work and heat

Heat and work are both components of what is known as the internal energy.  The first law of thermodynamics states that the total change of internal energy of a has a work and heat component: 

dU=dQ+dW where Q = heat, W = work$dU=dQ+dW\text{ where Q = heat, W = work}$

The work can be divided into many components. The mechanical work components involve:: 

• The hydrostatic pressure. -PdV$\text{-PdV}$ Or just the pressure.
• The surface tension and area. γdA$\gamma dA$
• The magnetic field and magnetization. H.dM$H.dM$

And etc. All of these are just products of the intensive and extensive variables. There is also the chemical work component: 

μdN ,where μ=chemical potential and$\mu dN\text{ ,where }\mu =\text{chemical potential and}$ N=number of mols$N=\text{number of mols}$ 
 
The second law of thermodynamics deals with the heat component, by relating it to entropy: 

dS≥dQT$dS\ge \frac{dQ}{T}$ 

Hence, if we rewrite the expression for the internal energy with the new expression for the change in heat and change in work: 

dU=TdS−PdV+μdN$dU=TdS-PdV+\mu dN$

(That is the change in U for a chemical reaction. Brian Bi's expression for U is the more complete definition) 

Heat is intrinsically related to the change in entropy and temperature: 

T=(∂U∂S)V,N

OR

 The original approaches to classical thermodynamics were to a large extent engineering approaches. They were based largely on the study of closed, cyclic heat engines.

That is to say, they were based on an empirical observation that it was possible in general to transfer something that was called "heat" to some system, and that it was easily possible to measure the amount of heat that was transferred. Heat was imagined at first to be some kind of fluid, the "caloric" fluid, and people thought that this fluid should be conserved and that some fixed amount of heat fluid should be contained within different bodies at different temperatures. Corresponding to this conception such things as "heat capacities" were defined.

The principle of conservation of energy was also known, in the form that the change in internal energy of a body should be equal to the work extracted.

This state of affairs led to an apparent contradiction, however, for if heat, having the same units as energy, were actually identified with internal energy, then it was easy to see that one could not always obtain all of the heat that was transferred to a body back in the form of useful work.

The amount of useful work one could extract from a system to which heat was transferred turned out to depend strongly on the precise process by which the heat was transferred and the work was done.

This idea that heat was a fundamental quantity in thermodynamics led to the formulations by Kelvin of the notions of absolute temperature, and the second law of thermodynamics in terms of the behavior of closed cycle heat engines. Other crucial contributors to this approach were of course Thompson and Carnot.

In their approach, the engineering approach, heat, energy, work and temperature were the fundamental quantities.

Entropy was a derived and rather mysterious quantity, and the absolute temperature played the role of an integrating factor for the heat transfer, the result of that integration yielding the change in entropy of a system.

However, at the turn of the twentieth century classical thermodynamics was given a far more elegant, and in my own view a far clearer axiomatic formulation by Constantine Carathéodory.

I will not go into many details, but I would advise anyone interested in the development of thermodynamics to study his formulation of the subject. He removed the emphasis on cycles and heat engines completely and derived the whole structure from just two axioms, plus a third relating to temperature which suspiciously resembles one of Euclid's axioms in plane geometry. He replaced the notion of a closed cycle with the notion of a reversible/irreversible process, reversible being simply a synonym for adiabatic.

Let me state the axioms as Carathéodory did. For a multiphase system, during an adiabatic process, the change in internal energy and the work extracted are related by:

Uf−Ui+W=0$U_f-U_i+W=0$

Second he states, very originally, that 

"In any neighborhood of any equilibrium state of a multi phase system there exist states of the system that are not reachable by any adiabatic process."

This formulation of thermodynamics doesn't ever involve heat in the first place. It involves instead temperature, work, internal energy, and reversible versus irreversible processes, and so it is much more appealing. It seems to correspond much better to what is actually observed.

Entropy emerges very naturally in this approach to the theory - it is a quantity that defines the level curves (or, rather, level surfaces) along which the macroscopic variables describing the equilibrium states of a system move during an adiabatic process - the so-called adiabats. These are intersected by another set of curves called the isotherms, or curves of constant temperature.

Heat is reduced, as it must be, to the status of a derived quantity, not directly related to the internal state of a thermodynamic system in equilibrium.

Carbohydrate's approach is very beautiful, I think, and it exposes the fundamentally geometric nature of classical thermodynamics.

Now of course, one gains a far, far clearer understanding of entropy when one adopts the microscopic approaches of Maxwell, Boltzmann and Gibbs.

But it should not be imagined that classical thermodynamics can't be put on a secure mathematical foundation, or that all definitions of entropy are circular without statistical mechanics.

I think that it would be a mistake to imagine that.

Entropy is actually a fundamental quantity in equilibrium thermodynamics - it is only an accident of history and of the complex nature of macroscopic systems that obscured this fact for so long.

The bottom line: there is no need at all to mention heat when formulating thermodynamics.