Typical values of a and P are listed in Table 2.1. For ideal gases, substitution of Equation (2.1) into Equations (1.2) shows that ot^ = 1/7 and Pgas = Hp. Also given in Table 2.1 are specific volumes of water and typical solids.
With the above definitions of a and P, Equation (2.18) becomes:
Because a and (3 are approximately independent of temperature and pressure, the above equation can be integrated while holding these properties constant In addition, a and P are small, so that volume changes due to changes in p and T are also small. Therefore, the term v on the right-hand side of Equation (2.18a) can also be taken as a constant and integration from initial state o to final state 1 yields:
For greater accuracy in determining volume changes of condensed phases, allowance must be made for the variation of a and P with temperature and pressure. In this case, integration of Equation (2.18a) between initial state zero (0) and final state 1, still assuming small changes in v, yields:
Even if a and p are known functions of T and p, the integrals of Equation (2.18c) require that a path between states 0 and 1 be specified. That is, the first
* Not to be confused with the nonideality measure of the same name in Equation (2.4).
integral on the right requires that the variation of p with T be known, as does the second integral. Even if the integrals on the right-hand side of Equation (2.18c) are path-dependent, their difference is not because Av represents a change in a thermodynamic property v. Thus, the relatively unimportant problem of calculating volume changes of a condensed phase when the properties involved are not constants serves as an illustration of a much more fundamental and general feature of thermodynamics: the independence of property changes on the path taken from the initial to the final state. This is explored in more detail in the following example.
Example: Liquid water at 20°C and atmospheric pressure (0.1 MPa) is heated to 30°C and 100 MPa. The detailed dependences of a and p on T and p are not known, but average values of these properties over ranges of one variable with the other held constant are given. With this sort of information available, natural choices of the path between the initial and final states are shown on the drawing below.
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Do we really want the one thing that gives us its resources unconditionally to suffer even more than it is suffering now? Nature, is a part of our being from the earliest human days. We respect Nature and it gives us its bounty, but in the recent past greedy money hungry corporations have made us all so destructive, so wasteful.