Explanation Write Equation 17-9 as in text book. c 2 = ( ∂ P ∂ ρ ) s (I) Write Equation 17-10 as in text book. c 2 = k ( ∂ P ∂ ρ ) T (II) Here, the velocity of sound is c , the partial derivative of the pressure with respect to the density is ∂ P ∂ ρ , the specific heat ratio is k , and the subscripts s is entropy and T is temperature. Write the thermodynamic relation for c p and c v as follows. c p T = ( ∂ s ∂ T ) P (III) c v T = ( ∂ s ∂ T ) v (IV) Here, the specific heat at constant pressure is c p and the specific heat at constant volume is c v . Write the partial derivative cyclic rule for P , T , s as follows. ( ∂ P ∂ T ) s ( ∂ T ∂ s ) P ( ∂ s ∂ P ) T = − 1 (V) Write the partial derivative cyclic rule for T , v , s as follows. ( ∂ T ∂ v ) s ( ∂ v ∂ s ) T ( ∂ s ∂ T ) v = − 1 (VI) Here, the temperature is T , the pressure is P , the entropy is s , the specific volume is v and the symbol ∂ indicates that the variables are subjected to partial derivation. Write the relation between density ( ρ ) and specific volume ( v ) . ρ = 1 v (VII) Conclusion: Partially differentiate the Equation (VII) as follows. ∂ ρ = − 1 v 2 ∂ v = − ∂ v v 2 Substitute − ∂ v v 2 for ∂ ρ in Equation (I). c 2 = ( ∂ P − ∂ v v 2 ) s = − v 2 ( ∂ P ∂ v ) s = − v 2 ( ∂ P ∂ T × ∂ T ∂ v ) s = − v 2 ( ∂ P ∂ T ) s ( ∂ T ∂ v ) s (VIII) Rearrange the Equation (V) to ( ∂ P ∂ T ) s . ( ∂ P ∂ T ) s = − 1 ( ∂ T ∂ s ) P ( ∂ s ∂ P ) T = − ( ∂ s ∂ T ) P ( ∂ P ∂ s ) T Rearrange the Equation (VI) to ( ∂ T ∂ v ) s

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Author: CENGEL

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Chapter 17.7, Problem 124RP

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Obtain the Equation 17-10 by starting with Equation 17-9 using cyclic rule and the thermodynamic property relations.

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Chapter 17.7, Problem 122RP

Chapter 17.7, Problem 125RP

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Chapter 17 Solutions

EBK THERMODYNAMICS: AN ENGINEERING APPR

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Obtain the Equation 17-10 by starting with Equation 17-9 using cyclic rule and the thermodynamic property relations.

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Obtain the Equation 17-10 by starting with Equation 17-9 using cyclic rule and the thermodynamic property relations.

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