Читать книгу Geochemistry - William M. White - Страница 47

A theorem of mathematics states that any inexact differential that is a function of only two variables can be converted to an exact differential. dW is an inexact differential, and dV is an exact differential. Since dW_rev = −PdV, dW_rev can be converted to a state function by dividing by P since

(2.56)

and V is a state function. Variables such as P that convert nonstate functions to state functions are termed integrating factors. Similarly, for a reversible reaction, heat can be converted to the state function entropy by dividing by T:

(2.57)

Thus temperature is the integrating factor of heat. Entropy is a state function and therefore an exact differential. Therefore, eqn. 2.57 is telling us that although the heat gained or lost in the transformation from state 1 to state 2 will depend on the path taken, for a reversible reaction the ratio of heat gained or lost to temperature will always be the same, regardless of path.

If we return to our example of the combustion of gasoline above, the second law also formalizes our experience that we cannot build a 100% efficient engine: the transformation from state 1 to state 2 cannot be made in such a way that all energy is extracted as work; some heat must be given up as well. In this sense, the second law necessitates the automobile radiator.

Where P–V work is the only work of interest, we can combine the first and second laws as:

The implication of this equation is that if equilibrium is approached at prescribed S and V, the energy of the system is minimized. For the specific situation of a reversible reaction

where dS = dQ/T, this becomes

(2.58)

This expresses energy in terms of its natural or characteristic variables, S and V. The characteristic variables of a function are those that give the simplest form of the exact differential. Since neither T nor P may have negative values, we can see from this equation that energy will always increase with increasing entropy (at constant volume) and that energy will decrease with increasing volume (at constant entropy). This equation also relates all the primary state variables of thermodynamics, U, S, T, P, and V. For this reason, it is sometimes called the fundamental equation of thermodynamics. We will introduce several other state variables derived from these five, but these will be simply a convenience.

By definition, an adiabatic system is one where dQ = 0. Since dQ_rev/T = dS_rev (eqn. 2.52), it follows that for a reversible process, an adiabatic change is one carried out at constant entropy, or in other words, an isoentropic change. For adiabatic expansion or compression, therefore, dU = –PdV.