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Chapter 5: Q 5.56 (page 191)
Prove that the entropy of mixing of an ideal mixture has an infinite slope, when plotted vs. x, at x = 0 and x= 1.
Therefore, it is proved that the entropy of mixing of an ideal mixture has an ideal slope.
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Repeat the preceding problem with T/TC=0.8
As you can see from Figure5.20,5.20,the critical point is the unique point on the original van der Walls isotherms (before the Maxwell construction) where both the first and second derivatives ofPPwith respect toVV(at fixedTT) are zero. Use this fact to show that
When carbon dioxide "dissolves" in water, essentially all of it reacts to form carbonic acid, H2CO3:
The carbonic acid can then dissociate into H* and bicarbonate ions,
(The table at the back of this book gives thermodynamic data for both of these reactions.) Consider a body of otherwise pure water (or perhaps a raindrop) that is in equilibrium with the atmosphere near sea level, where the partial pressure of carbon dioxide is 3.4 x 10-4 bar (or 340 parts per million). Calculate the molality of carbonic acid and of bicarbonate ions in the water, and determine the pH of the solution. Note that even "natural" precipitation is somewhat acidic.
Problem 5.35. The Clausius-Clapeyron relation 5.47 is a differential equation that can, in principle, be solved to find the shape of the entire phase-boundary curve. To solve it, however, you have to know how both L and V depend on temperature and pressure. Often, over a reasonably small section of the curve, you can take L to be constant. Moreover, if one of the phases is a gas, you can usually neglect the volume of the condensed phase and just take V to be the volume of the gas, expressed in terms of temperature and pressure using the ideal gas law. Making all these assumptions, solve the differential equation explicitly to obtain the following formula for the phase boundary curve:
P= (constant) x e-L/RT
This result is called the vapour pressure equation. Caution: Be sure to use this formula only when all the assumptions just listed are valid.
Functions encountered in physics are generally well enough behaved that their mixed partial derivatives do not depend on which derivative is taken first. Therefore, for instance,
where each is taken with fixed, each is taken with fixed, and is always held fixed. From the thermodynamic identity (for) you can evaluate the partial derivatives in parentheses to obtain
a nontrivial identity called a Maxwell relation. Go through the derivation of this relation step by step. Then derive an analogous Maxwell relation from each of the other three thermodynamic identities discussed in the text (for ). Hold fixed in all the partial derivatives; other Maxwell relations can be derived by considering partial derivatives with respect to , but after you've done four of them the novelty begins to wear off. For applications of these Maxwell relations, see the next four problems.
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