# Can The Expression Dm =De/c^2 Be Derived From Classic Thermodynamics?

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### #1 rhertz

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Posted 05 May 2019 - 10:35 PM

Given that fact and using an isolated system (but no enclosures) with an infrared laser of 10 Watts pointing to an almost perfect black body of rectangular shape (10cm x 10cm x 1cm), is it possible to derive a non-relativistic relationship between mass increase and radiant energy?

Edited by rhertz, 18 June 2019 - 08:37 PM.

### #2 exchemist

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Posted 06 May 2019 - 02:51 AM

It's well known that iron and other metals expand when heated, and in a visible amount.

Given that fact and using an isolated system (but no enclosures) with an infrared laser of 10 Watts pointing to an almost perfect black body of rectangular shape (10cm x 10cm x 1cm), is it possible to derive a non-relativistic relationship between mass increase

The black body is a metal with low emissivity covered by a black coat of a composite elastic material. The body absorbs a net energy dE, which is transformed into internal heat, increasing the body's temperature (and its size).

I mean, can be proven that there is a mass gain dm = dE/c2and, in this case, is this fundamental equation related to temperature T of the black body?

Also, it this process reversible when the laser is turned off? (the object cool off with time).

I don't know if anyone has tried this. The sensitivity required would be extremely high, as for each joule of energy added the mass increase would be about 10⁻¹⁴ grammes. Chemical balances can only get down to microgrammes, so even if you added 1kJ of energy one would still be 5 orders of magnitude from what would be needed.

Normally, physicists take the mass defect in the atomic masses of the elements, and the measured energy changes in radioactive decay processes, as demonstrations of E=mc².  The heat output from a commercial nuclear reactor can be calculated from these figures, which thus becomes a demonstration of E=mc² that does not rely on assuming relativity.

Edited by exchemist, 06 May 2019 - 02:53 AM.

### #3 exchemist

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Posted 06 May 2019 - 09:07 AM

Well, if you take 10 Watts in one day, it delivers 864,000 Joules to the body. If you assume that thermal equilibrium is reached in 1 day,

then the equivalent mass:F(energy) would be 9.6*10-9 grams or aprox. 0.01 ugrams.

Assuming that I use Beryllium (emissivity = 0.18) coated with a black polymer (emissivity = 0.999), the body could absorb most of this energy.

With Beryllium melting point at 1,287°C, and assuming that a thermal equilibrium at the isolated system is reached at 27°C (300°K),

and using Stefan's Law:

j = T4  , with = 5.67x10-8 Watt.m-2.K-4

J = 4.59 Watts

So, the body would have a net absorption of 5.41 watts OR 476,424 Joules/day to mantain thermal equilibrium (while the laser is ON).

It gives an amount of 5.19 nanograms (using dm = dE/c2).

I'm wondering if this "gedankeexperiment" has any sense at all.

It's because I introduced a couple of fallacies, starting with "a-priori" assumption that  dm = dE/c2   is valid.

This doesn't seem to make much sense at all.

What are you trying to do? If you want to measure the mass gained by a body due to heating, the last thing you want is a radiating system, because you will have a devil of a job working out how much heat has been lost, while you have been putting heat in. You want something that can be thermally isolated: a calorimeter, in fact. Don't you? Why use a laser and futz about with Stefan's Law and all the problem of radiative loss? You can heat it electrically.

And if you think the system comes to equilibrium at only 27C (2C above standard "room temperature"), what temperature are you assuming the block has when you start? Have you made a typo somewhere?

No, the predicted mass gain from thermal effects is too small to measure in practice, so I think you are barking up the wrong tree with this idea. You are far better off to consider the nuclear mass defect. Now that is something that confirms E=mc² in a very obvious and indisputable way.

Edited by exchemist, 06 May 2019 - 09:14 AM.

### #4 Dubbelosix

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Posted 06 May 2019 - 09:12 AM

I am not sure why you have taken the differential of the equation, but energy is related to thermodynamic activity.

$\frac{1}{2}mv^2 = \frac{3}{2}k_BT$

So it is not so much you derive one from the other, but they are related to systems and define the temperature through kinetic motion of the systems constituents.

### #5 exchemist

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Posted 06 May 2019 - 11:02 AM

I know, it's wrong but I let myself enter into my own trap.

I used black bodies and lasers to have at hand the speed of light "c" in some way.

I used Stefan's law to introduce body's temperature, but it didn't work. When I used 700°K as temperature at

thermal equilibrium, it gave me that the body irradiated 11.5 Watts, while being heated with a 10 Watts laser!

Because of that I lowered the temperature. I cheated but failed anyways.

Mea culpa: I presented the problem in a hurry, without proper considerations.

Still, I can say that "in the internet" there is consensus that a heated body gain mass due to energy absorption.

I didn't explore it any further, so I started to play with simple theories.

This "feeling" that there is a relationship between mass and energy is older than written history.

Once, I imagined Newton observing how a piece of paper burn with flames and smoke, and I thought

that we might be asking himself: "Where did the matter of the paper go?. Some of it converts into

radiant heat (the concept was known by then); some is converted into convected heat; some part

is converted into smoke and there are carbonized remains. How do all of those parts relate to the whole?"

But this is just me. Don't pay attention and let the topic die.

Fair enough. I would not myself have expected it to have been be possible to see the practical effect of the mass-energy relation before the advent of  mass spectrometers able to measure the masses of nuclei accurately. For that you do not need to assume anything about relativity.

I have had a look but cannot see when the mass defect was first reported. Mass spectrographs appear to have been around since the end of the c.19th so one might have thought someone would have commented on the nuclear masses not being exact whole number multiples of that of hydrogen. But it seems that the discovery of the neutron was in practice needed first, before all the compositions of the nuclei could be worked out and the mass defect highlighted unambiguously.