# Circular argument in the first law of thermodynamics ?

According to Caratheodory, the first law of thermodynamics (dE=dQ-dW) could be derived from the definition of heat (dQ=dE-dW), whether this form a circular argument or tautology ? How to clarify the confusion between the first law of thermodynamics and the definition of heat，and capture the true meaning of the first law of thermodynamics?

Mapes
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It's not circular if the definition of heat is taken to be axiomatic (provided without proof as a starting point for defining the rules of a system). It would be circular if the definition of heat were simply acquired from the first law.

The first law is phenomenological; it captures the observation that energy never seems to arise on its own in the middle of nowhere. Note that the work part can contain a variety of terms, though: chemical, mechanical, electrical, magnetic, etc.

Heat is defined by specifying what degrees of freedom of the system are going to be treated in a statistical way. Usually we specify one or two degees of freedom that are not treated statistically. The energy associated with changing these degrees of freedom is then, by definition, work.

But in principle, the partitioning of energy change between heat and work is completely arbitrary. If you keep track of the exact state of a system then the entropy is zero and unitary time evolution then causes the entropy to stay zero at all times. This is called the "fine grained entropy". In practice we use what is called the "coarse grained entropy", which is larger than zero as a result of not keeping track of all the degrees of freedom in the system.

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