Maxwell_relations_and_measurable_quantities

Maxwell_relations_and_measurable_quantities - © M. S....

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Unformatted text preview: © M. S. Shell 2009 1/9 last modified 10/13/2009 Maxwell relations and measurable quantities ChE210A We saw previously that we could derive other thermodynamic potentials using Legendre transforms. These allowed us to switch independent variables and gave rise to alternate, but ultimately equivalent, equilibrium conditions: gG¡,¢,£¤ entropy fundamental equation maximum at equilibrium ¡Gg,¢,£¤ energy fundamental equation minimum at equilibrium ¥Gg,¦,£¤ enthalpy minimum at equilibrium §G¨,¢, £¤ Helmholtz free energy minimum at equilibrium ©G¨,¦,£¤ Gibbs free energy minimum at equilibrium Here, we will discuss some of the mathematical properties of these functions, and the conse- quences they have for relationships between thermodynamic variables. These relationships will allow us to connect quantities that are difficult to measure directly—like the entropy and the chemical potential—to variables that we can easily access using experiments. Maxwell relations Recall from calculus that one of the properties of well-behaved multivariate functions is that second derivatives don’t depend on the order in which they are taken, ª « ¬ ª­ª® ¯ ª « ¬ ª®ª­ Or, writing out the derivatives explicitly, ª ª­ °± ª¬ ª® ² ³ ´ µ ¯ ª ª® °± ª¬ ª­ ² µ ´ ³ This fact has very important consequences for the potentials that we have developed, since their derivatives involve other thermodynamic quantities. As an example, let us consider the energy equation, ª¡ ¯ ¨ªg ¶ ¦ª¢ · ¸ª£ © M. S. Shell 2009 2/9 last modified 10/13/2009 We will examine the second derivative g G ¡ g¢g£ at constant ¤ conditions: ¥ ¦ § ¥¨¥© ª ¥ ¦ § ¥©¥¨ « ¥ ¥¨ ¬­ ¥§ ¥© ® ¢,¯ ° £,¯ ª ¥ ¥© ¬­ ¥§ ¥¨ ® £,¯ ° ¢,¯ Substituting for the inner derivatives using the fundamental equation above, ¥ ¥¨ ±²³´ £,¯ ª ¥ ¥© ±µ´ ¢,¯ Or, after simplifying the notation, ²­ ¥³ ¥¨ ® £,¯ ª ­ ¥µ ¥© ® ¢,¯ There is, therefore, a relationship between the derivatives of ³ and µ that simply emerges from the fact that these two quantities are related by a common second derivative of a thermody- namic potential. Such relationships, based on second derivatives of potentials, are called Maxwell relations, after James Maxwell, one of the early founders of modern thermodynamics and electromagnetism. There are, in fact, many Maxwell relations, depending on which poten- tial is used and which pair of independent variables is examined. Here is another example, based on the Gibbs free energy: ¥¶ ª ²¨¥µ · ©¥³ · ¸¥¤ ¥ ¦ ¶ ¥³¥¤ ª ¥ ¦ ¶ ¥¤¥³ « ¥ ¥³ ¬­ ¥¶ ¥¤ ® ¹,º ° ¹,¯ ª ¥ ¥¤ ¬­ ¥¶ ¥³ ® ¹,¯ ° ¹,º Substituting for the inner derivatives, ­ ¥¸ ¥³ ® ¹,¯ ª ­ ¥© ¥¤ ® ¹,º Therefore, the essence of this procedure is the following: pick a potential and a second deriva- tive involving two of its independent variables, and then substitute first derivative definitions to...
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This note was uploaded on 12/29/2011 for the course CHE 210a taught by Professor Staff during the Fall '08 term at UCSB.

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Maxwell_relations_and_measurable_quantities - © M. S....

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