# Boltzmann's constant validity domain

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• jostpuur
In summary: This is because the constants that govern the behavior of particles in matter and radiation are the same in all of these units.)So, the numerical value of the Boltzmann's constant is independent of the temperature convention, but the Boltzmann's constant has a nontrivial effect on the shapes of the gas speed distribution and the black body radiation energy density distribution.
jostpuur
Boltzmann's constant appears in many distributions of statistical physics, and I have been left confused whether it is always the same constant with certainty. For example, suppose we define the Boltzmann's constant so that it is the constant that works with certainty for gases, i.e. it gives the Maxwell's speed distribution right. Could it be then that the same Boltzmann's constant would no longer give the Planck's distribution for black body radiation right, and some "other Boltzmann's constant" would be needed?

The question is reasonable, since it is an empirical fact that the Planck's distribution usually does not approximate the real empirical radiation distributions very well. This issue has not been seen as a serious flaw, since it has been explained by the fact that real hot objects are not ideal black bodies. Based on this alone one might think it would be reasonable to speculate that the Planck's distribution for black body radiation might actually need a different Boltzmann's constant that the Maxwell's speed distribution for gases.

The question is affected by a claim that Max Planck actually produced an estimate for the Boltzmann's constant by studying the black body radiation. I have the book Introductory Statistical Mechanics by Bowley and Sanchez, and it says this concerning Planck's achievements:

Since the ratios h/k_B and h^3/k_B^4 involve different powers of h and k_B we can solve for both. Planck worked them out from the data then available. With the wisdom of hindsight we know that he got a far more accurate value of k_B than any of his contemporaries. The evaluation of k_B was one the outstanding problems of the period.

If Planck estimated Boltzmann's constant accurately from data concerning black body radiation, that would imply that the Boltzmann's constant is the same for gases and black body radiation after all. But how is that possible, since at the same time the real radiation distributions are usually not very close to the ideal black body radiation? What was that data that Planck used really?

Boltzmann's constant is just a conversion factor between temperature and energy. Just like the speed of light is just a conversion factor between time and length.

If the Boltzmann's constant was different for gas speed distribution and for black body radiation, it would have a measurable consequence (assuming that our radiating body would be sufficiently close to an ideal black body).

For example consider an experiment where a hot solid object is placed inside a very large room filled with hot gas, and assume that we wait until the solid object and the surrounding gas are in equilibrium. We could then measure independently the gas speed distribution and the black body radiation energy distribution from the solid object, and independently fit the Maxwell and Planck distributions to those. The both fits would rely on finding the best value for the product $k_{\textrm{B}}T$. Under the assumption that $T$ is the same in both fits, it would then follow that $k_{\textrm{B}}$ could get either same or different values.

I understand that the numerical value of the Boltzmann's constant is dependent on the temperature convention, because if in the product $k_{\textrm{B}}T$ one scales the constant and the temperature in the opposite directions, there would be no effect on anything relevant, but this does not mean that the Boltzmann's constant would be purely a matter of convention. The Boltzmann's constant has a nontrivial effect on the shapes of the gas speed distribution and the black body radiation energy density distribution, and for this reason my question is still relevant.

jostpuur said:
For example consider an experiment where a hot solid object is placed inside a very large room filled with hot gas, and assume that we wait until the solid object and the surrounding gas are in equilibrium. We could then measure independently the gas speed distribution and the black body radiation energy distribution from the solid object, and independently fit the Maxwell and Planck distributions to those. The both fits would rely on finding the best value for the product kBTkBTk_{\textrm{B}}T. Under the assumption that TTT is the same in both fits, it would then follow that kBkBk_{\textrm{B}} could get either same or different values.
You could measure them independently, but they will have the same values if they are in thermal equilibrium. Where do you think the energy in the blackbody radiation is coming from and what sets the typical value?

In fact, many physicists select to work in units where ##k_B = 1##, i.e., measure temperature in units of energy. Just like many physicists select to work in units where ##c = 1##. (Actually, it is very common to set ##c = \hbar = k_B = 1## - and some times also other constants.)

Orodruin said:
You could measure them independently, but they will have the same values if they are in thermal equilibrium.

So that is what you seem to believe. Do you know of a real experiment where that would have been verified? I am under impression that if that kind of experiments really are carried out, the result will be that the Boltzmann's constants do turn out very much different. This is then explained away by the fact that real solid objects are not very close to being ideal black bodies, and that additional explanation helps maintaining the one Boltzmann's constant.

jostpuur said:
So that is what you seem to believe. Do you know of a real experiment where that would have been verified? I am under impression that if that kind of experiments really are carried out, the result will be that the Boltzmann's constants do turn out very much different. This is then explained away by the fact that real solid objects are not very close to being ideal black bodies, and that additional explanation helps maintaining the one Boltzmann's constant.
so your implying that everything should be treated as a blackbody with a different Boltzmann's constant. think about it, it is called a constant for a reason. by saying that, the concept of emissivity is wrong, and I have read way too much on that and done too much work with that concept to accept it as incorrect.

Dr Transport said:
so your implying that everything should be treated as a blackbody with a different Boltzmann's constant. think about it, it is called a constant for a reason.

That would be one hypothesis, but of course the truth could be somewhere between the two opposing hypotheses.

When I look at the derivation of the Maxwell's speed distribution, to me it looks like the Boltzmann's constant has its origins in the way in which the gas particles interact are scatter from each other. So it doesn't look like a same kind of constant like the speed of light or the Planck's constant.

jostpuur said:
I am under impression that if that kind of experiments really are carried out, the result will be that the Boltzmann's constants do turn out very much different

jostpuur said:
This is then explained away by the fact that real solid objects are not very close to being ideal black bodies, and that additional explanation helps maintaining the one Boltzmann's constant.

Because real solid objects are very far away from being ideal black bodies. I am sorry, but this is a basic fundamental principle regarding how temperature relates to the energy per degree of freedom. This is what temperature is.
jostpuur said:
When I look at the derivation of the Maxwell's speed distribution, to me it looks like the Boltzmann's constant has its origins in the way in which the gas particles interact are scatter from each other. So it doesn't look like a same kind of constant like the speed of light or the Planck's constant.
You could substitute temperature for energy per degree of freedom everywhere in those computations and never talk about temperature at all. It is exactly on the same level as c and hbar. In fact, the last TA I trained for my relativity course had the opposite problem from you in the beginning - not being able to see that c was a unit conversion on the same level as hbar or kB. He is doing his thesis in statistical mechanics.

Orodruin said:

My claim was equivalent with the claim that real radiating objects are not close to being ideal black bodies, so there is no need for reference.

It has become clear that you are not really interested in physics, you are unable to think logically, and I've started to waste my time with this thread.

jostpuur said:
My claim was equivalent with the claim that real radiating objects are not close to being ideal black bodies, so there is no need for reference.

It has become clear that you are not really interested in physics, you are unable to think logically, and I've started to waste my time with this thread.
Excuse me? I am sorry, but it is you who are being stubborn and not really interested in the actual physics. If you were familiar with the actual underlying physics, you would see that what I am saying is true. Yes, 200 years ago people might have argued as you do, physics has evolved since then.

## 1. What is Boltzmann's constant validity domain?

Boltzmann's constant validity domain refers to the range of conditions in which Boltzmann's constant, represented by the symbol k, can accurately describe the relationship between temperature and the average kinetic energy of particles in a system. It is commonly used in statistical mechanics to calculate the macroscopic properties of systems consisting of a large number of particles.

## 2. Why is Boltzmann's constant important in science?

Boltzmann's constant is important because it relates the microscopic behavior of particles to the macroscopic properties of a system. It allows scientists to understand and predict the behavior of matter at different temperatures, pressures, and energies.

## 3. What is the value of Boltzmann's constant?

The value of Boltzmann's constant is approximately 1.38 x 10^-23 joules per kelvin (J/K). It is a fundamental physical constant and its value is derived from the Boltzmann distribution law.

## 4. Does Boltzmann's constant have any limitations?

Yes, Boltzmann's constant has limitations in certain conditions such as at very low temperatures or in systems with strong interactions between particles. In these cases, other statistical mechanics equations may be necessary to accurately describe the behavior of the system.

## 5. How is Boltzmann's constant related to entropy?

Boltzmann's constant is intimately related to entropy, a thermodynamic quantity that measures the disorder or randomness of a system. It is represented in the famous equation S = k ln(W), where S is entropy, k is Boltzmann's constant, and W is the number of microstates available to a system at a given macrostate. This equation is known as the Boltzmann entropy formula.

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