hw3_solutions - MS&E 303 – Fall 2003 Homework #3 –...

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Unformatted text preview: MS&E 303 – Fall 2003 Homework #3 – Due Wednesday, September 24 1. Yet more practice. Reduce the following thermodynamic questions to their simplest forms involving only co- ordinates P,V,T,S , physical parameters α and β (expansion coefficient and the isothermal compressibility), and the constant pressure specific heat c P . (a) ∂V ∂T ¶ S ∂V ∂T ¶ S =- ∂S ∂T ¶ V ∂S ∂V ¶ T =- ∂U ∂T ¶ V ` ∂U ∂S ¶ V ∂P ∂T ¶ V =- C v /T ∂T ∂P ¶ V = C v T ∂V ∂P ¶ T ∂V ∂T ¶ P = C v T- βV αV =- C v β αT =- ( C p- α 2 V T β ) β αT = αV- C p β αT (b) ∂H ∂S ¶ V ∂H ∂S ¶ V = ∂H ∂S ¶ P + ∂H ∂P ¶ S ∂P ∂S ¶ V = T- V ∂T ∂V ¶ S = T + V ∂S ∂V ¶ T ∂S ∂T ¶ V = T + V ∂P ∂T ¶ V ∂U ∂T ¶ V – ∂U ∂S ¶ V = T- V ∂V ∂T ¶ P – ∂V ∂P ¶ T C v /T = T + αV T βC v = T + αV T ( βC p- α 2 V T ) = T + 1 ‡ β αV T · C p- α (c) At constant temperature, show that c p varies with pressure proportional to the curvature of the volume with temperature ∂ 2 V ∂T 2 ¶ P . Find the constant of proportionality (ie. solve the problem exactly). 1 The question is just English for the derivative ∂C p ∂P ¶ T . ∂C p ∂P ¶ T = ∂ ∂H ∂T ¶ P ∂P T = ∂ 2 H ∂T∂P ¶ = ∂ ∂H ∂P ¶ T ∂T P = ∂ ∂H ∂P ¶ S + ∂H ∂S ¶ P ∂S ∂P ¶ T ¶ ∂T P = ∂ V- T ∂V ∂T ¶ P ¶ ∂T P = ∂V ∂T ¶ P- ∂T ∂T ¶ P ∂V ∂T ¶ P- T ∂ 2 V ∂T 2 ¶ P =- T ∂ 2 V ∂T 2 ¶ P It is much less “obvious”, but turns out easier to use the definition of C p related to the derivative of S . ∂C p ∂P ¶ T = ∂ T ∂S ∂T ¶ P ¶ ∂P T = T ∂ ∂S ∂T ¶ P ∂P T = T ∂ 2 S ∂T∂P ¶ = T ∂ ∂S ∂P ¶ T ∂T P =- T ∂ ∂V ∂T ¶ P ∂T P =- T ∂ 2 V ∂T 2 ¶ P (d) Under adiabatic and reversible conditions, the pressure of a system is increased. Obtain an expression for the rate that the temperature increases with pressure. Use this result to determine the temperature increase for adiabatically and reversibly compressing water from 1 atm to 10 atm at 3.98 o C. Adiabatic reversible is just a euphanism for constant entropy (second law) – no heat flow under re- versible equals no entropy flow. Thus the appropriate partial derivative has constant entropy as the condition. So the mathematical translation is ∂T ∂P ¶ S ....
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This note was uploaded on 09/21/2008 for the course MSE 303 taught by Professor Thompson during the Fall '04 term at Cornell University (Engineering School).

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hw3_solutions - MS&E 303 – Fall 2003 Homework #3 –...

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