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Boltzmann's Entropy Formula
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0:00
Boltzmann's entropy formula is probably one of the most difficult equations in all of physics not because the equation itself is that confusing it isn't it's just two variables one constant and a trig function but because it relates two things entropy and the probability of being in different energy states that are both really difficult to understand so as I usually do I looked into the history of this equation who made it and why was it
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made and I found it was created by a scientist named Max Planck in 1900 and written almost identically to the modern form in 1901 but wait why is it called Boltzmann equation and why is the constant it called Boltzmann's constant and why is it even on Boltzmann's gravestone if Planck created it? Good questions ready for the answer let's go!
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I would like to start with the origin of the idea of entropy in 1854 a German scientist named Rudolf Clausius noted that absorbing less heat at lower temperature was equivalent to absorbing more heat at higher temperature He therefore he called the heat over the temperature the equivalence value and he would later call the equivalence value the entropy. By 1862 Clausius found that any decrease of equivalence value on one object would require an increase of
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equivalence value or entropy on the other objects so that the total equivalence value for any process for closed system had to be greater than or equal to zero. By 1865 Clausius renamed the equivalence value the entropy and gave it the letter s and defined the second law of thermodynamics to be the entropy of the universe can only increase which is a modern definition. Now I want to take a moment to talk a little bit about the history
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of probability in thermodynamics and for that I'm gonna start with, ready for it, a german man named Rudolf Clausius. In 1857 three years after Clausius's first paper introducing the equivalence value Clausius wrote a paper on what the temperature meant about the motion of molecules in this paper Clausius became the first person to introduce rotational and vibrational motion of molecules as well as linear motion even with these limitations Clausius found that molecules move at
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incredibly fast speeds. For example hydrogen gas at zero degrees celsius should move at slightly more than five times the speed of sound! After reading Clausius's paper another scientist published an objection, if molecules are moving that fast how come cigar smoke doesn't just fill the room faster than the speed of sound? Clausius thought it was a very interesting objection but instead of invalidating his theories Clausius
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thought it would all work out if the gas molecules are moving very fast but not very far. In other words in a cigar smoke or in any other gas there are a ridiculous number of molecules moving in all directions but they don't get very far before bouncing off another molecule and changing direction so that even though the individual gas molecules are moving very quickly the gas itself diffuses slowly. Clausius then came up with a term that he called the mean
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length of path which was "how far on average can the molecule move before it's center of gravity comes into the sphere of action of another molecule" Conveniently an English scientist named Frederick Guthrie was a fan of Clausius and was also bilingual and Guthrie translated Clausius's work into English and published it in February of 1859. Three months later a 27 year old scottish scientist named James Clerk Maxwell wrote a friend
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that Clausius's paper had inspired him to determine the mean path by "comparing it with phenomenon that seemed to depend on this mean path" Between 1860 and 1866, Maxwell published a series of articles on what he called the dynamical theory of gases although he admitted it was Professor Clausius to whom we owe the most extensive developments of the dynamical theory of gases. During the same time the busy Maxwell also published a couple of
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papers some electricity and magnetism the results of which are known as Maxwell's equations. Funny story, in 1861 Maxwell was once stuck in a crowd of people and Michael Faraday, referring to his work in statistics, yelled out, "Ho, Maxwell cannot you get out? If any man can find his way through a crowd it should be you!" Anyway Clausius had some minor complaints about Maxwell's theory but there was another German scientist who was entranced and his name is Ludwig
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Boltzmann, yes he is involved, Boltzmann was thirteen years younger than Maxwell and went to college in 1863 while still an undergraduate one of his professors immediately appreciated Boltzmann's brilliance and according to Boltzmann he decided, "to hand me a copy of Maxwell's papers on gases however at the time I didn't understand a word of English so he also gave me an English grammar book"! Boltzmann managed to translate Maxwell and he soon was
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publishing his own papers on the theory of gases and three years later he earned his PhD on the dynamical theory as gases and by the time he was 25 he was made a full professor. Aside from his eerie intelligence Boltzmann was also known for his great teaching prowess, Lisa Meitner described him as the best teacher she ever had, his humor like when he said he was tolerably good at mathematics except "when I'm counting beer glasses" and his oddly
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naive view of reality like when he consulted the professor of zoology to determine how to milk a cow! Maxwell and Boltzmann work on statistics in thermodynamics led to the Maxwell Boltzmann distribution or equations for the probability of different speeds of gases still used today. Maxwell started off being confused about entropy but once his confusion was cleared up by an American named Josiah Gibbs he decided to use his methods of studying statistics in molecules on entropy
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this led Maxwell to decide that the truth of the second law is therefore a "statistical not a mathematical truth for it depends on the fact that the bodies we deal with consists of millions of molecules and we can never get hold of single molecules" or as Maxwell amusingly put it to a friend, "the second law of thermodynamics has the same degree of truth as the statement that if you throw a cup of water into the sea you cannot get the same cup of water out again"
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Not surprisingly Clausius was not pleased with Maxwell for downgrading his law like that however he had bigger issues first in 1870 Clausius was injured in the franco-prussian war when he organized an ambulance corps second in 1875, Clausius's wife died in childbirth and Clausius took time off his research to focus on raising his six children and teaching. Clausius always taught even on his deathbed! Young Boltzmann however was
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motivated once again by Maxwell and now dived into the relationship between statistics and entropy. In 1872 Boltzmann wrote that the molecules of a body are indeed so numerous and their motion is so rapid that we can perceive nothing more than the average values hence the problems of mechanical theory of heat are also problems of probability theory. By 1877 Boltzmann started to work on the relationship between probability and entropy. He started with
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the idea that "in the game of lotto any individual set of five numbers is just as improbably as a set one two three four five is only because there are many more uniform distributions than non-uniform ones that the distribution of states will become uniform in the course of time". Boltzmann even declared that one could even calculate from the relative numbers of the different state distributions their probabilities so in October of 1877 Boltzmann set out to find equations to figure out how many
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distributions there could be. Boltzmann ended up writing over 50 pages of dense equation rich material (still not translated out of German) where he broke up the energy of molecules into different sections and predicted the probability of ending up with different scenarios. He wrote "applying this to the second law we can identify the quantity which is usually called the entropy with the probability of the condition in question"
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However Boltzmann didn't make an equation that related the entropy to this newly defined probability. Fascinatingly both Maxwell and Boltzmann could have had the help of their scientifically minded wives. Katherine Maxwell had no scientific or mathematical training but it was recorded that she helped her husband with his experiments. In fact, in 1877 a friend wrote Maxwell asking for data on viscosity and Maxwell wrote back
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"my better half who did all the real work of kinetic theory is it present engaged in other researches when she is done I will let you know the answer to your inquiry" Boltzmann's wife Henriette Boltzmann actually tried to study physics and mathematics at school and she even basically sued to be able to audit classes on Boltzmann suggestion! In his marriage proposal Boltzmann referred to her as a mathematician and said, "it seems to me
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that permanent love cannot exist if a wife has no understanding and enthusiasm for her husband's efforts and it's just his maid and not the companion who struggles alongside him" However she was probably pretty busy raising their five children and dealing with his depressed moods. The Maxwell's and the Boltzmann's worked together for several years but then James Clerk Maxwell fell ill in the spring of 1877 and he died of stomach cancer when he was only 48 years old in
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November of 1879. Henriette and Ludwig Boltzmann continued to publish work on their theories of statistics and heat and probability and entropy however they felt increasingly outnumbered in Germany as the existence of molecules was kind of an avant-garde idea that was deeply unpopular among the scientific establishment. Now we finally get to Max Planck Planck was 14 years younger than Boltzmann and in 1877, the same year Maxwell fell ill and Boltzmann wrote his
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very long paper and probability, Planck went to graduate school and found to his disappointment that the great researchers there were pretty awful teachers! He decided was better off just reading original papers years later he recalled, "one day I happened to come across the treatises of Rudolf Clausius whose lucid style and enlightening clarity of reasoning made an enormous impression on me and I became deeply absorbed in his articles
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with an ever increasing enthusiasm I appreciated especially his exact formulation of the two laws of thermodynamics" By 1879 Planck got his PhD in the second law of thermodynamics "however the effect of my dissertation on the physicist of those days was nil none of my professors at the University had any understanding for its contents I found no interest let alone approval even among the very physicists who were closely connected with the topic" Even
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Clausius was indifferent and Planck went all the way over to his house but with no reply. Despite that Planck managed to become a professor in Berlin in 1889. Planck then got into a debate with people called energetics who believed in energy but not entropy and also did not believe in molecules or believe in the absolute zero scale for temperature. In this debate Planck found a strange defender Ludwig Boltzmann. In 1896 Boltzmann had a public debate with an
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energetic on the issue which seemed to convince at least the younger generation of the validity of the second law but Planck also felt the Boltzmann was attacking him. See Planck never liked Boltzmann's theory of statistics saying "I regarded the principle of the increase of entropy as no less immutably valid than the principle of conservation of energy itself whereas Boltzmann treated the former merely as a law of probabilities and for that reason
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was not only indifferent but to a certain extent even hostile to the atomic theory which was the foundation of his entire research" Meanwhile Planck had moved on to a new subject to eventually force him to accept Boltzmann theory about probability. See in 1894 an acquaintance of Planks named Wilhelm Wien had made an equation to describe the distribution of radiation from a perfectly absorptive object called the blackbody that was based on experimental
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data but had very little theoretical background. Planck thought he had a unique approach to the theory because he was worried about entropy and no one else was years later he recalled "it was an odd jest of fate the lack of interest of my colleagues in [entropy] now turned out to be an outright boon." By 1899 Planck published his results and for a short while it was called the the Planck-Wien law. However there was a problem with the law it didn't work at low
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energies and in October of 1900 Planck told by an experimentalist that he needed to tweak his equation. Without any time to prove anything Planck made up a new equation that would work both at low frequencies and also would look like the Wien law at higher frequencies. The experimentalists were delighted but Planck was distraught a theoretician is not supposed to just guess an equation from the experimental data they're supposed to drive it from fundamental ideas. He recalled, "for six
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years I had struggled with the blackbody theory I knew the problem was fundamental and I knew the answer I had to find a theoretical explanation at any cost except the invulnerability of the two laws of thermodynamics" so in a self-described "act of desperation" he turned to Boltzmann's method of statistics with entropy after a few weeks of intense work Planck decided that, "since the entropy s is an additive magnitude but the probability W
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is a multiplicative one I simply postulated that S equals k log W where k is a universal constant. A quick note about the log function it is the inverse of the exponent function so if they multiply the input by 10 the value of the log base 10 increases by 1 that's what Planck meant about additive and multiplicative. Here is the famous equation but what does it mean? First it means that entropy has an absolute value that one can calculate from the
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properties of the molecules in the material. Although not as easy to measure as temperature, entropy has a definite value for any object with any arrangement of molecules, the more complicated the arrangements are the more entropy the object has. Secondly, the constant k has a value above and beyond just a relationship between probability and entropy. K is used in many different equations including the heat capacity and the relationship between kinetic
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energy of particles and the temperature. With an accurate measure of K one could predict many microscopic things that weren't known accurately at the time like the mass of a hydrogen atom and even the charge of an electron. In fact, as of 2019, k is only one of seven basic constants like the speed of light that all of our units of everything is currently based on! But then Planck had a problem if the energy was continuous then it could be split up into
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infinitely many arrangements so therefore the probability would be infinite and entropy would also be infinite as Planck put it "if the energy is considered to be continuously divisible quantity this distribution is possible in infinitely many ways" For that reason Planck constrained the energy to be created in little energy packets with energy equal to a constant h times the frequency. This is the origin of quantum mechanics so you could argue
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the Boltzmann's entropy equation started the quantum revolution! In addition h is another one of the seven fundamental constants the we base our units on. Then, with the voluminous data he had on blackbody radiation Planck managed to get values for his two constants h and k startlingly h is around 1% of its current value and k was about to 1.5% percent off of its current value. Even more impressively with his value for k, Planck managed to
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get a value for the charge an electron within 2% of its current value when the other two accepted values at the time were 73% too low and 35% too high! You might ask how this equation and the constant got named after Boltzmann and how they were put on Boltzmann's grave? Well, Planck had a lot to do with this. When Planck first introduced the entropy equation he was using a new method for him and therefore said he was "introducing probability
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considerations the importance of which for the second law of thermodynamics was first of all discovered by Mr. Boltzmann" Which implied to people who hadn't poured through Boltzmann's 50 page long paper about probability that Boltzmann had originated the equation. Then, in September of 1906 Boltzmann's depression got the better of him and he committed suicide. At the same time Planck was giving a series of lectures on heat radiation and learned through some of
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the scientific memorials of Boltzmann that 4 years before Planck had written his equation Boltzmann had written that a function that was related to the entropy was also related to the log of probability. (I know it's a little confusing) Anyway, for that reason during these lectures Planck said that, "The logarithmic connection between the entropy and probability was first stated by Boltzmann [in 1896]" and then he added that it didn't really count because
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Boltzmann never had a constant and as Boltzmann never directly linked entropy to the log of probability Boltzmann had created an unspecified value for the entropy. Planck added that the absolute value for the entropy is "a step of fundamental importance as it leads necessarily to the hypothesis of quanta" A few years later Planck published these talks as a book which had the opposite of the desired effect as that one quote is still to
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this very day used to justify the naming of the equation that Planck created to Boltzmann! Also, during the 1910s it became very popular to name constants after the scientists who discovered them. For example, in 1913 Niels Bohr wrote a paper about the arrangement of atoms where he named the constant h in e =hf Planck's constant. Bohr's paper was a hit, years later Einstein recalled that Bohr's paper "Appeared to me like a miracle and
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appears to me as a miracle even today this is the highest form of musicality in the sphere of thought". Six months after Bohr's paper English scientists Ezzor and Ernest Griffiths wrote a paper on the specific heat of metals of very low temperature and as far as I can tell they became the first people to call k Boltzmann's constant. This paper was not revolutionary like Bohr's but it did include a lot of very interesting data and it also included an overview of
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quantum mechanical view of solids so it was widely read pretty soon most scientists were talking about quantum mechanics and they were using Planck's constant and Boltzmann's constant to do so. By 1920, when Max Planck's won his Nobel Prize he complained that k was "often referred to as Boltzmann constant" which he found to be "a peculiar state of affairs" as "to my knowledge Boltzmann himself never introduced it" By the 1930s it became trendy to name the equations after their inventors and soon
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S= k log W was referred to as Boltzmann entropy equation because k was known as Boltzmann's constant. At around the same time the alderman of Vienna realized that Boltzmann's grave was kind of decrepit. The aldermen decided that "the city of Vienna and its people are proud of the man of genius who lived here and they have endeavored to find a burial place worthy of him" for that reason they moved Boltzmann's body to place of honor near Brahms and Beethoven
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and they created a marble bust of Boltzmann and a local scientist named Hans Thirring had them add what he called Boltzmann's greatest scientific achievement to his grave ie S = k log W. So that's how Boltzmann's equation ended up on Boltzmann's grave even though Boltzmann's entropy equation should probably be called Planck's entropy equation and Boltzmann's constant should definitely be called Planck's constant. Except we already have a Planck's constant so that's super confusing so I
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guess we aren't going to fix it! So now I'm at a bit of an impasse of what I should do for my next video so I'm gonna do something I've never done before I'm gonna let you guys vote on what you would like to see for the next video. Should I make a video about the tragic and sometimes heroic life of Max Planck? or should I make a video about Niels Bohr's musical quantum model of nature that we still study even though we know it's wrong? Or should I go back in time
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and talk about amazing woman I missed, a scientist in the 1700s who brought electricity to Italy and whose marriage was considered a travesty against science? Or should I skip all the theory and jump into the wild story of the teenager who invented television? Comment on the bottom to let me know or if you want more of a vote you can become one of my patrons (thank you patrons) and then for just $1 of video or $5 a video you get thanked you get the videos early and your vote
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gets more of an influence. There's a link down below remember the options are max Planck's life, Niels Bohr's model, Laura Bassi's influence or Farnsworth's TV. I do want to hear from you. Also, if you have any questions or went too fast or too slow or move my arms too much please tell me in the comments. I can't stop moving my arms, that's part of it. Thank you so much for watching Remember, subscribe, hit the silly bell thing I don't make that many videos so
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it won't bother you, share it on social media give me thumbs up all that jazz! okay have a nice day :) gonna get this I'm gonna get it I believe in myself gotcha
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