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What *is* an inexact differential. I believe I was the person that put that redlink up a long while ago in Entropy (thermodynamic views), and I had hoped that someone would fill that space with an explanation. I still have no idea what the inexact differential symbol δ *means*. Does anyone know? Fresheneesz 09:34, 5 May 2006 (UTC)
I think "inexact differential" just means a differential (i.e. an expression of the form ) which is not exact. For example, is an inexact differential. Line integrals of this will depend on path. Cjao ( talk) 03:48, 22 December 2009 (UTC)
Please do not attmept to merge this article into exact differential. These are seperate but related terms. See: Talk:First law of thermodynamics for example. Thanks:-- Sadi Carnot 05:48, 25 July 2006 (UTC)
I stumbled over the first sentence of Inexact differential, because it does not say at all how the "inexact differential" of a function f is defined, and I could not make any sense of it. Since the rest of the article only says that the properties of an inexact differential are the opposite of the properties of an exact differential, which is obvious, I thought it would be best to merge it with exact differential. However, you reverted this. Why? -- Jitse Niesen ( talk) 05:54, 25 July 2006 (UTC)
i will change "mathematics" to "physics" in the intro, as it is not accurate. it is not a math term or object. as above conversation indicates, a trained mathematician may very well not recognize it. Mct mht 20:29, 19 August 2006 (UTC)
I'm not sure how to make a re-direct, so could someone make a redirect from imperfect differential to here? Over in this locale we call it an " imperfect differential". Fephisto 21:36, 24 August 2006 (UTC)
it says: The symbol Inexact differential symbol.PNG (d-bar), or δ (in the modern sense)
what is the diff between the two? What is meant by "in the modern sense"? I was taught the d-bar symbol in my physical chemistry course... —Preceding unsigned comment added by Mikejones2255 ( talk • contribs) 03:34, 17 December 2009 (UTC)
Would it be correct to say (as I conclude from the scattered comments above) that is not a symbol of its own, but just part of a mnemotechnic convention for constructing symbols? It seems that "" must be understood as a single atomic symbol; it cannot be understood as some operation applied to . It is implied that we can write , but this defines in terms of (and an implied path through the state space), not the other way around. – Henning Makholm ( talk) 22:19, 28 March 2010 (UTC)
Would it be correct to say (as I conclude from the scattered comments above) that is not a symbol of its own, but just part of a mnemotechnic convention for constructing symbols? It seems that "" must be understood as a single atomic symbol; it cannot be understood as some operation applied to . (Apparently nitpick-inducing clarification of question omitted) – Henning Makholm ( talk) 06:19, 3 April 2010 (UTC)
I'm afraid "a differential for which the integral is path specific" does not really tell me, operationally, what to do with an equation that asks me to construct one. I can grasp the meaning "a function whose path integrals depend on the path", but a free-floating differential that is not being quotiented into a derivative or used to complete an integral makes me queasy enough (being primarily a mathematician) that I'm going to insist on finding an explicit definition what an equation that involves one means. So let me go back a few steps and start anew:
I came here from the Enthalpy article. It presents a long equation with a legend that goes like:
I do not know how what an inexact differential is, so I come here and expect to find an explanation. Finding no explanation in the article itself, I go to the talk page and try to piece one together. I reach the tentative conclusion that the Enthalpy article is in error because there is no mathematical object or operation named "the inexact differential" that the symbol alone can represent in the equation. And now I'm asking: Is this tentative conclusion correct?
If it is correct, then I'll go and remove the misleading legend from enthalpy. But if it's not correct, then I'm trying to figure out what is, such that I can make this say it.
In other words, my hypothesis was that "" is just one compound symbol that cannot be understood as the operation applied to the quantity . If this hypothesis is true, if we pick a new letter, say X and replace all instances of "" with "X", then the mathematics means exactly what it did before, and the only loss is that it is less intuitive to remember that X and Q are related by a certain equation.
But this cannot be true, because is meaningless (it lacks a variable of integration) and you sound like has meaning. So apparently the does mean something by itself, and I'm back to trying to understand what its meaning is. And presently it seems that there must be two definitions – first one that applies when appears in an ordinary algebraic context, and a definition that says how to compute an integral where the variable is introduced with a , as in .
Hmm... or is just sloppy notation for ? That would seem to save the integral from meaninglessness – but then we'd need a definition for . How does it differ from plain old ?
Does this make my question clearer? – Henning Makholm ( talk) 01:48, 4 April 2010 (UTC)
Your PDF proposes this definition (on page 9, not 10):
This seems to me to be mathematically vacuous, because according to it every differential is exact, because for every choice of F, we can simply take , , and .
Is it meant to define a notion of "exactness" only relative to some already (implicitly) given set of 's? If that is so, it seems rather unfair to blame on the innocent a "failure" that is wholly caused by your choice of 's. – Henning Makholm ( talk) 21:56, 4 April 2010 (UTC)
The only place I've ever seen this talk of "exact" vs "inexact" differentials is in the study of thermodynamics. However, I only saw this way of framing it in my book targeting chemists. In Sandler's "Chemical, Biochemical, and Engineering Thermodynamics," Stanley Sandler dispenses with the notion altogether. Personally the idea of talking about "differentials" makes me uncomfortable and it seems less than rigorous, and frankly in thermodynamics you don't need them, just plain old calculus.
Rather than talking about some infinitesimal change , specify a control volume and talk about the accumulation of internal energy in time, . To avoid the added complexity of energy associated with flow of mass, assume it's a closed system, and you can write the first law as , where Q represents the rate of heat flowing into the system. Similarly, a closed-system entropy balance is written as , with the second law stating that in a reversible process, and in an irreversible process. If the process is reversible, i.e. no entropy is generated, then you can solve for Q and insert into the first law to obtain , the familiar fundamental equation for internal energy expressed using derivatives that actually make sense in the context of calculus. Also, you can evaluate the change in internal energy of the system by simply invoking the 2nd fundamental theorem of calculus, so
I think this formulation is much better, as you don't have to talk about such things as δQ and δw, everything just makes sense both physically and mathematically. Unfortunately, the nonsense about differentials is as old as the science itself and will probably never go away. Ryan ChemE ( talk) 09:12, 29 January 2015 (UTC)
I have (almost) rewritten the article to make it more accessible to non-engineers (I am one myself though).
I have shortened the description part (it was way too long and thereby confusing). Also the part with integrals was quite confusing:
This expression really does not make sense ("it is not even wrong"). Assuming that a and b are the initial and final states of the system, the point of the inexact differential is that you cannot get a unique primitive f. Rewriting in variables of the first principle of thermodynamics:
The reason this does not make any sense is that we cannot figure out how Q changes with the state U without knowing W. You can phrase it like there is an unsaturated degree of freedom, I suppose.
This concept is really simple, but it seems that thermodynamics textbooks are written by some scribes that love keeping everything difficult to understand. 193.175.53.21 ( talk) 11:11, 4 August 2010 (UTC)
I don't see the relevance of the Vanuatuan movie to this topic. It seems nothing more than a slightly comical (lewd) video, with a post-hoc rationalisation. — Preceding unsigned comment added by 129.67.105.62 ( talk) 13:18, 9 June 2011 (UTC)
This article looks weird. What is the exact mathematical definition of inexact differentia? –– 虞海 (Yú Hǎi) ✍ 17:45, 17 January 2012 (UTC)
It seems to me that the temperature as an integrating factor is a neutral factor in this business. What makes the entropy an exact differential is the specification of the path of integration: a reversible path (Q_rev). With the specification of a path all inexact differentials become exact. Am I totally missing the point or am I right? 109.93.162.246 ( talk) 11:31, 4 January 2014 (UTC)
I would like to propose that the very idea of talking about thermodynamics in the context of differential changes makes absolutely no sense. It may very well be how the concepts are discussed in textbooks (some textbooks), but it forces you into twisted contortions of logic and to abandon the rigorous rules of mathematics. That is why there are so many mathematicians on this talk page who have no idea what the heck we are talking about here. When the mathematics we use to try to describe the physics of reality baffles mathematicians, that should be a red flag.
The first problem we need to address is: heat does not have units of energy, it has units of energy/time. Heat is a flow of energy. It doesn't make sense to talk about an infinitesimal amount of heat δQ. We can certainly talk about a heat flux q, which has units of energy/(time*area), which is what appears in Fourier's Law of Heat Conduction . Or the same quantity appears in Newton's Law of Cooling . We can then get a total heat flow by multiplying by the area across which heat flows, or if the temperature gradient varies we can chop the area up into small pieces and apply the Riemann sum definition of an integral. The point is it makes no sense to talk about an infinitesimal amount of heat energy which we would somehow "integrate" through state space in a path-dependent way. We should change our perspective and understand that heat is simply a flow of energy into or out of a system.
The people who talk about trying to find the change in energy of a system by integrating the "inexact differential" δQ are taking a convoluted look at the problem. Internal energy and enthalpy are state functions - all you need to know is where you start and where you end. You can then perform an energy balance and recognize that energy is conserved, so if the system has more energy than it started with, it must have come from somewhere. Equivalently, if you have less energy than you started with, it must have gone somewhere. You don't even have to use the normal variables found in the fundamental equations. For example, if I want to know what is the change in internal energy of a material if I take it from one temperature and pressure to another, I can perfectly well do that.
Formally, you can simply integrate the above equation with respect to time to find the change in U from t1 to t2. It would probably be easier, however, to invoke the substitution rule and integrate the first term with respect to temperature and the second with respect to pressure. You can also view this as an example of the fundamental theorem of line integrals, and break this into a path of constant pressure followed by a path of constant temperature for easy computation. (As an aside, the above equation also proves that the internal energy of an ideal gas only depends on temperature) But if an energy balance shows that the change in the internal energy of a system is equal to the heat flow, now we also know what the heat flow has to have been, without having to figure out what in the world it means to integrate an inexact differential. I am confident that any mathematician would look at the above equations and find them perfectly understandable and justified rigorously, and I would definitely welcome any feedback to the contrary! Just my humble opinion, but I think all the thermodynamics pages that talk in terms of exact and inexact differentials should be completely recast in terms of balance equations. This framework is easier to understand physically and mathematically rigorous. Ryan ChemE ( talk) 00:06, 3 February 2015 (UTC)
There are a few issues with the "Heat and work" section under "Examples." The first sentence, that a fire requires heat, fuel, and an oxidizing agent, is more related to the "fire triangle" concept in fire safety, and I think is off topic here. Also the discussion regarding activation energy is misleading. The activation energy of a reaction has nothing to do with energy flows into or out of the system. Activation energy simply allows one to relate the speed of a reaction with the system's temperature, and any reaction will occur (however slowly) at any finite temperature, so long as it is thermodynamically favorable. Heat is not consumed in a reaction to overcome the activation energy, it would only be transferred to maintain the temperature of a reaction with a positive enthalpy change of reaction. The reason you need to add heat (or work to generate friction) is because you need to raise the temperature high enough to allow the reaction to proceed at a high enough rate to sustain itself. The physical law relating reactions and activation energy to temperature is the Arrhenius equation: where k is the reaction rate constant. Finally, the last sentence about integration needing to account for the path taken - what integration? What is being integrated here, and with respect to what? Ryan ChemE ( talk) 14:00, 27 June 2015 (UTC)
There were several incorrect descriptions of the first part of the Carnot cycle. I made changes, but the section is rendered not very coherent. I hope someone can clarify that section. Dhh787 ( talk) 18:25, 12 July 2024 (UTC)
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What *is* an inexact differential. I believe I was the person that put that redlink up a long while ago in Entropy (thermodynamic views), and I had hoped that someone would fill that space with an explanation. I still have no idea what the inexact differential symbol δ *means*. Does anyone know? Fresheneesz 09:34, 5 May 2006 (UTC)
I think "inexact differential" just means a differential (i.e. an expression of the form ) which is not exact. For example, is an inexact differential. Line integrals of this will depend on path. Cjao ( talk) 03:48, 22 December 2009 (UTC)
Please do not attmept to merge this article into exact differential. These are seperate but related terms. See: Talk:First law of thermodynamics for example. Thanks:-- Sadi Carnot 05:48, 25 July 2006 (UTC)
I stumbled over the first sentence of Inexact differential, because it does not say at all how the "inexact differential" of a function f is defined, and I could not make any sense of it. Since the rest of the article only says that the properties of an inexact differential are the opposite of the properties of an exact differential, which is obvious, I thought it would be best to merge it with exact differential. However, you reverted this. Why? -- Jitse Niesen ( talk) 05:54, 25 July 2006 (UTC)
i will change "mathematics" to "physics" in the intro, as it is not accurate. it is not a math term or object. as above conversation indicates, a trained mathematician may very well not recognize it. Mct mht 20:29, 19 August 2006 (UTC)
I'm not sure how to make a re-direct, so could someone make a redirect from imperfect differential to here? Over in this locale we call it an " imperfect differential". Fephisto 21:36, 24 August 2006 (UTC)
it says: The symbol Inexact differential symbol.PNG (d-bar), or δ (in the modern sense)
what is the diff between the two? What is meant by "in the modern sense"? I was taught the d-bar symbol in my physical chemistry course... —Preceding unsigned comment added by Mikejones2255 ( talk • contribs) 03:34, 17 December 2009 (UTC)
Would it be correct to say (as I conclude from the scattered comments above) that is not a symbol of its own, but just part of a mnemotechnic convention for constructing symbols? It seems that "" must be understood as a single atomic symbol; it cannot be understood as some operation applied to . It is implied that we can write , but this defines in terms of (and an implied path through the state space), not the other way around. – Henning Makholm ( talk) 22:19, 28 March 2010 (UTC)
Would it be correct to say (as I conclude from the scattered comments above) that is not a symbol of its own, but just part of a mnemotechnic convention for constructing symbols? It seems that "" must be understood as a single atomic symbol; it cannot be understood as some operation applied to . (Apparently nitpick-inducing clarification of question omitted) – Henning Makholm ( talk) 06:19, 3 April 2010 (UTC)
I'm afraid "a differential for which the integral is path specific" does not really tell me, operationally, what to do with an equation that asks me to construct one. I can grasp the meaning "a function whose path integrals depend on the path", but a free-floating differential that is not being quotiented into a derivative or used to complete an integral makes me queasy enough (being primarily a mathematician) that I'm going to insist on finding an explicit definition what an equation that involves one means. So let me go back a few steps and start anew:
I came here from the Enthalpy article. It presents a long equation with a legend that goes like:
I do not know how what an inexact differential is, so I come here and expect to find an explanation. Finding no explanation in the article itself, I go to the talk page and try to piece one together. I reach the tentative conclusion that the Enthalpy article is in error because there is no mathematical object or operation named "the inexact differential" that the symbol alone can represent in the equation. And now I'm asking: Is this tentative conclusion correct?
If it is correct, then I'll go and remove the misleading legend from enthalpy. But if it's not correct, then I'm trying to figure out what is, such that I can make this say it.
In other words, my hypothesis was that "" is just one compound symbol that cannot be understood as the operation applied to the quantity . If this hypothesis is true, if we pick a new letter, say X and replace all instances of "" with "X", then the mathematics means exactly what it did before, and the only loss is that it is less intuitive to remember that X and Q are related by a certain equation.
But this cannot be true, because is meaningless (it lacks a variable of integration) and you sound like has meaning. So apparently the does mean something by itself, and I'm back to trying to understand what its meaning is. And presently it seems that there must be two definitions – first one that applies when appears in an ordinary algebraic context, and a definition that says how to compute an integral where the variable is introduced with a , as in .
Hmm... or is just sloppy notation for ? That would seem to save the integral from meaninglessness – but then we'd need a definition for . How does it differ from plain old ?
Does this make my question clearer? – Henning Makholm ( talk) 01:48, 4 April 2010 (UTC)
Your PDF proposes this definition (on page 9, not 10):
This seems to me to be mathematically vacuous, because according to it every differential is exact, because for every choice of F, we can simply take , , and .
Is it meant to define a notion of "exactness" only relative to some already (implicitly) given set of 's? If that is so, it seems rather unfair to blame on the innocent a "failure" that is wholly caused by your choice of 's. – Henning Makholm ( talk) 21:56, 4 April 2010 (UTC)
The only place I've ever seen this talk of "exact" vs "inexact" differentials is in the study of thermodynamics. However, I only saw this way of framing it in my book targeting chemists. In Sandler's "Chemical, Biochemical, and Engineering Thermodynamics," Stanley Sandler dispenses with the notion altogether. Personally the idea of talking about "differentials" makes me uncomfortable and it seems less than rigorous, and frankly in thermodynamics you don't need them, just plain old calculus.
Rather than talking about some infinitesimal change , specify a control volume and talk about the accumulation of internal energy in time, . To avoid the added complexity of energy associated with flow of mass, assume it's a closed system, and you can write the first law as , where Q represents the rate of heat flowing into the system. Similarly, a closed-system entropy balance is written as , with the second law stating that in a reversible process, and in an irreversible process. If the process is reversible, i.e. no entropy is generated, then you can solve for Q and insert into the first law to obtain , the familiar fundamental equation for internal energy expressed using derivatives that actually make sense in the context of calculus. Also, you can evaluate the change in internal energy of the system by simply invoking the 2nd fundamental theorem of calculus, so
I think this formulation is much better, as you don't have to talk about such things as δQ and δw, everything just makes sense both physically and mathematically. Unfortunately, the nonsense about differentials is as old as the science itself and will probably never go away. Ryan ChemE ( talk) 09:12, 29 January 2015 (UTC)
I have (almost) rewritten the article to make it more accessible to non-engineers (I am one myself though).
I have shortened the description part (it was way too long and thereby confusing). Also the part with integrals was quite confusing:
This expression really does not make sense ("it is not even wrong"). Assuming that a and b are the initial and final states of the system, the point of the inexact differential is that you cannot get a unique primitive f. Rewriting in variables of the first principle of thermodynamics:
The reason this does not make any sense is that we cannot figure out how Q changes with the state U without knowing W. You can phrase it like there is an unsaturated degree of freedom, I suppose.
This concept is really simple, but it seems that thermodynamics textbooks are written by some scribes that love keeping everything difficult to understand. 193.175.53.21 ( talk) 11:11, 4 August 2010 (UTC)
I don't see the relevance of the Vanuatuan movie to this topic. It seems nothing more than a slightly comical (lewd) video, with a post-hoc rationalisation. — Preceding unsigned comment added by 129.67.105.62 ( talk) 13:18, 9 June 2011 (UTC)
This article looks weird. What is the exact mathematical definition of inexact differentia? –– 虞海 (Yú Hǎi) ✍ 17:45, 17 January 2012 (UTC)
It seems to me that the temperature as an integrating factor is a neutral factor in this business. What makes the entropy an exact differential is the specification of the path of integration: a reversible path (Q_rev). With the specification of a path all inexact differentials become exact. Am I totally missing the point or am I right? 109.93.162.246 ( talk) 11:31, 4 January 2014 (UTC)
I would like to propose that the very idea of talking about thermodynamics in the context of differential changes makes absolutely no sense. It may very well be how the concepts are discussed in textbooks (some textbooks), but it forces you into twisted contortions of logic and to abandon the rigorous rules of mathematics. That is why there are so many mathematicians on this talk page who have no idea what the heck we are talking about here. When the mathematics we use to try to describe the physics of reality baffles mathematicians, that should be a red flag.
The first problem we need to address is: heat does not have units of energy, it has units of energy/time. Heat is a flow of energy. It doesn't make sense to talk about an infinitesimal amount of heat δQ. We can certainly talk about a heat flux q, which has units of energy/(time*area), which is what appears in Fourier's Law of Heat Conduction . Or the same quantity appears in Newton's Law of Cooling . We can then get a total heat flow by multiplying by the area across which heat flows, or if the temperature gradient varies we can chop the area up into small pieces and apply the Riemann sum definition of an integral. The point is it makes no sense to talk about an infinitesimal amount of heat energy which we would somehow "integrate" through state space in a path-dependent way. We should change our perspective and understand that heat is simply a flow of energy into or out of a system.
The people who talk about trying to find the change in energy of a system by integrating the "inexact differential" δQ are taking a convoluted look at the problem. Internal energy and enthalpy are state functions - all you need to know is where you start and where you end. You can then perform an energy balance and recognize that energy is conserved, so if the system has more energy than it started with, it must have come from somewhere. Equivalently, if you have less energy than you started with, it must have gone somewhere. You don't even have to use the normal variables found in the fundamental equations. For example, if I want to know what is the change in internal energy of a material if I take it from one temperature and pressure to another, I can perfectly well do that.
Formally, you can simply integrate the above equation with respect to time to find the change in U from t1 to t2. It would probably be easier, however, to invoke the substitution rule and integrate the first term with respect to temperature and the second with respect to pressure. You can also view this as an example of the fundamental theorem of line integrals, and break this into a path of constant pressure followed by a path of constant temperature for easy computation. (As an aside, the above equation also proves that the internal energy of an ideal gas only depends on temperature) But if an energy balance shows that the change in the internal energy of a system is equal to the heat flow, now we also know what the heat flow has to have been, without having to figure out what in the world it means to integrate an inexact differential. I am confident that any mathematician would look at the above equations and find them perfectly understandable and justified rigorously, and I would definitely welcome any feedback to the contrary! Just my humble opinion, but I think all the thermodynamics pages that talk in terms of exact and inexact differentials should be completely recast in terms of balance equations. This framework is easier to understand physically and mathematically rigorous. Ryan ChemE ( talk) 00:06, 3 February 2015 (UTC)
There are a few issues with the "Heat and work" section under "Examples." The first sentence, that a fire requires heat, fuel, and an oxidizing agent, is more related to the "fire triangle" concept in fire safety, and I think is off topic here. Also the discussion regarding activation energy is misleading. The activation energy of a reaction has nothing to do with energy flows into or out of the system. Activation energy simply allows one to relate the speed of a reaction with the system's temperature, and any reaction will occur (however slowly) at any finite temperature, so long as it is thermodynamically favorable. Heat is not consumed in a reaction to overcome the activation energy, it would only be transferred to maintain the temperature of a reaction with a positive enthalpy change of reaction. The reason you need to add heat (or work to generate friction) is because you need to raise the temperature high enough to allow the reaction to proceed at a high enough rate to sustain itself. The physical law relating reactions and activation energy to temperature is the Arrhenius equation: where k is the reaction rate constant. Finally, the last sentence about integration needing to account for the path taken - what integration? What is being integrated here, and with respect to what? Ryan ChemE ( talk) 14:00, 27 June 2015 (UTC)
There were several incorrect descriptions of the first part of the Carnot cycle. I made changes, but the section is rendered not very coherent. I hope someone can clarify that section. Dhh787 ( talk) 18:25, 12 July 2024 (UTC)