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Apart of the mathematical technicallity of a inexact differential, the real meaning of the usage of inexact differentials is just to translate, mathematically, that a system can contains just Energy (internal energy) but cannot contain work and heat? In other words, inexact differentials captures this concept of "cannot contain work and heat"?

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2 Answers 2

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In other words, inexact differentials captures this concept of "cannot contain work and heat"?

Correct.

The inexact differential ($\delta$) for heat and work is meant to describe in incremental transfer of energy in the form of heat or work between a system and surroundings. A system does not "contain" heat or work. So there is no "change" in the heat and work of a system. Moreover the amount of heat and work done depends on the process (path) between two states. There is potentially a change in internal energy as a consequence of heat and work.

The exact differential ($d$) used for internal energy means a differential change in the amount of internal energy contained in a system. The change in internal energy between two states is independent of the process (path) between the states. So it only has one value between states, making it an exact differential.

Consequently in differential form we write the first law as

$$dU=\delta q-\delta w$$

Hope this helps.

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Yes. A system contains energy. Work and heat transfer are two ways in which energy can be added or removed.

So, if a system undergoes some process, the change in energy can always be split into heat transfer and work. But you can't split the system's energy into "total work" and "total heat" and say that a heat transfer is a change in the "total heat" et cetera. Why? Because it is possible to find processes where the system starts and ends in exactly the same state, but where the work and heat added are both nonzero. Their sum, however, is the change in energy and is always zero. So heat and work are properties of processes, while energy is a property of states.

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