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### 2001AugQualTHERM

Course: PHYSICS 767, Spring 2012
School: Abu Dhabi University
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Word Count: 251

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Sauerwine Practice Ben for Qualifying Exams Thanks to Ben B. for this solution. Near absolute temperature T0 , the tension F in a rubber band is given by: F = aT 2 (L L0 ) where L is the stretched length, L0 is the unstretched length, and a is a constant. When L = L0 , the heat capacity at constant length is: C L = bT (a) Find the entropy S (L, T ) for temperatures T near T0 in terms of its value at T0 and L0...

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Sauerwine Practice Ben for Qualifying Exams Thanks to Ben B. for this solution. Near absolute temperature T0 , the tension F in a rubber band is given by: F = aT 2 (L L0 ) where L is the stretched length, L0 is the unstretched length, and a is a constant. When L = L0 , the heat capacity at constant length is: C L = bT (a) Find the entropy S (L, T ) for temperatures T near T0 in terms of its value at T0 and L0 . P F (T , L ) S S (T , L ) S S dS = dT + dL T L L T C S =L T T L Maxwell : F F dA = dL = SdT PdL = SdT FdL dT + T L L T S F 0 = L T T L S F = L T T L Then, F = 2aT (L L0 ) T L C dS = L dT + 2aT (L L0 )dL T Integrate, ( S = S (T0 , L0 ) + bT + aT L2 2 LL0 ) (b) If one starts at T0 and L0 and stretches the insulated rubber band quasistatically to L, what will the temperature T be of the rubber band? T= CL S T L T= b b + a L 2 LL0 ( 2 ) (c) Calculate the heat capacity at constant tension C F (L, T ) . S CF T T F F = aT 2 (L L0 ) ( F 2 = a 2T 4 L2 2 LL0 + L2 0 ( ) F2 aTL2 = aT L2 2 LL0 0 aT 3 ( ) ) S = S (T0 , L0 ) + bT + aT L2 2 LL0 = S (T0 , L0 ) + bT + 4F 2 S aL2 CF = T = T b 0 4 aT T F F2 aTL2 0 3 aT
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