Explanation Write the given equation. c p , g − c p , f = T ( ∂ ( h f g / T ) ∂ T ) P + v f g ( ∂ P ∂ T ) sat (I) Here, the temperature is T , the pressure is P , the specific volume is v , the specific heat at constant pressure is c P , the vaporization enthalpy is h f g , the vaporization specific volume is v f g , the subscripts g , f indicates the gaseous and fluid states and the symbol ∂ indicates the partial derivative of variables. Write the formula for specific heat at constant pressure ( c P ) . c p = ( ∂ h ∂ T ) P (II) Write the relation of Clapeyron equation. ( d P d T ) sat = h f g T v f g (III) The vaporization enthalpy ( h f g ) is expressed as follows. h f g = h g − h f (IV) Conclusion: Divide the Equation (IV) by T on both sides. h f g T = h g − h f T = h g T − h f T (V) Partially differentiate the Equation (V) with respect to temperature ( T ) at constant pressure. ∂ ∂ T ( h f g T ) P = ∂ ∂ T ( h g T − h f T ) P ( ∂ ( h f g / T ) ∂ T ) P = [ ∂ ∂ T ( h g T ) ] P − [ ∂ ∂ T ( h f T ) ] P = 1 T ( ∂ h g ∂ T ) P − h g ( 1 T 2 ) − [ 1 T ( ∂ h f ∂ T ) P − h f ( 1 T 2 ) ] = 1 T ( ∂ h g ∂ T ) P − h g T 2 − 1 T ( ∂ h f ∂ T ) P + h f T 2 = 1 T ( ∂ h g ∂ T ) P − 1 T ( ∂ h f ∂ T ) P − h g T 2 + h f T 2 = 1 T ( ∂ h g ∂ T ) P − 1 T ( ∂ h f ∂ T ) P − h g − h f T 2 (VI) Refer Equation (II) and (IV). Substitute c p , g for ( ∂ h g ∂ T ) P , c p , f for ( ∂ h f ∂ T ) P , and h f g for h g − h f in Equation (VI)

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Chapter 12.6, Problem 31P

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To prove that cp,g−cp,f=T(∂(hfg/T)∂T)P+vfg(∂P∂T)sat.

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Chapter 12.6, Problem 30P

Chapter 12.6, Problem 32P

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To prove that c p , g − c p , f = T ( ∂ ( h f g / T ) ∂ T ) P + v f g ( ∂ P ∂ T ) sat .

Solution Summary: The author explains the formula for specific heat at constant pressure (c_P).

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To prove that cp,g−cp,f=T(∂(hfg/T)∂T)P+vfg(∂P∂T)sat.

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