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My background is in statistics, so physics is rather new to me. But I was reading Reichl's book on Statistical Mechanics and had a question about the first law.

So the basic idea is that the internal energy is:

$$ \mathrm{d} U=\delta Q+\delta W $$ where $Q,W$ are inexact differentials because they are path dependent. So I can see how this language of "inexact" can be confusing, for how can two inexact quantities equal an exact quantity. But it seems that other folks have answered this in part.

But my question is what constraint does this equation put on the possible functions that define $Q, W$. So imagine that Q and W were exact differentials, then the first law would already impose constraints on those two functions because they have to sum to the change in internal energy $U$. So what further constraint is imposed on the form of these two function if they are inexact, as opposed to exact? Hopefully the question make sense. Just trying to tease out the impact of their inexactness.

I might be way off in my intuition, so please correct me if I am off.

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So what further constraint is imposed on the form of these two function if they are inexact, as opposed to exact? Hopefully the question make sense. Just trying to tease out the impact of their inexactness.

Work and heat are inexact differentials because integration has to account for the path taken. There are an infinite number of possible heat and work paths between two states and only one difference in internal energy between the points. For example, I might get between two points by a constant pressure path followed by a constant volume path. Or I might start with a constant volume path followed by a constant pressure path. Both get me where I want to go, but the values for heat and work will be different because the integrals of the paths will be different. So the constraint is you need to define the specific path(s) going between the two points to determine the heat and work involved.

Once the particular heat and work paths are specified they can be integrated and thus become exact differentials.

Hope this helps.

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  • $\begingroup$ Thanks so much. Yes, this makes sense. So the constraint added by inexactness is just that the heat and work functions will depend on the path chosen. So a priori we don't know the functional form of heat and work. But once the path is chosen, then the functional form is known and we can calibrate the parameters to sum to the change in internal energy. That makes sense. Thanks for your help. $\endgroup$
    – krishnab
    Commented Jun 3, 2019 at 22:01
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    $\begingroup$ @krishnab That is a precisely correct interpretation, i.e., a priori we don't know the functional form of heat and work. $\endgroup$
    – Bob D
    Commented Jun 3, 2019 at 22:07
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    $\begingroup$ I think it also helps to mention that, no matter which path you take, the difference between the heat integral and the work integral is always the same. $\endgroup$
    – Chet Miller
    Commented Jun 3, 2019 at 22:12
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    $\begingroup$ @ChetMiller Duly noted as it is the fact that the difference is always the same is what makes the change in internal energy always the same. I will pass that on to the OP $\endgroup$
    – Bob D
    Commented Jun 4, 2019 at 0:37
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    $\begingroup$ @krishnab See Chet Miller's comment. Since $\Delta U=Q-W$ the difference between the heat added to the system and the work done by the system will always be the same regardless of the specific path(s) taken. $\endgroup$
    – Bob D
    Commented Jun 4, 2019 at 0:39

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