HW__5_solutions - Chemical Engineering Thermodynamics 141...

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Chemical Engineering Thermodynamics 141 Fall 2005 Homework 5 Solutions Problem 1. 1. This problem asks us to show how an isotherm and isobar behave on a Mollier diagram. We must realize that an isobar or isotherm is the derivative of enthalpy with respect to entropy at constant pressure or temperature, respectively. We first want to prove that isobars within the vapor phase have constant slope and constant curvature, starting with the fundamental relationship for enthalpy. Then we try to calculate these partial derivatives explicitly in terms of the quality, to find out whether the partial derivatives depend on quality. Mollier Diagram To begin with, we are asked to prove that 0 0 2 2 P P dS H d dS dH (1) 1
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Since the problem asks us to, let us start with the fundamental relationship for enthalpy: vdP TdS dH + = (2) We want things that look like relations (1). So divide (2) by dS, holding P constant: T dS dH P = (3) Since T > 0 always, we have proven the first relation. It remains to prove the second. Let’s substitute in T into the second relation now, so we get: P P dS dT dS H d = 2 2 (4) It appears we’re stuck. If you try using a Maxwell relation, it will not work because T and S are apposed. We need to recognize that we can write P P dT dS dS dT = 1 (5) Now, we can find the term on the right hand side of the equation. Let start again with (1) and divide through by dT, holding P constant: P P dT dS T dT dH = (6) By now you should be able to recognize the term on the left hand side as Cp. Rearranging the result, 0 0 2 2 = = = P P p p P P P dS dT dS H d C T C T dS dT T C dT dS (7) 2
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