Explanation Write the differential of internal energy ( d u ) as the functions of temperature, pressure, specific volume and specific heat at constant volume c v . d u = c v d T + [ T ( ∂ P ∂ T ) v − P ] d v (I) Write the differential of enthalpy ( d h ) as the functions of temperature, pressure, specific volume and specific heat at constant pressure c p . d h = c p d T + [ v − T ( ∂ v ∂ T ) P ] d P (II) Here, the temperature is T , the specific volume is v , the pressure is P , and the symbol ∂ indicates the partial derivative of variables. Write the cyclic relation of P , T , v as follows. ( ∂ P ∂ T ) v ( ∂ T ∂ v ) P ( ∂ v ∂ P ) T = − 1 (III) Conclusion: Partially differentiate the Equation (I) with respect to pressure by keeping temperature as constant. ( ∂ u ∂ P ) T = ∂ ∂ P ( c v d T + [ T ( ∂ P ∂ T ) v − P ] d v ) T = ∂ ∂ P ( c v d T ) T + ∂ ∂ P { [ T ( ∂ P ∂ T ) v − P ] d v } T = 0 + [ T ( ∂ P ∂ T ) v − P ] ( ∂ v ∂ P ) T = [ T ( ∂ P ∂ T ) v − P ] ( ∂ v ∂ P ) T = T ( ∂ P ∂ T ) v ( ∂ v ∂ P ) T − P ( ∂ v ∂ P ) T (IV) Refer Equation (III) and rearrange it to obtain ( ∂ P ∂ T ) v ( ∂ v ∂ P ) T . ( ∂ T ∂ v ) P ( ∂ v ∂ P ) T ( ∂ P ∂ T ) v = − 1 ( ∂ P ∂ T ) v ( ∂ v ∂ P ) T = − 1 ( ∂ T ∂ v ) P ( ∂ P ∂ T ) v ( ∂ v ∂ P ) T = − ( ∂ v ∂ T ) P Substitute − ( ∂ v ∂ T ) P for ( ∂ P ∂ T ) v ( ∂ v ∂ P ) T in Equation (IV). ( ∂ u ∂ P ) T = T [ − ( ∂ v ∂ T ) P ] − P ( ∂ v ∂ P ) T = − T ( ∂ v ∂ T ) P − P ( ∂ v ∂ P ) T Thus, the expressions for ( ∂ u ∂ P ) T in terms of P , v and T only is ( ∂ u ∂ P ) T = − T ( ∂ v ∂ T ) P − P ( ∂ v ∂ P ) T

CONNECT FOR THERMODYNAMICS: AN ENGINEERI

9th Edition

ISBN: 9781260048636

Author: CENGEL

Publisher: MCG

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Chapter 12.6, Problem 51P

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To derive an expressions for (∂u∂P)T and (∂h∂v)T in terms of P, v and T only.

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Chapter 12.6, Problem 50P

Chapter 12.6, Problem 52P

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To derive an expressions for ( ∂ u ∂ P ) T and ( ∂ h ∂ v ) T in terms of P , v and T only.

Solution Summary: The author explains how to derive expressions for (partial P)_T in terms of P,v and T only.

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To derive an expressions for (∂u∂P)T and (∂h∂v)T in terms of P, v and T only.

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Chapter 12 Solutions