Notices
Results 1 to 15 of 15

Thread: Inexact differentials

  1. #1 Inexact differentials 
    Forum Freshman
    Join Date
    Jul 2014
    Posts
    37

    both A and B give the area under the integral curve for y(x) where y is zero at x = 0 and dy/dx is always positive.
    dA = y dx
    dB =
    We can call that area also U but
    dU = where stands for inexact differential. Inexact differentials are used in thermodynamics where only the sum of the two differentials have meaning, not their individual values. The individual values are path dependant. B stands for mechanical and A for thermal energy.


    Last edited by Rudiger von Massow; August 16th, 2014 at 10:37 AM.
    Reply With Quote  
     

  2.  
     

  3. #2  
    KJW
    KJW is offline
    Forum Professor
    Join Date
    Jun 2013
    Posts
    1,507
    Quote Originally Posted by Rudiger von Massow View Post
    Inexact differentials are used in thermodynamics where only the sum of the two differentials have meaning, not their individual values.
    This appears to me to be a misunderstanding of the nature of inexact differentials. An exact differential is a differential of a function of the variables of the relevant space (in thermodynamics, the state). A path-integral evaluates to the difference of the function at the two endpoints and is independent of the path itself. The key issue is the existence of the function itself. One may have a function over paths in a space but the function over the space itself does not exist. This would be the case if the path-integral of the corresponding differential evaluates to different values for different paths between the same two endpoints. A simple example is distance. The distance between two points depends on the path between those two points. Therefore, there is no distance function over space. But one can still have a differential of distance for any given path. Such a differential will be inexact.

    If such a function exists, then the chain rule says:



    and the differential is exact. But for the differential , if the partial derivatives , , etc do not exist, then is inexact.


    There are no paradoxes in relativity, just people's misunderstandings of it.
    Reply With Quote  
     

  4. #3  
    Forum Freshman
    Join Date
    Jul 2014
    Posts
    37
    The two differentials dA and dB are both limited to one dimension, so the other partial derivatives are missing. That, according to you, makes them inexact.
    Reply With Quote  
     

  5. #4  
    KJW
    KJW is offline
    Forum Professor
    Join Date
    Jun 2013
    Posts
    1,507
    Quote Originally Posted by Rudiger von Massow View Post
    The two differentials dA and dB are both limited to one dimension, so the other partial derivatives are missing. That, according to you, makes them inexact.
    Well, no. I was assuming the space in which the differentials exist was at least two-dimensional. This is the usual context for discussing inexact differentials. One normally doesn't refer to the definite integral of a function of the real numbers as a "path integral". Furthermore, the space of states in thermodynamics is typically two-dimensional (pressure and temperature).

    What would make dA inexact in any case is the non-existence of A (and similarly for dB).

    A point I was attempting to correct with my original reply is the suggestion in the opening post that the sum of two inexact differentials is exact. In other words, the first law of thermodynamics is not the result of a property of inexact differentials.
    There are no paradoxes in relativity, just people's misunderstandings of it.
    Reply With Quote  
     

  6. #5  
    Forum Freshman
    Join Date
    Jul 2014
    Posts
    37
    Sorry, I have to go to thermodynamics to prevent confusing myself and others. In thermodynamics
    dU =
    is a well accepted formula supported by steam tables. Putting the steam into a tube with movable plug I can plot U versus entropy and column length using steam table data in the region of steam stability.
    Sorry, I don't know how to include a figure.
    I can read of that figure that:



    This is based the equation above. It means that mechanical energy only alters the temperatures, heat requirements of entropy differentials, of the steam, not the entropy. Heat addition increases entropy. Thus mechanical energy and heat are inexact quantities since their differentials are inexact.
    I tried to get confirmation from a mathematician.
    Reply With Quote  
     

  7. #6  
    Forum Freshman
    Join Date
    Jul 2014
    Posts
    37
    I admit the double integral

    is problematic, since
    has the form of a curved splinter with its point at the origin. Laying these next to one another until the wide ends sum up to T makes:

    That interpretation may not be quite kosher. What do you think?
    By the way: Can I send you an email with the figures to demonstrate ?
    The in post 5 should also be
    Last edited by Rudiger von Massow; August 24th, 2014 at 05:36 PM.
    Reply With Quote  
     

  8. #7  
    Forum Freshman
    Join Date
    Jul 2014
    Posts
    37
    It is so simple: The energy of a steam is

    That integral extended to more S is
    dQ = TdS if no work is exchanged
    That integral extended to higher T is
    dW = FdX = if no heat is exchanged.
    Both work W and heat Q are inexact if any time during the process the other energy is also varied:
    dU =

    U =
    can be reached by any combination of one or the other energy increases.
    Last edited by Rudiger von Massow; August 27th, 2014 at 10:09 PM.
    Reply With Quote  
     

  9. #8  
    KJW
    KJW is offline
    Forum Professor
    Join Date
    Jun 2013
    Posts
    1,507
    Quote Originally Posted by Rudiger von Massow View Post
    In thermodynamics
    dU =
    is a well accepted formula supported by steam tables.
    Yes. That is the First Law of Thermodynamics. And you are correct in saying that dQ and dW are inexact and dU is exact. But you seem to be suggesting that dU is exact because dQ and dW are inexact, implying that the sum of two inexact differentials is exact. This is not mathematically true. The First Law of Thermodynamics is a law of physics and not the mathematical consequence of properties of inexact differentials.


    Quote Originally Posted by Rudiger von Massow View Post
    Thus mechanical energy and heat are inexact quantities since their differentials are inexact.
    This is a misrepresentation of the terms "exact" and "inexact" as they are applied to differentials.


    It's worth noting that in thermodynamics, one defines dQreversible and dWreversible as the corresponding differentials along paths in state space that are reversible (the direction can be reversed by an infinitesimal change of state). The restriction to reversible paths allows functions of state to be defined from the differentials and therefore these differentials are exact. They are used to define functions of state such as enthalpy, entropy, Gibbs free energy, etc.
    There are no paradoxes in relativity, just people's misunderstandings of it.
    Reply With Quote  
     

  10. #9  
    Forum Freshman
    Join Date
    Jul 2014
    Posts
    37
    In my above arguments I have always implied that dQ and dW are reversible. The first law dU = is applicable to reversible and irreversible processes. A steam locomotive can operate even if the process may be almost reversible. Even if reversible the change in mechanical energy and heat are path dependant and therefore convertible.
    Enthalpy, the Helmholtz function and the Gibbs function are functions which result from the partial or total application of Legendre transformation on energy. They do not enter the above discussion.
    Reply With Quote  
     

  11. #10  
    Forum Freshman
    Join Date
    Jul 2014
    Posts
    37
    The definition of 'inexact', though that is poor nomenclature, is that it applies to a pair such that dU = The two individually are variable and in their variation define a different path.
    Reply With Quote  
     

  12. #11  
    Forum Freshman
    Join Date
    Jul 2014
    Posts
    37
    The first Law of thermodynamics is the result of numerous experimental data. If mathematics can not handle it, that is too bad. I would like to prove that if are expressed as:
    and

    then they are inexact differentials and follow all paths possible in dU = . I can do that graphically but I do not know how to transfer a drawing to a reply.
    Reply With Quote  
     

  13. #12  
    Moderator Moderator
    Join Date
    Jun 2005
    Posts
    1,620
    I am not at all sure that discussion of Carnot cycles has a real place in this sub-forum, especially as you are committing one the major sins in marhetmatics - failure to declare your symbols. Naughty!!

    However I do agree with this
    Quote Originally Posted by KJW View Post
    This is a misrepresentation of the terms "exact" and "inexact" as they are applied to differentials.
    Let's see. Suppose a differential p-form. Then if one says this is a closed differential (form)

    Recall that the operator maps p-forms onto (p+1)-forms.

    So if, for some and a p-form there exists a (p - 1)-form such that one calls this an exact differential (form). Obviously for an inexact differential form this is not true

    Clearly all exact differential forms are closed - . This is the Poincare Lemma

    The converse is true - for every closed p-form there exists a (p+1)-form - only if the space over which these differential forms are defined is topologically trivial (or only locally otherwise).

    The implications are profound, especially in applications
    Reply With Quote  
     

  14. #13  
    Forum Freshman
    Join Date
    Jul 2014
    Posts
    37
    We do not have to be scared of 'inexact differentials' defining an exact quantity. My joined bank account has two inexact differentials.
    . As the money is paid in it looses its identity and when it is taken out it gets a new unrelated identity in my or my wife's wallet. This is only a very simplified comparison. The BA depends only on the sum, not on the individual values.
    Reply With Quote  
     

  15. #14  
    Forum Freshman
    Join Date
    Jul 2014
    Posts
    37
    It is interesting that the most important and best proven law of thermodynamics gets poohooed by two mathematicians. Interesting.
    Reply With Quote  
     

  16. #15  
    exchemist
    Join Date
    May 2013
    Location
    London
    Posts
    3,342
    Quote Originally Posted by Rudiger von Massow View Post
    It is interesting that the most important and best proven law of thermodynamics gets poohooed by two mathematicians. Interesting.
    I don't think they are saying any such thing. And, from the way he writes, I would guess KJW is a very competent physical chemist or physicist.
    Reply With Quote  
     

Similar Threads

  1. Does economics' inexact nature invalid it as a science?
    By sarnamluvu in forum Business & Economics
    Replies: 39
    Last Post: November 24th, 2013, 09:43 AM
  2. dumb question about differentials...
    By Stranger in forum Mathematics
    Replies: 13
    Last Post: January 31st, 2010, 06:43 PM
Tags for this Thread

View Tag Cloud

Bookmarks
Bookmarks
Posting Permissions
  • You may not post new threads
  • You may not post replies
  • You may not post attachments
  • You may not edit your posts
  •