Rigorous mathematical definition of quantity of energy transferred as heat
It is sometimes convenient to have a very rigorous mathematically stated definition of quantity of energy transferred as heat. Such definition is customarily based on the work of Carathéodory (1909), referring to processes in a closed system, as follows.
The internal energy UX of a body in an arbitrary state X can be determined by amounts of work adiabatically performed by the body on its surrounds when it starts from a reference state O, allowing that sometimes the amount of work is calculated by assuming that some adiabatic process is virtually though not actually reversible. Adiabatic work is defined in terms of adiabatic walls, which allow the frictionless performance of work but no other transfer, of energy or matter. In particular they do not allow the passage of energy as heat. According to Carathéodory (1909), passage of energy as heat is allowed, by walls which are "permeable only to heat".
For the definition of quantity of energy transferred as heat, it is customarily envisaged that an arbitrary state of interest Y is reached from state O by a process with two components, one adiabatic and the other not adiabatic. For convenience one may say that the adiabatic component was the sum of work done by the body through volume change through movement of the walls while the non-adiabatic partition was excluded, and of isochoric adiabatic work. Then the non-adiabatic component is a process of energy transfer through the wall that passes only heat, newly made accessible for the purpose of this transfer, from the surroundings to the body. The change in internal energy to reach the state Y from the state O is the difference of the two amounts of energy transferred.
Although Carathéodory himself did not state such a definition, following his work it is customary in theoretical studies to define the quantity of energy transferred as heat, Q, to the body from its surroundings, in the combined process of change to state Y from the state O, as the change in internal energy, ΔUY, minus the amount of work, W, done by the body on its surrounds by the adiabatic process, so that Q = ΔUY − W.
In this definition, for the sake of mathematical rigour, the quantity of energy transferred as heat is not specified directly in terms of the non-adiabatic process. It is defined through knowledge of precisely two variables, the change of internal energy and the amount of adiabatic work done, for the combined process of change from the reference state O to the arbitrary state Y. It is important that this does not explicitly involve the amount of energy transferred in the non-adiabatic component of the combined process. It is assumed here that the amount of energy required to pass from state O to state Y, the change of internal energy, is known, independently of the combined process, by a determination through a purely adiabatic process, like that for the determination of the internal energy of state X above. The mathematical rigour that is prized in this definition is that there is one and only one kind of energy transfer admitted as fundamental: energy transferred as work. Energy transfer as heat is considered as a derived quantity. The uniqueness of work in this scheme is considered to provide purity of conception, which is considered as guaranteeing mathematical rigour. The conceptual purity of this definition, based on the concept of energy transferred as work as an ideal notion, relies on the idea that some frictionless and otherwise non-dissipative processes of energy transfer can be realized in physical actuality. The second law of thermodynamics, on the other hand, assures us that such processes are not found in nature.