Talk:Boltzmann constant

Order of magnitude

The statement:-

Given a thermodynamic system at an absolute temperature T, the thermal energy carried by each microscopic "degree of freedom" in the system is on the order of magnitude of kT/2 (i. e., about 2.07×10−21 J, or 0.013 eV, at room temperature).

is unsupported, neither is it self evident. The part about the order of magnitude is far to generalised, and probably incorrect, to be allowed to remain as it is. --Damorbel (talk) 17:00, 9 December 2012 (UTC)

Well, here is the calculation for room temperature at about 25°C:

${\frac {1}{2}}kT\approx {\frac {1}{2}}\times (1.38\times 10^{-23})\times (273+25)=2.0355\times 10^{-21}$

What's your point about it being far too "generalized"? Maschen (talk) 14:07, 10 December 2012 (UTC)

Maschen, if your calculation of the energy in a particle (1/2kT - above ) gave a result within an order of magnitude it would read like this:-

${\frac {1}{2}}kT\approx {\frac {1}{2}}\times (1.38\times 10^{-23})\times (273+25)=2.0355\times 10$ ^-21(±1)

i.e. there would be a factor of uncertainty of 10 in the value of 1/2kT, and yet you quote it to four decimal places. --Damorbel (talk) 11:44, 11 December 2012 (UTC)

What I did was approximate the answer using the numbers shown - the answer to 4 dp is the exact answer using those numbers, but still an approximation, not far off the quote you give, which does not say "within an order of magnitude", it says "on (of?) the order of magnitude", hence your "±1" is irrelevant. Maschen (talk) 13:54, 11 December 2012 (UTC)

Maschen, Is this what you mean by order of magnitude, where it has:-

"We say two numbers have the same order of magnitude of a number if the big one divided by the little one is less than 10. For example, 23 and 82 have the same order of magnitude, but 23 and 820 do not."? --Damorbel (talk) 15:58, 11 December 2012 (UTC)

Yes, although it's simpler to see if two numbers are of the same magnitude if written in scientific notation, and comparing the powers of 10 - the power of 10 is the order of magnitude.

Back to the main topic of this section, there doesn't seem to be anything to fix (WP:CALC). Maschen (talk) 17:21, 11 December 2012 (UTC)

Reorganize article?

This article places too much emphasis on gases. It would be better to start with a definition of temperature from statistical mechanics (as in Statistical Physics by Landau and Lifshitz):

$1/T=dS/dE,$

point out that T has dimensions of energy, and identify k as the conversion factor between temperature and other energy units. Much of the material in the section "Role in the statistical definition of entropy" could go here. The rest of the article could be organized on these lines:

• Model systems that are used to measure k: ideal gases, black-body radiation and spin paramagnetism (Appendix B of Kittel's Thermal Physics)
• Equipartition of energy
• Role in other systems
• History

Comments? RockMagnetist (talk) 18:42, 14 December 2012 (UTC)

Sounds good, but shouldn't that be a partial derivative? Ref:F. Mandl (2008). Statistical physics (2nd ed.). Wiley. p. 88. ISBN 9-780-471-915331. (... also by considering dE = TdS − PdV). M∧Ŝc²ħεИ_τlk 19:00, 14 December 2012 (UTC)

I just transcribed an equation in L&L. Yes, in general it should be a partial derivative. RockMagnetist (talk) 19:08, 14 December 2012 (UTC)

Since the Boltzmann constant is defined as the energy of a particle per kelvin or perhaps energy per degree of freedom per kelvin, I don't see the relevance to multiparticle energy partitions as in equipartition? Surely this is the domaine of the Equipartition theorem? --Damorbel (talk) 19:21, 14 December 2012 (UTC)

Except "multiparticle energy partitions" are already the case by taking averages over the number of particles to get average energy (which equates to multiples of kT per degree of freedom, which is equipartition) etc... The point RockMagnetist is trying to say is to state the connection between energy and entropy first (since energy and entropy are fundamental), then discuss equipartition and so on... M∧Ŝc²ħεИ_τlk 19:38, 14 December 2012 (UTC)

I don't know who defines the Boltzmann constant as energy of a particle per Kelvin. Landau and Lifshitz don't - in the edition I have, they define k as I describe above on page 35 and don't get to equipartition of energy until page 131. RockMagnetist (talk) 19:49, 14 December 2012 (UTC)

Similarly Kittel defines k in chapter 2 and discusses equipartition in chapter 3. Both books discuss the Boltzmann constant when they first define the temperature. RockMagnetist (talk) 19:53, 14 December 2012 (UTC)

maschen you write:-

The point RockMagnetist is trying to say....

I'd rather hear it from RockMagnetist, if you don't mind.

Oh, and while you are at it, would you mind putting:-

Except "multiparticle energy partitions" are already the case by taking averages over the number of particles to get average energy (which equates to multiples of kT per degree of freedom, which is equipartition) etc..

in better language?

I mean ..."which equates to multiples of kT per degree of freedom, which is equipartition"... is meaningless as it stands. --Damorbel (talk) 20:00, 14 December 2012 (UTC)

"Except "multiparticle energy partitions" are already the case..." As in, we're not just talking about one particle, we take averages over many particles...

"Multiples of kT for each degree of freedom" meaning the energy of each degree of freedom is proportional to kT.

...F. Mandl discusses the ideal gas law in the first introduction, giving the definition k_B = R/N_A (at the very beginning of this article - which is presumably what Damorbel was referring to) and pV = nRT, then mentions equipartition (but explains this in detail later)... So Damorbel is correct in terms of units (if referring to k_B = R/N_A).

We seem to be conflicted with sources, Mandl's approach is more or less in the style of the current article...M∧Ŝc²ħεИ_τlk 20:11, 14 December 2012 (UTC)

There is some pedagogical value in discussing the ideal gas law first. I'd be fine with using it as an introductory example and then moving on to a more general definition. RockMagnetist (talk) 20:18, 14 December 2012 (UTC)

Re: I don't know who defines the Boltzmann constant as energy of a particle per Kelvin.

The Boltzmann constant will (probably) replace the kelvin as the fundamental standard of energy density because it can be determined more accurately (than the kelvin) by measuring Johnson noise, see here section 2.3.5 3rd para. where it has:-

Thus we have the exact relation k = 1.380 x 10⁻²³ J/K. The effect of this definition is that the kelvin is equal to the change of thermodynamic temperature that results in a change of energy per degree of freedom kT by 1.380 x 10⁻²³ J.

The quotation is from a document of the International Bureau of Weights and Measures where international agreement is sought for physical standards. --Damorbel (talk) 20:24, 14 December 2012 (UTC)

Ok... So does this mean reorganizing reduces to rewriting the section Role in the statistical definition of entropy? Then some possible shuffling of sections? M∧Ŝc²ħεИ_τlk 20:28, 14 December 2012 (UTC)

The Boltzmann constant pops up all over the place where particle energy is involved, what is needed is to show how it is unique and that it has many applications; but in no way is it defined by multi particle averages!

PS The reason Johnson noise is used to define the Boltzmann constant is that electrons are all identical but the freezing point of water is dependent on the isotopic purity of the water, something that is exceptionally difficult to establish.--Damorbel (talk) 20:49, 14 December 2012 (UTC)

<<Personal attack by Damorbel removed>> M∧Ŝc²ħεИ_τlk 21:05, 14 December 2012 (UTC)

Nobody is attacking another contributor see here. --Damorbel (talk) 09:28, 15 December 2012 (UTC)

It doesn't matter if it's not another WP editor. What you wrote was simply rude and uncalled for (hopefully - you know what those terms mean). M∧Ŝc²ħεИ_τlk 14:55, 15 December 2012 (UTC)

There is an important point to be observed when citing your lecturer's professor's book a "reliable source". Wikipedia help specifically classifies textbooks as a tertiary source, the reason being that these are effectively self-published and certainly are not subjected to any critical review. Personally I will not contest a citation if it is consistent with other material, but then the citation must be justified to show the consistency with basic physics. A good example of inconsistency is Kroemer & Kittel - "Thermal Physics", the references to the conservation laws in this, as with many textbooks, are quite inadequate. I did not know of your connection to Mandl but this certainly does not improve its reliability as a source. --Damorbel (talk) 08:05, 15 December 2012 (UTC)

Of course that doesn't "improve" the validity of the source - but Mandl, Kittel etc ARE reliable secondary sources (which includes textbooks, and research monographs and papers), not just tertiary sources. You seem to think you’re greater and more "consistent" than professors that do research that you can dismissively say their books/papers are "unimportant"?? Really? M∧Ŝc²ħεИ_τlk 14:55, 15 December 2012 (UTC)

The first author of the other book I mentioned, Lev Landau, is one of the greatest physicists of the 20th century. That sounds reliable to me. RockMagnetist (talk) 16:40, 15 December 2012 (UTC)

Re The first author ... Lev Landau, is... greatest physicists of the 20th century.

I absolutely do not see any merit in this statement. You may be right or you may be wrong, it is utterly subjective. But the truth of the matter is that many great scientists have been quite resolutely wrong about important matters; Joseph Priestley carried the phlogiston theory to his grave, denying the caloric theory all the way(!). If you read textbooks or lecture notes by notable physicist youshould be prepared for quite a number of errors. Textbooks and lecture notes, even when written by the famous, normally contain quite a number of errors.

It is quite worthless to refer to a book as a reliable source without including the text of the citation, the reader will have to find the book which may not be available or hidden away in a library somewhere; this is not what Wikipedia is about. --Damorbel (talk) 11:19, 16 December 2012 (UTC)

Damorbel, have you forgotten that we're discussing an outline that I have proposed on a talk page? It's a bit early to be providing quotes. Do you have any specific objections to the outline? You can access previews of both Landau and Lifshitz and Kittel online. RockMagnetist (talk) 22:45, 16 December 2012 (UTC)

The BIPM draft defines k as a particular quantity and then describes an effect. I agree that each degree of freedom has energy kT, and that needs to be in the article. However, the equipartition theorem is generally derived after the Boltzmann distribution is introduced, but the Boltzmann constant can be defined at the same time as the temperature is defined. It may not be the only way to do it, but it is a sensible way. And Johnson noise should be included in the methods of measuring k. RockMagnetist (talk) 21:31, 14 December 2012 (UTC)

By all means proceed, RockMagnetist. I've made my edits... M∧Ŝc²ħεИ_τlk 14:55, 15 December 2012 (UTC)

"The BIPM draft defines k as a particular quantity and then describes an effect"

Surely this is the purpose of an international standard? The value is given in kelvins but it is readily converted to other temperature scales by the usual arithmetic. The Boltzmann constant can be determined more accurately than temperature so it is logical to use it as the base reference. --Damorbel (talk) 08:15, 15 December 2012 (UTC)

I am proposing that we follow some of the best textbooks in defining k along with temperature and then deriving the equipartition theorem. Is there anything in that program that contradicts the BIPM draft? Conversely, if you started the article with the statement that k is defined as a particular exact value (only a proposal at present), where would you go from there? I am open to alternative organizations of the article, if you tell me what they are. RockMagnetist (talk) 16:37, 15 December 2012 (UTC)

Only makes sense?

The section Bridge from macroscopic to microscopic physics contains the assertion:-

temperature (T) makes sense only in the macroscopic world

Which is not correct. Temperature is an intensive property, it is very real at the microscopic (atomic, molecular or even 'degree of freedom') level. For a macroscopic system (with multiple particles) temperature has only one value when it is in equilibrium i.e. all its degrees of freedom have the same energy. --Damorbel (talk) 09:29, 17 January 2013 (UTC)

Actually, temperature is a property of an ensemble -- of a probability distribution.

Talking about the temperature of a single atom or molecule is tricky, because a single state of an atom or molecule doesn't have a temperature. (It might be associated with all sorts of temperatures). It's only when you have a probability distribution for the states of that atom or molecule, with an entropy, and can say how that distribution would change with changes in the average energy that you can define 1/T = dS/dE.

More usually temperature is defined for a collection of atoms or molecules, where one uses the energy distribution across that collection as a proxy for the probability distribution.

Note also that when a substance is in equilibrium, all its degrees of freedom do not have the same energy -- they have a distribution of energies, characteristic of the temperature. Jheald (talk) 12:57, 17 January 2013 (UTC)

While an ensemble of themodynamic entities (all must be in mutual equilibrium) has a temperature, the individual thermodynamic entities also have the same temperature, unless there is something there to stop the equalaisation. The point you should include is that the lower limit of the thermodynamic entities is the particle (or the degree of freedom).

Looking at the problem from the naive viewpoint, does it not seem obvious that a bunch of particles in a box will exchange energy in one way or another until they have the same temperature? The lower limit on particle number is surely one. --Damorbel (talk) 13:14, 17 January 2013 (UTC)

But when they have reached equilibrium, they won't all have the same energy. They will have a distribution of energies. So you can't say, just from the energy of one particle, what the temperature is. You need to know the energy distribution over all of the particles. That is what makes the temperature a property of the many particles, not the one particle. Jheald (talk) 13:25, 17 January 2013 (UTC)

I don't quite understand what you mean by But when they have reached equilibrium, they won't all have the same energy.

1/ 'Equilibrium' does not mean that, at a given moment, all particles have the same energy (temperature). Particles in a thermal system are continually exchanging energy, with an individual particle experiencing large energy fluctuations in time. These 'fluctuations' or 'distribution' of random energy follows the Maxwell-Boltzmann distribution.

2/ Neither, when measured with a suitable device, does an assembly of particles have exactly (with time) the average energy. Any given suitable device, such as a pressure sensor or perhaps a microphone, will give an output 'noise' proportional to the random motion of the particles. A very sensitive microphone will produce an output from the motions of air particles but thermometers are mostly far too slow to respond to fluctuations of particle energy, however that doesn't mean there are none! --Damorbel (talk) 14:07, 17 January 2013 (UTC)

There you go then. The energy of any one particle at any moment in time doesn't tell you the temperature. Jheald (talk) 14:50, 17 January 2013 (UTC)

Indeed it does! The relationship of the energy of one particle to its temperature is called the Boltzmann constant and plenty of people are making a lot of effort to measure it with ever increasing accuracy. --Damorbel (talk) 15:01, 17 January 2013 (UTC)

Tautology

"quantity of energy kT which is of a similar scale to the average energy of a given atom " - this phrase is scientific bullshitting: it explains nothing. 2.4*k*T is also "of a similar scale". So what? Please find a good source ans say something useful in this place. Otherwise I am gioing to delete the phrase. Staszek Lem (talk) 21:33, 12 February 2013 (UTC)

The precise meaning is clarified in the next line, and indeed throughout the rest of the article. kT is not necessarily exactly twice the average energy of a degree of freedom, but it is on that scale, so a similar scale to the average energy "of a given atom", and it's not a bad way to start trying to build up readers' intuition about it. Jheald (talk) 21:59, 12 February 2013 (UTC)

Sorry, sloppy way. Once again, reference, please. In particular, what does the word "scale" mean? Especially "similar scale"? Staszek Lem (talk) 22:15, 12 February 2013 (UTC)

Scale just means 'of roughly the same order of magnitude'.

The phrase is just trying to give people a qualitative motivation of what the Boltzmann constant actually means, intuitively.

No, it is not very rigorous. But per WP:UPFRONT in WP:MTAA, it is often a better thing to try to give readers something broadly accessible, intuitive, qualitative, motivating first; and leave off being strictly rigorous till a little later.

Besides, how would you try to qualitatively motivate what the Boltzmann constant means, if not something like this? Jheald (talk) 22:48, 12 February 2013 (UTC)

"order of magnitude" will do nicely, thank you. I am not against informal intros. Still, regardless alphabet soup, a reference for an informal description is just as well necessary, since there is more ways to introduce confusion informally, rather than with formal definition. Staszek Lem (talk) 23:04, 12 February 2013 (UTC)

Now, lets dig further. (Please keep in mind, I am not trying to be nasty; I am trying to read and understand, after briefly looking into a mess happening in the "temperature" article). The section in question begins with of a given atom, but the subsequent derivation ends with "for each gas molecule". I understand informal and such, but this really borders with sloppiness. I hope this is just as easily fisable as with "scale". Staszek Lem (talk) 23:10, 12 February 2013 (UTC)

Added Boltzmann's constant in J/K and hyperlinked it.

I Added Boltzmann's constant in J/K and hyperlinked it, to make it easier. 71.139.169.27 (talk) 04:59, 10 October 2013 (UTC)

Conversion factor

Boltzmann constant according to New SI proposals is only a conversion factor from joule to kelvin units. Joule is already linked to kg, m, s, and so K is becoming: kelvin was measured independently, now it will not be a fundamental unit anymore and this will allow saving a lot of teaching and experimental effort. I can provide also an example: in fusion research it is already common practice measuring temperature in eV, where eV was already linked to J. in this sense temperature actually is the same quantity as the energy per dof, it is not only proportional to the latter by the boltzmann constant, at the same way that rpms are a common measure of an engine frequency proportional to the same frequency in hertz. In my opinion this leading concept would extremely simplify understanding by common people and this should inspire definition also on wikipedia: one should concentrate on real problems, rather than on unit conversions. --37.100.183.187 (talk) 20:04, 17 February 2014 (UTC)

It's right. There is no such a fundamental constant. No experiment is able to measure it. Just the product kT is measurable. So this number is just a conversion factor with no physics involved. It make sense on a historical view only. All formulas involving thermodynamics and regarding temperature effects just make use of the product kT. I suggest to change article's name to Boltzmann conversion factor. — Preceding unsigned comment added by Graphen* (talk • contribs) 19:19, 22 November 2015 (UTC)

I encourage adding an additional section, explaining the role for the Boltzmann constant as a conversion factor. Changing the title is to radical for the time being. 217.234.107.130 (talk) 21:16, 9 January 2016 (UTC)

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((Planck Mass *(Planck Length)^2/(Planck Temp *((Planck Time)^2) = (Bolzmanns Constant)

((Planck Mass *(Planck Length)^2/(Planck Temp *((Planck Time)^2) = (Bolzmanns Constant)

(2.176470e-8 kg * ((1.616229e-35 m)^2)) / (1.416808e+32 K * ((5.39116e-44 s)^2) * (1.38064852e-23 (m^2) (kg (s^(-2))) (K^(-1)))) = 1

(6.03676081927e+23 Avogadro) , (Gas Constant 8.33464489072) , (Boltzmann's Constant) , (Mole)

((((376.730313/299792458) (m^2)) / (pi s)) / (6.03676081927e+23 * ((mol / 1000)^(-1)))) / Planck's constant = 1 mol / kg

1.38064852e-23 * 6.03676081927e+23 = 8.33464489072

((2.176470e-8 kg) * ((1.616229e-35 m)^2)) / ((1.416808e+32 K) * ((5.39116e-44 s)^2) * (1 gram)) = 1.38064821e-23 Boltzmann

((2.176470e-8 kg) * ((1.616229e-35 m)^2)) / ((1.416808e+32 K) * ((5.39116e-44 s)^2)) = 1.38064821e-23 = Boltzmann per kilogram

Mol is per Gram = (Mol * 1000) per kilogram

(Moles, Gas Constant & Avogadro Constant) should be Adjusted to units of Kilograms & Boltzmann's Constant for Clarity

1.38064821e-23 * 6.03676081927e+26 = 8334.64301932

http://en.wiki.x.io/wiki/Mole_(unit)

The number of molecules per mole is known as Avogadro's constant, and is defined such that the mass of one mole of a substance, ((expressed in grams)), is equal to the mean relative molecular mass of the substance.

(Planck Mass * (1 Planck Length)^2) / (Planck Temp * (Planck Time)^2) / (hbar *2 ) * 2.45 K = 1.604e-11 Hz CMBR

(2.176470e-8 kg * (1.616229e-35 m)^2) / (1.416808e+32 K * (5.39116e-44 s)^2) / (hbar *2 ) * 2.45 K = 1.604e-11 Hz

(2.176470e-8 kg * (1.616229e-35 m)^2) / (1.416808e+32 K * (5.39116e-44 s)^2) / (hbar *2 ) * 2.45 K = 1.604e-11 Hz

(Moles, Gas Constant & Avogadro Constant) should be Adjusted to (Units of kg & kB) for Clarity

(6.0e+26 / 6.25) * 8333.3333333 * (375 / 3e+8) = 1e+24

(Avogadro / 2pi) * Gas Constant * (Impedance / c) = 1e+24

8333.333333 / 6.0e+26 = 1.3888889e-23 Boltzmann's

Gas Constant / Avogadro = 1.3888889e-23 Boltzmann's

Fuller.david (talk) 02:59, 24 September 2017 (UTC)

Boltzmann's Avogadro's Gas Constants

((2.176470e-8 kg *(1.616229e-35 m)^2/(1.416808e+32 K *((5.39116e-44 s)^2) =(1.38064852e-23 m^2 kg s^-2 K^-1)

http://en.wiki.x.io/wiki/Boltzmann_constant

(1 / ((c^3) * ((1.616229e-35 meters) / (pi / (((1 / (2 * pi)) + 1)^0.5))) * ((5.39116e-44 seconds) / (pi / (((1 / (2 * pi)) + 1)^0.5)))))^0.5 = 6.02220471e+26

http://en.wiki.x.io/wiki/Avogadro_constant

((1 / ((c^3) * ((1.616229e-35 meters) / (pi / (((1 / (2 * pi)) + 1)^0.5))) * ((5.39116e-44 seconds) / (pi / (((1 / (2 * pi)) + 1)^0.5)))))^0.5) * 1.38064852e-23 = 8314.54802

http://en.wiki.x.io/wiki/Gas_constant

pi / (1 / (2pi) + 1) = 2.71024393443899972

Fuller.david (talk) 16:33, 25 September 2017 (UTC)

"The value is exact but not expressible as a finite decimal"

The values table is annotated with a heap of notifications that the values are unexpressible as decimals. That's clearly wrong in e.g. the kJ/(mol.K) case (Avogadro constant is exact). Do any of these cases sensibly require this warning? 131.236.42.201 (talk) 02:13, 29 April 2024 (UTC)

1/7 is an exact number that you cannot express as a finite decimal. So is √2. Headbomb {t · c · p · b} 05:25, 29 April 2024 (UTC)

what is "F" in /* Thermal voltage */

Boltzmann constant#Thermal voltage presents the following equation:$$V_{\mathrm {T} }={kT \over q}={RT \over F},$$but the letter "F" isn't defined and needs to be. Presumably "R" is the molar gas constant from earlier, and so could figure out that ${\textstyle F={Rq \over k}}$ using algebra, but that still doesn't tell us much about the meaning of F. Em3rgent0rdr (talk) 19:32, 24 September 2024 (UTC)