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@Danielj We have learned to allow probability and other statistical techniques to exist alongside deterministic formulae and to pick and choose the most expedient for a particular job in hand. Engineers even manage to mix the two with so called 'Limit State Design Theory'. Remember also that there are two types of variable, discrete and continuous. Probabilities defining orbitals is a fine example of mixing the two since the variables in the Schrodinger equation are continuous, but we are trying to describe 'an electron', which is discrete. Hi, MigL I can't cut and paste or quote using this machine so just to say 'No Problemo :-)'

Thank you for your input, MigL. I'm not sure what part, in any, CPT plays in (particularly classical) thermodynamics. But yes, in classical thermodynamics it is possible to construct theoretical systems/processes where there is no entropy change. This prevents universal use of the maximum entropy criterion as a 'necessary and sufficient' criterion for equilibrium. This situation is the problem with Callen. An example construct (similar to the recent piston question in homework help) is presented and discussed in Carrington. You mentioned reversibility as though we had a satisfactory (mathematical) definition. Achieving one is the point of this thread. I do not have an answer to pull out of a hat. Do you have any suggestions? One of the difficulties is that folks often rattle of thermodynamic statements without properly distinguishing between the system and the surroundings and changes in each. These variables are not described directly in the laws of thermodynamics, but are connected by the exchange variables across the system boundary. Proper specification of the system boundary and the system process need also to be made to complete a thermodynamic analysis.

Discussion of the 'Entropy of the Universe' is for philosophers and mystics, not classical physicists, so we must agree to differ on the value of such. I would have welcomed some discussion about my topic though, since in the 50 years or so I have been successfully doing thermodynamics I have never come across a purely mathematical definition of reversibility.

Cosine rule? Wow you really are doing things the difficult way. I also think you have a typo in your posted angle QPR. triangle QPR. <QPR = Perhaps you are not comfortable using bearings? I have prepared an 8 step guide to completing this problem. 1) Position P 2) Draw a North line through P 3) Mark on Q, 20km north of P. Note the North line through Q is also the same as the North line through P, since Q is due North of P. 4) At Q mark out a line at 140 deg from the North Line at Q. 5) Mark R 8km along this line. Note also that smaller angle PQR = (180 - 40) = 40 6) Join PR to form triangle PQR, Call the angle QPR theta. This angle is the bearing of R from P because it is the clockwise angle of PR from the north line PQ, at P. 7) From R draw line RS at right angles to PQ, meeting PQ at S. You now have two smaller right angled triangles and in triangle SQR the hypotenuse is 8. Calculate SR = 8sin40 Calculate QS = 8cos40 8) S is the same distance north of P as R and PQ= 20 = PS + QS So calculate the distance R is north of P = SP = QP-QS tan (Theta) in triangle SRP is SR/SP so calculate the bearing of R from P = theta = tan-1(SR/SP)

If you were less condescending and more open minded you would realise that all I have stated is purely conventional, and that you yourself are under some misunderstanding. You didn't like the references I gave from full and famous professors at Oxford or Cambridge, and referred me to Van Ness. Well I looked at my old and battered 4th edition and on page 39 found the following standard definition How does that compare to your claim that my post #10 attempts to overturn standard definitions? I would say that it is essentially the same as my post#10. So I looked further (in Van Ness) at your claim that the Clausius inequality was not defined by a cyclic process. Well from page 148, he develops the conventional Carnot theory for cyclic processes, leading to a cyclic integral, specifically The above statement is reasonable and correct. But it is not what you stated. Clearly post#2 claims that all reversible processes have no (zero) entropy change, and post#4 confirms you are asserting this. Compare this with your Van Ness reference when he summaries his 2nd Law discussion This in general will not be zero. Indeed, Van Ness offers various methods of calculation, depending upon the circumstances. Unfortunately Van Ness does not provide a T-S indicator diagram for a Carnot cycle in which there are four reversible processes in series to form a cyclic overall process as per the first quote. I therefore display one here. The T-S diagram has the form of a rectangle with four reversible processes AB in which entropy increases, at constant temperature BC in which entropy remains constant but temperature falls CD in which entropy decreasess DA completes the cycle and raise the temperature, at constant entropy. I know you did not say that entropy cannot decrease, but many believe this so please note that it must decrease somewhere in any cycle for which it's change sums to zero. The Carnot indicator diagram shows this very clearly. As a matter of interest, the originator of the inequality and cyclic integral, Clausius actually said (translated : Wilson : Cambridge University) All this is, of course, an interesting digression from the topic. In thermodynamics three conditions (and their opposites) are recognised, reversible, equilibrium and quasi-static. They are all subtly different, but usually coincide.

C'mon what you said was neither mathematical, nor correct, and in particular should not involve a quantity called entropy, unless you first define that. Further you have offered no answer to my post#3, that I made in response to your post#2. Here is a non mathematical definition. A thermodynamically reversible process is one that is reversed by an infinitesimal change in the conditions of the surroundings. For instance consider a light piston being pushed out under pressure against a lower resisting pressure. An infinitesimal change in the surrounding pressure will not reverse the motion of the piston. But if the external pressure is equal to the internal pressure then an infinitesimal change, in either direction to the external pressure will result in the piston moving in or out. We can also observe that the system is in equilibrium with its surroundings, which is why thermodynamicists often say that reversible means equilibrium. But that is in words, not mathematics.

Well I don't agree with this interpretation. 1) Any heat flow in either the first or second law, by definition, crosses the system boundary. You are suggesting otherwise. 2) Bignose's definition should have read for a cyclic process. Therefore we have to melt and refreeze the water. then the entropy change is indeed zero. 3) Why only closed systems I asked for any general system, without special conditions. 4) I was looking for a mathematical statement or formulae or calculation I could perform on any process that would have as its outcome "this process is or is not reversible" None of the foregoing discussion shows this. This last is a really difficult issue that I have not seen supplied anywhere, as I said at the outset.

Surely that depends upon your specification of the system. I find that getting this specification is the key to success or failure in thermodynamics. Further I note that entropy has been mentioned twice now. How does one approach that before one has defined 'reversible'?

Given that there are no truly reversible processes in Nature, do you think melting and evaporation are irreversible, if carried out sufficiently slowly? Neither are isentropic.

Many posters here argue against a proposition on the basis that it is 'non mathematical' rather than against the merits or demerits of the proposition itself. So who can offer a mathematical definition of thermodynamic reversibility? The only attempt (not even wikipedia tries http://en.wikipedia.org/wiki/Reversible_process_(thermodynamics ) that I can find is in Callen and that fails counterexamples in both Atkins and Carrington. Remember any proposed definition must be applicable to any conceivable set of circumstances, not just specially constructed ones.

I may not do your homework for you, only help you do it for yourself. Do you understand this statement I made in post #2 You have an example bearing drawn in my sketch. Do you have a North line through P?

Yes indeed, that was my point. So what about my second example? As to the heliograph you would still have to look to see the flashes and know where to look. Perhaps this was not such a good example because I think that simple heliographs went back to Alexander so actually predated Monte. However the person communicating was trying to communicate with his friend not me or to make a general broadcast. Perhaps the aliens, if they exist, are communicating with each other but not directing their space-graphs at us.

Since you haven't specified an equation, I am guessing that you are confusing reference frames. You need to start by specifying the reference frame and if that frame centre is always the centre of rotation of the orbiting body then why would there be any change to 'The equation' ?

Would Montezuma have recognised flashes from a heliograph as a signal? And what would he have made of signals from a Napoleonic semaphore?

I don't see anything remotely offensive in post#4. Surely if someone doesn't like the subject the simple answer is not to visit the thread?

Actually it is, but it is only an analogy to help explain where probabilities fit into QM. If you take the one dimensional Schrodinger equation the particle must be somewhere along the single axis you have. But that axis is homeomorphic to a segment of itself so it is legitimate to use a finite interval to represent the full line. The process of normalisation maps the whole line to the interval (0.1). Are you not thinking of boundary conditions here? Whole number waves have no meaning on an infinite line.

You can improve the model by noting that the ball must be somewhere so the total probability of it being on the channel equals one. Further if the channel is tilted down the ball will accelerate so the speed of the ball will increase as it rolls down and therefore the probability of hitting it will decrease. Alternatively the channel may be tilted upwards so the ball will slow. The model, of course is to compare the probability of hitting the ball with the probability of finding the electron in a given segment of space. You can even extend the model to probabilities curves that rise and fall with a hump, like an electron orbital.

Probabilities are just a very useful interpretation of the square of the wavefunction. There are other interpretations. It's similar to the following: Suppose your friend rolls a baseball along a channel from one end to the other, Now suppose that at some time t after he has set the ball in motion you whack down with a baseball bat. You can calculate a probability that you will hit the ball, which decreases with the speed of the ball and increases with the time your bat spends down on the channel. Wave function probabilities are like that, for a given segment of space.

Thank you for a better viewpoint JC, your numbers certainly stack up. I didn't think of that explanation, but it's so obvious when you put it like that, and really helps the thread along.

Thank you Sensei, yes I know it was programmed to not do what it was instructed. Would you expect your children to think they know better and not do what they were told?

The label states the standard dose. Your maths is adrift. 0.025% [math] = \frac{{.025}}{{100}}g/L[/math] [math] = .00025g/L[/math] [math] = 0.250mg/L[/math] [math] = 250\mu g/L[/math] If you want to know why the 0.025% and 0.035%, fumaric acid is diprotic and the salt is some mixture of ketotifen hydrogen fumarate and ketotifen fumarate that makes 250 microgrms of it equivalent to 350 micrograms of pure fumarate. You will need to ask pharmacist for more details than that.

According to the BNF (British National Formulary) the standard dose for eye drops is 250 microgrms per mL = 0.25 milligrams per mL

I note that the OP is a geoinformatics engineer and is perhaps looking for more of an operational view. I am guessing that some satellite is gathering data using one of these techniques. Anyway it is good that several interpretations are available. There are also techniques known as resonance fluorescence and resonance Raman spectroscopy. Indeed here is a quote from Whiffen : spectroscopy Edit Why are computers so stupid? On the posting entry editor I can write a c in parenthesis, and see it that way there on the editing editor afterwards. Yet it appears in my full screen as the copyright symbol. I make more than enough genuine spelling mistakes, without the system inventing them.

Because I am not drawing Feynman diagrams How is that caustic comment helpful to the thread?

A simpler answer than swansont's is that flourescence is a light emission and Ramam scattering is the result of light subtraction (absorbtion). With flourescence some of the incident light is first absorbed by the substance. But what is left of the incident beam is simply reduced in intensity, its frequency/wavelength is unchanged. That is the individual photon's energy remains unchanged, there are just fewer of them left in the incident beam. Each absorbed photon is fully absorbed. Additionally other light is now observed, at particular frequencies, different from those of the incident light. This additional light is due to the absorbed light being re-emitted, almost instantaneously and is called the flourescence. These are new photons due to the emission. With Raman scattering some of the incident light is again absorbed but in a different manner so that the photons actually loose some energy, but none are fully absorbed. So this time the intensity remains the same (ie number of photons) but they have lost some energy so have lower frequency/longer wavelength. Further they are partly deflected in direction hence so the beam is broadened or scattered. None of the observed photons are new photons as with flourescence.