Explanation Proof for P v k = constant : Write the given expression for isentropic expansion exponent ( k ) . k = − v P ( ∂ P ∂ v ) s (I) Here, the specific volume is v , the pressure is P , the entropy is s and the symbol ∂ indicates the partial derivative of variables. Let assume the entropy ( s ) is the functions of pressure ( P ) and specific volume ( v ) . s = s ( P , v ) (II) Partially differentiate the Equation (II) with respect to pressure and specific volume as follows. d s = ( ∂ s ∂ P ) v d P + ( ∂ s ∂ v ) P d v (III) It is given that the process is said to be isentropic i.e. the entropy is kept as constant. The change in entropy becomes zero. s = constant d s = 0 Substitute 0 for d s in Equation (III). 0 = ( ∂ s ∂ P ) v d P + ( ∂ s ∂ v ) P d v ( ∂ s ∂ P ) v d P + ( ∂ s ∂ v ) P d v = 0 (IV) Write the first Maxwell relation and second Maxell relation as follows. ( ∂ s ∂ P ) v = − ( ∂ v ∂ T ) s (V) ( ∂ s ∂ v ) P = ( ∂ P ∂ T ) s (VI) Here, the temperature is T . Substitute − ( ∂ v ∂ T ) s for ( ∂ s ∂ P ) v and ( ∂ P ∂ T ) s for ( ∂ s ∂ v ) P in Equation (IV). [ − ( ∂ v ∂ T ) s ] d P + [ ( ∂ P ∂ T ) s ] d v = 0 − ( ∂ v ∂ T ) s d P + ( ∂ P ∂ T ) s d v = 0 − ( ∂ v ∂ T ) s [ d P − ( ∂ P ∂ T ) s d v ( ∂ v ∂ T ) s ] = 0 d P − ( ∂ P ∂ T ) s d v ( ∂ v ∂ T ) s = 0 d P − ( ∂ P ∂ T ) s ( ∂ T ∂ v ) s d v = 0 d P − ( ∂ P ∂ v ) s d v = 0 (VII) Divide the Equation (VII) by the pressure P . d P − ( ∂ P ∂ v ) s d v P = 0 P d P P − 1 P ( ∂ P ∂ v ) s d v = 0 (VIII) Refer Equation (I). Substitute − k v for 1 P ( ∂ P ∂ v ) s in Equation (VIII). d P − ( ∂ P ∂ v ) s d v P = 0 P d P P − ( − k v ) d v = 0 d P P + k d v v = 0 (IX) Consider the k as constant and integrate the Equation (IX). ∫ d P P + k ∫ d v v = ∫ 0 ln P + k ln v = constant k ln ( P v ) = constant ln ( P v ) k = constant P v k = constant Thus, it has been shown that P v k = constant

9th Edition

ISBN: 9781259822674

Author: Yunus A. Cengel Dr., Michael A. Boles

Publisher: McGraw-Hill Education

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Chapter 12.6, Problem 91RP

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Show that Pvk=constant and also the isentropic expansion exponent k reduces to the specific heat relation cp/cv for an ideal gas.

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Chapter 12.6, Problem 90RP

Chapter 12.6, Problem 92RP

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Show that P v k = constant and also the isentropic expansion exponent k reduces to the specific heat relation c p / c v for an ideal gas.

Solution Summary: The author demonstrates that the isentropic expansion exponent k reduces to the specific heat ratio for an ideal gas.

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Show that Pvk=constant and also the isentropic expansion exponent k reduces to the specific heat relation cp/cv for an ideal gas.

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