To determine The specific heat at constant volume at final state. Explanation Write the expression for the Berthelot equation. p = R T v − b − a T v 2 (I) Differentiate the Equation (I) with respect to temperature. p = R T v − b − a T v 2 ( ∂ p ∂ T ) v = R v − b + a T 2 v 2 ( ∂ 2 p ∂ T 2 ) v = − 2 a T 3 v 2 (II) Here, the gas constant is R , the temperature is T , the volume of the liquid is v , the pressure is p and the constant are a and b . Write the expression for the constant a . a = 27 64 R 2 T c 3 p c (III) Here, the critical pressure is p c and the critical temperature is T c . Write the expression for the constant b . b = 1 8 R T c p c (IV) Write the expression for the specific heat at constant volume. C v = ∂ u ∂ T | v ( ∂ u ∂ v ) T = T ( ∂ u ∂ T ) v − p ∂ ∂ v [ ( ∂ u ∂ T ) v ] T = T ( ∂ 2 p ∂ T 2 ) v + ( ∂ p ∂ T ) v − ( ∂ p ∂ T ) v ( ∂ C v ∂ v ) T = T ( ∂ 2 p ∂ T 2 ) v (V) Substitute − 2 a T 3 v 2 for ( ∂ 2 p ∂ T 2 ) v in Equation (V). ( ∂ C v ∂ v ) T = T ( ∂ 2 p ∂ T 2 ) v C v ( T , v 2 ) − C v ( T , v 1 ) = 2 a T 2 v ] v 1 v 2 C v ( T , v 2 ) = C v + 2 a T 2 ( 1 v 2 − 1 v 1 ) (VI) Conclusion: Refer to Table A-1 “Atomic or molecular weights and critical properties of the selected elements and compounds” to obtain the critical properties of air as T c = 239 °R and p c = 37

Fundamentals of Engineering Thermodynamics

8th Edition

ISBN: 9781118832301

Author: SHAPIRO

Publisher: JOHN WILEY+SONS,INC.-CONSIGNMENT

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Chapter 11.9, Problem 63P

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The specific heat at constant volume at final state.

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