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(a) Argue that the entropy S has the form S = f(l) + g(E - U (l)), where f and g are, so far, arbitrary functions. [Hint: For a given arrangement of folds with a given length f, the number of ways of arranging the system in phase space just depends on the total kinetic energy of the system.] (b) Show that the quantity E-U() is a function of the temperature T only. (c) Argue that S decreases when the rubber band is stretched isothermally.

(a) Argue that the entropy S has the form S = f(l) + g(E - U (l)), where f and g are, so far, arbitrary functions. [Hint: For a given arrangement of folds with a given length f, the number of ways of arranging the system in phase space just depends on the total kinetic energy of the system.] (b) Show that the quantity E-U() is a function of the temperature T only. (c) Argue that S decreases when the rubber band is stretched isothermally.

Introduction to Chemical Engineering Thermodynamics
Introduction to Chemical Engineering Thermodynamics
8th Edition
ISBN:9781259696527
Author:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart
Publisher:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart
Chapter1: Introduction
Section: Chapter Questions
Problem 1.1P
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A rubber band is made out of long molecular chains ("polymers") which are folded back over themselves in an arbitrary way, as indicated by the examples shown. Both of these examples have 16 links in the chain and a net length of six links, but they are folded up differently. The potential energy of the rubber band is due to the (weak) attraction between links which are lying next to cach other because of the folding. To first approximation, the potential energy in a particular state does not depend on the particular arrangement of folds but only on the net length of the chain and the total number of links. Let l be the length of the rubber band; U(e), its potential energy; E, its mean total energy; and t, its mean tension. (a) Argue that the entropy S has the form S = f(e) + g(E – U(0)), where f and g are, so far, arbitrary functions. [Hint: For a given arrangement of folds with a given length &, the number of ways of arranging the system in phase space just depends on the total kinetic energy of the system.] (b) Show that the quantity E – U(t) is a function of the temperature T only. (c) Argue that S decreases when the rubber band is stretched isothermally. (d) Show that the rubber band must get warmer when it is stretched adiabatically. You may assume that the specific heat is “normal," i.e., that the heat capacity at fixed length Ct, is positive. (e) Show that the tension t increases when the temperature T is increased at fixed &. [Hint: Use a Maxwell relation with the replacements V + l,p → -t.]
Transcribed Image Text:A rubber band is made out of long molecular chains ("polymers") which are folded back over themselves in an arbitrary way, as indicated by the examples shown. Both of these examples have 16 links in the chain and a net length of six links, but they are folded up differently. The potential energy of the rubber band is due to the (weak) attraction between links which are lying next to cach other because of the folding. To first approximation, the potential energy in a particular state does not depend on the particular arrangement of folds but only on the net length of the chain and the total number of links. Let l be the length of the rubber band; U(e), its potential energy; E, its mean total energy; and t, its mean tension. (a) Argue that the entropy S has the form S = f(e) + g(E – U(0)), where f and g are, so far, arbitrary functions. [Hint: For a given arrangement of folds with a given length &, the number of ways of arranging the system in phase space just depends on the total kinetic energy of the system.] (b) Show that the quantity E – U(t) is a function of the temperature T only. (c) Argue that S decreases when the rubber band is stretched isothermally. (d) Show that the rubber band must get warmer when it is stretched adiabatically. You may assume that the specific heat is “normal," i.e., that the heat capacity at fixed length Ct, is positive. (e) Show that the tension t increases when the temperature T is increased at fixed &. [Hint: Use a Maxwell relation with the replacements V + l,p → -t.]
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