In textbooks, heat is usually defined as the energy transfer due to temperature difference.
However, we don't know what temperature is in the first place. I think it's better to define heat first and then define temperature $1/T$ as the integrating factor so that one has
$$\oint\frac{\delta Q}{T}=0$$
and hence define entropy as the potential
$$\Delta S=\int\frac{\delta Q}{T}$$
So my problem is I cannot use temperature or entropy when defining heat.
So what is the definition of heat?
I think my problem can be solved if one can define either heat or work without mentioning temperature and entropy. For instance, if we can somehow define work, then heat can be define as the energy change not in the form of work. And vice versa.
So in summary, my question is how to define work or heat without mentioning temperature and entropy in thermodynamics (without referring to statistical mechanics)?
Best Answer
Essentially, Callen gave the following description in the first chapter of his book, "An Introduction To Thermostatics":
There are say $10^{23}$ atoms in a system and each atom can be described by 3 coordinates. Now out of all possible combinations of the motion(also called modes) of these atoms, only a few particular combinations of atomic coordinates that are essentially time-independent are macroscopically observable. For example, a combination can be in which all the atoms are moving in one direction. Then the system as a whole moves in that direction and that is macroscopically observable. Or perhaps, the atoms are moving away from each other such that the volume of the system increases.
Thus macroscopic observations sense only coarse spatial averages of atomic coordinates. He further says that thermodynamics describes only static states of macroscopic systems.
Finally, energy transferred via a "mechanical mode"(mechanical macroscopic coordinate like volume, stress, etc.) is called mechanical work.
Whereas, energy transfer via "hidden atomic modes" is called "heat".
Hope this helps. Refer to Callen for details. This topic is discussed in the beginning of the first chapter. Here is a link to an online version of the book:
http://keszei.chem.elte.hu/1alapFizkem/H.B.Callen-Thermodynamics.pdf