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Unformatted text preview: MS&amp;E 303 Fall 2003 Homework #2 Due Wednesday, September 17 1. Consider the exact differential for U ( S,V ) where dU = T dS- P dV , and its Legendre transforms H = U + PV , A = U- TS and G = U- TS + PV . For all but the first part, reduce the derivatives to the simplest form involving only the coordinates P,V,T,S , the physical parameters and (expansion coefficient and the isothermal compressibility), and the constant pressure specific heat C P (if you find C V occurs, just use the result derived in class where C P = C V + 2 V T . These problems though should not require either heat capacity derivatives.) As this is pure math, there is still no need to understand the meaning of S . (a) Write the differentials dH , dA , and dG . H = U + PV dH = T dS + V dP A = U- TS dA =- S dT- P dV H = H- TS dG =- S dT + V dP (b) A T P Use the generalized chain rule to change the path from P to V corresponding to the A potential. Then left only with derivatives of the equation of state (EOS). A T P = A T V + A V T V T P =- S- P V T P =- S- PV (c) U P T This problem uses all three rules. The variables P,T correspond to neither of the natural variables of U . Have to first use the chain rule to switch out P for V , then the generalized chain rule to change the parth from T to S , a Maxwell relation based on dA to get rid of the derivative involving S , and finally a triplet rule to get to derivatives of the EOS. U P T = U V T V P T = U V S + U S V S V T V P T = - P + T P T V V P T = - P + T - V T P V P T V P T =- P V P T- T V T P = PV- V T 1 (d) T P G Relatively direct application of the triplet rule. T P G =- G P T G T P = V S (e) T V A Again, a simple triplet rule. T V A =- A V T A T V =- P S (f) Show that 2 G P 2 T =- 1 2 A V 2 T Only challenge here is working from left to the right. A very non-trivial relationship between the potentials that results from the mathematical linkages of the Legendre transformations. 2 G P 2 T = G P T P T = V P T = 1 P V T = 1 - A V T V T =- 1 2 A V 2 T 2. Simple values and numbers time to learn/memorize these important constants and typical values. These are considered fundamentals and it will be assumed that you know these values on exams, in life, etc. Those that are not known, you will need to learn where to find in the library (or as abslolute last resort on the WEB)....
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