Tetrode 1915.02 1915 (e) k de constante van Planck
Richardson 1915.04 1915 k Boltzmann’s constant
Richardson 1915 k Boltzmann’s constant
Guillaume 1915 k R la constante d’énergie moléculaire
von Laue 1915.05 1915a k S
von Laue● 1915.07 1915b k S
Born 1915.07 1915 k die Boltzmannsche Konstante
Rubens 1915.07 1915 k die Boltzmannsche Konstante
Planck 1915.11 1915 k E
Mandersloot 1915.11 1916 K Boltzmannsche Konstante
Kroò 1915.11 1915 k U
Planck 1915.11 1915 k
Jeans● 1916.01 1916 R S the universal gas-constant
Stern 1916.02 1916 k R
Gerlach 1916.04 1916 k U die universelle Konstante k
Wolfke 1916.04 1916 k die Boltzmannsche Konstante
Stern 1916.08 1916 k die Boltzmannsche Konstante
Einstein 1916.08 1916 ϰ S
Debye 1916 k die Boltzmann’sche Konstante
Einstein 1916 k die bekannte Boltzmann’sche Konstante
Hertz 1916 k die Boltzmann-Plancksche Konstante

$=$ textbook or reference book. (e|f|g) $=$ English | French | German translation.
Letters S|U|R|E tell the formula each article not naming $k$ uses to first introduce it: $$ S=k\log W\qquad\quad U=\frac{h\nu}{e^{\frac{h\nu}{kT}}-1}\qquad\quad R=Nk\qquad\quad E=\tfrac32kT. $$

Notes (in progress, very much open to debate / correction):

1 First known occurrence of the principle. (Quoted by Planck 1900b.)
2 First known occurrence of the constant. (Ever quoted?)
3 First known occurrence of the formula. (Ever quoted?)
4 Earli(est?) to call the principle Boltzmann’s.
5 Earli(est?) to call the constant Boltzmann’s.
6 Earli(est?) to call the formula Boltzmann’s.
7 Earli(est?) to call the constant Planck’s.
8 Earli(est?) to call the constant Boltzmann-Planck’s.
9 Calls the constant not Boltzmann’s.

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    $\begingroup$ I have found an earlier (earliest?) reference to Boltzmann's constant: Paul Ehrenfest, Zur Planckschen Strahlungstheorie, Physikalische Zeitschrift 7: 528-532 (1906): In paragraph 3 he writes Boltzmannschen Konstante k when referring to the average kinetic energy per molecule $\frac{3}{2}kT$. $\endgroup$ Commented Jan 17, 2018 at 18:11
  • $\begingroup$ Great — you may very well have found the original proponent of “Boltzmann’s constant”. Interestingly, Ehrenfest seems to have rarely repeated this, e.g. it’s not in his Boltzmann obituary 3 months later. Nor did anyone else (that I could find) until much later: Debye (1912), Born, von Laue (1913); experimentalists even slower. I added some — note especially who said what at the Wolfskehl & Solvay conferences (April & October 1913). I wonder when Sommerfeld switched, and whether his Atombau und Spektrallinien was the first textbook to do so. Can anyone find a copy of the first edition (1919)? $\endgroup$ Commented Jan 20, 2018 at 7:19
  • $\begingroup$ It is of interest to note that Mehra and Rechenberg (volume 5, page 98) refer to this paper of Ehrenfest as "the official point of view of Boltzmann's institute in 1906". Note also that it was submitted by Ehrenfest in July, before Boltzmann's death in September, so the name "Boltzmann's constant" was not coined in response to that tragic event. $\endgroup$ Commented Jan 20, 2018 at 12:18
  • $\begingroup$ Speaking of representing Vienna: Hasenöhrl did that at the Solvay meetings of both 1911.11 and 1913.10. One might hope that the campaign described in his booklet left possibly interesting traces in letters by some of the protagonists... but I don’t know where to look for it. (So far only found these by Debye, Ehrenfest.) $\endgroup$ Commented Jan 27, 2018 at 13:39
  • $\begingroup$ Am I right to question the translation of Einstein 1912.01? To me Einführung der Planckschen Konstanten sounds plural, i.e. he’s introducing h = 𝜀/𝜈 and 𝜘 = R/N and calling both Planck’s. (That surprised me, as I was already picturing him as likely behind his friend Kowalski’s switch between 1908 and 1909.) Then again, it seems it could also be singular? $\endgroup$ Commented Jan 29, 2018 at 23:10
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Q: When did the name “Boltzmann constant” prevail, and how?

In Herzfeld's review article from 1920 for the Encyklopädie der Mathematischen Wissenschaften the symbol $k$ is called Boltzmann's constant without further discussion, so this must have been common usage at that time. This implies that there is no connection with Boltzmann's tombstone, which is from 1933 (as described here).

Tetrode in his 1912 article still refers to Planck's constant (and makes a point that his $\ln$ is Planck's $\log$), so the formula $S=k\log W$ was still a novelty.

I would conclude that the transition from Planck's constant to Boltzmann's constant happened in the 1912-1920 time frame.

Concerning the role of Lorentz in the naming of the constant, I note that in a 1917 paper entitled Some remarks on the theory of monatomic gases, Lorentz refers to $k$ as "Planck's well-known coefficient" (online, page 47). I did not find a later reference from Lorentz, however, in the Ph.D. thesis of his student Van de Sande Bakhuyzen, from 1921, the constant $k$ is called "Boltzmann's constant" (online, page 39).$^*$

So this is consistent with the idea that the name Boltzmann constant emerged within a few years before 1920.


One might ask what prompted the switch from Planck constant to Boltzmann constant. This screen shot from Van de Sande Bakhuyzen's Ph.D. thesis suggests the obvious answer: once quantum mechanics and statistical mechanics came together, a single formula would contain both $h$ and $k$ and different names were needed.


$^*$ I also searched for references to Boltzmann's constant in other Ph.D. theses of Lorentz's students (online here):

  • Fokker (1913): calls $k$ the single-molecule gas constant (page 20)
  • Bruins (1918): uses the symbol $s$ instead of $k$ and calls it the gas constant of a molecule (page 92)
  • van Slingelandt (1919): on page 39 writes $\eta =\frac{R}{N}\log W$ and calls it the "principle of Boltzmann", but without introducing a separate notation for $R/N$.
  • van de Sande Bakhuyzen (1921) finally calls $k$ the "constant of Boltzmann" (page 39).

So 1920 seems to be the "phase transition" year, at least for Lorentz's students.

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  • $\begingroup$ Ha, I had no idea the tombstone formula was this late. There is another monument (1912) with remarks by his successor Hasenöhrl (1913), apparently not found online. From Leiden, can you say more on ter Haar’s claim that Lorentz kept calling both constants Planck’s? What I found (e.g. Solvay 1913, footnote p. 283) is not so clear cut, but his collected works may have more... $\endgroup$ Commented Jan 11, 2018 at 15:46
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    $\begingroup$ I found a reference to Planck's coefficient in a 1917 paper by Lorentz, and a reference to Boltzmann's constant in the 1923 Ph.D. thesis of his last student, so the switch must have happened in that time frame. $\endgroup$ Commented Jan 11, 2018 at 17:27
  • $\begingroup$ Ah, that Hasenöhrl booklet of 1913 is actually online (linked by German Wikipedia). He writes on p.15: “When the idea arose to build Boltzmann a monument in our university’s arcade courtyard, a distinguished expert proposed to decorate it with the formula $S=k.\log W$. That would go against our university’s customs and may be superfluous, as the discovery itself is a monumentum aere perennius.” Also reproduced is an early 1911 “call to build” signed by 100+ physicists including Einstein, Lorentz, Nernst, Planck, Rutherford, Sommerfeld, Zeeman .../... $\endgroup$ Commented Jan 17, 2018 at 3:38
  • $\begingroup$ ... at least some of whom must have read, and perhaps been (one way or the other) influenced by, the idea Hasenöhrl is floating here. This and your other references put me on track to more, that I will next try to post in some sort of chronology. Do you know what (if anything) Lorentz’s earlier students were calling $k$? $\endgroup$ Commented Jan 17, 2018 at 3:43
  • $\begingroup$ I listed the Tetrode paper as not naming the constant, because unless I missed it, he’s only crediting Planck for its numerical value. Also, van Slingelandt actually introduces k — pp. 5, 8, 13, as “the constant which determines the average kinetic energy of a molecule at temperature T, which energy equals 3/2 kT.” (So far as I can tell, he doesn’t call it a name, nor mentions that R/N of page 39 is the same thing.) $\endgroup$ Commented Jan 25, 2018 at 1:16
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Community-wiki timeline (continued, 1917–):

After 1920, a vast majority of papers call $k$ Boltzmann’s constant. So (with few exceptions) mention is only made of sources calling it another name, plus some textbooks and reference books.

Author Dated Published Denoted Called
Millikan 1917.01 1917 k S the entropy constant
Schrödinger 1917 R/N R
Gerlach 1917.08 1918 k S die Konstante der Entropie
Flamm 1917.09 1917 k S die Entropiekonstante
Born 1917.10 1917 k die Maxwell-Boltzmannsche Konstante
Sommerfeld 1918 k S die Konstante k des Boltzmannschen Prinzips
Wien 1918 k S die Entropiekonstante
von Laue 1918.07 1918 k E
Schrödinger 1918.07 1919 k R
Bruins 1918.07 1918 s R de gasconstante van een molecuul
von Laue 1918.08 1918 k der Boltzmannsche k
Stern 1918.11 1919 k die Boltzmannsche Konstante
Reiche 1918.12 1919 k die Boltzmannsche Konstante
von Laue 1918.12 1919 k die Boltzmannsche Konstante
Lorentz● 1919a k E 2/3 der translatie-energie per graad
Lorentz● 1919b (e) k constante van Planck
Nernst-Wulf 1919.04 1919 k R Gaskonstante pro Molekül
Lindemann 1919.05 1919a k Boltzmann’s constant
Lindemann 1919.07 1919b k Boltzmann’s constant
Birge 1919.07 1919 k the Boltzmann entropy constant
Sommerfeld● 1919.09 1919 ? ?
Lindemann 1919.09 1920 k Boltzmann’s constant
van Slingelandt 1919.12 1919 k E constante, die kinetische energie bepaalt
Stern 1919.12 1920 k die Boltzmannsche Konstante
Silberstein 1920.02 1920 k U constant in Planck’ formula
Lyon-Wolfram 1920.03 1920 𝔎 die Boltzmannsche Konstante
Stern 1920.04 1920 k die Boltzmannsche Konstante
Ladenburg 1920.05 1920 k die Boltzmannsche Entropiekonstante
Planck 1920.06 1920 (e) k U die erste Strahlungskonstante 9
Sommerfeld● 1920.09 1921b (e) k die Boltzmannsche Konstante
Reiche● 1920.10 1921 (e) k die Boltzmannsche Konstante
Schaefer● 1920.10 1921 k R
Herzfeld● 1920.12 1920 k Boltzmannsche Konstante
Haas● 1921.03 1921 k die Boltzmannsche Konstante
Gerlach 1921 k die Plancksche Entropiekonstante
V.d.S.Bakhuyzen 1921.12 1921 k constante van Boltzmann
Gerlach-Stern 1922.04 1922 k E
Hertz 1922 k die Boltzmann-Plancksche Konstante
Eucken 1922 k die Boltzmannsche Konstante
Henning 1922 k die Boltzmannsche Konstante
Born● 1922.09 1923 k die Boltzmannsche Konstante
Lorentz● 1923 (e) k constante van Planck
Darwin● 1923 k E the atomic gas constant
Coblentz● 1923 k Boltzmann gas constant
Berliner & al● 1924.04 1924 k Boltzmannsche Konstante
Einstein 1924.07 1924 (e) ϰ S
Schrödinger 1924.11 1924 k Boltzmannkonstante
Einstein 1925.01 1925a ϰ
Einstein 1925.01 1925b ϰ
Smekal● 1925.06 1926 k Boltzmann-Plancksche Konstante
Smekal● 1925.06 1926 k Boltzmannsche Konstante
Schrödinger 1925.12 1926 (e) k
Geiger-Scheel● 1926 k Boltzmannsche Konstante
Henning-Jaeger● 1926 k Boltzmannsche Konstante
Bennewitz● 1926 k Boltzmannsche Konstante
Smekal● 1926 k die sog. Boltzmann-Plancksche Konstante
Byk● 1926 k die molekulare Gaskonstante
Jäger● 1926 k Boltzmannsche Konstante
Henning● 1926 k Boltzmannsche Konstante
Henning● 1926 k Boltzmann-Plancksche Konstante
Grüneisen● 1926 k Plancksche Strahlungskonstante
van der Waals● 1926 k
Schrödinger● 1926 k
Simon● 1926 k Boltzmannsche Konstante
Gerlach● 1926 R/N
Pauli● 1926 k die Boltzmannsche Konstante
Ladenburg● 1926 k U die universelle Entropiekonstante 9
Franck-Jordan● 1926 k Boltzmannsche Konstante
Guillaume & al● 1926 k R molecular gas constant
Tolman● 1926.08 1927 k R the Boltzmann constant
Uhlenbeck 1927.07 1927 k constante van Boltzmann
Planck● 1930.03 1930 (e) k S der universelle Faktor k 9
Berliner & al● 1931.11 1932 k Boltzmann-Plancksche Konstante
(tombstone) 1933.07 1933 k S
Planck 1943 k S meine Konstante k 9
van Dantzig 1939.06 1939 k Boltzmann's number
Planck 1945.03 1948 (e) k S die sog. absolute Gaskonstante 9
Meissner 1951 k the Boltzmann-Planck constant 9
Sommerfeld● 1952 (e) k die Boltzmannsche Konstante 9,10

$=$ textbook or reference book. (e) $=$ English translation. Letters S|U|R|E tell the formula each article not naming $k$ uses to first introduce it: $$ S=k\log W\qquad\quad U=\frac{h\nu}{e^{\frac{h\nu}{kT}}-1}\qquad\quad R=Nk\qquad\quad E=\tfrac32kT. $$

Notes:

9 Calls the constant not Boltzmann’s.
10 Calls the formula not Boltzmann’s.

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    $\begingroup$ Split off from the other CW answer, as it was reaching the 30k character limit. $\endgroup$ Commented Jan 29, 2018 at 23:09

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