Consider the internal energy to be a function of temperature and specific volume, or u(T,v). The total differential is:
The coefficient of dT is by definition the heat capacity at constant volume, CV. The coefficient of dv is obtained from the fundamental differential du = Tds - pdv by dividing by dv and holding T constant:
Where the partial derivative involving the entropy has been replaced by the Maxwell relation of Eq (6.19). The final result for du is:
This form is applicable to nonideal gases for which an equation of state p(T,v) is available. For solids, replacement of (ôp/ôT)v by a/ß (see Eq (6.8)) provides a more useful form.
The relation analogous to Eq (6.21a) for the enthalpy is obtained from the total differential of h(T,p) and following a procedure similar to that used above for the internal energy. The fundamental differential dh = Tds + vdp is divided by dp at constant T to arrive at (ôh/ôp)T in terms of (ds/dp)T, which is then eliminated by the Maxwell relation given by Eq (6.20). The result is:
For solids, av is substituted for the partial derivative.
For an ideal gas, the bracketed terms in Eqs (6.21b) and (6.22b) are identically zero. For solids, the second terms on the right hand sides of these equations are essential if the process is isothermal and involves changes of v or p.
Other derivatives of the internal energy and enthalpy can be obtained by manipulations of the fundamental differentials of du and dh different from those employed above to give Eqs (6.21a) and (6.22a). Problem 6.2 applies this method to the derivatives (öu/öp)T and (öu/öT)p, wherein the "off-natural" variable p replaces the "natural" variable v associated with u (see Sect. 6.2).
For an ideal gas, internal energy and enthalpy are independent of specific volume or pressure. For a nonideal gas such as one obeying the Van der Waals equation of state, both u and h depend on v, as shown in Problem 6.4
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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.