Heat Capacity relations for 1st order phase transition

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SUMMARY

The discussion centers on proving the relation for heat capacity during a first-order phase transition, specifically Cs = Cp - αV(ΔH/ΔV). Here, Cs represents the heat capacity at constant entropy, while Cp is the heat capacity at constant pressure. The user seeks clarification on the behavior of heat (q) under constant entropy conditions and the implications of the Clausius equation. Key equations referenced include dq = dU + dW and dP/dT = ΔH/(ΔV*T), which are essential for deriving the relationship.

PREREQUISITES
  • Understanding of Clausius equation and its application in thermodynamics.
  • Familiarity with first-order phase transitions and their characteristics.
  • Knowledge of heat capacities, specifically Cp and Cs.
  • Basic principles of thermodynamic differentiation and partial derivatives.
NEXT STEPS
  • Study the derivation of the Clausius-Clapeyron equation in detail.
  • Explore the concept of thermal expansion coefficient (α) and its significance in phase transitions.
  • Learn about the relationship between enthalpy (H) and internal energy (U) in thermodynamic systems.
  • Investigate the implications of heat capacity behavior during phase transitions, particularly near coexistence lines.
USEFUL FOR

This discussion is beneficial for students and professionals in thermodynamics, particularly those studying phase transitions, heat capacity, and related physical chemistry concepts.

Carbon123
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Homework Statement


Prove the following relation for which clausius equation holds :
Cs=Cp-αV(ΔH/ΔV)
Where Cs=∂q/∂T at constant S and is the heat capacity in the coexistence line of 2 phases

Homework Equations


dq=dU+dW
dP/dT=ΔH/(ΔV*T)

The Attempt at a Solution


I do not fully understand why q could change even though S is constant .Is the problem correct ? I thought dqrev=Tds .Isn't Cp in first order transition in coexistence line near infinite ?
Anyway trying to differentiate by dT to dq=dU+dW where dw =pdv gives ∂q/∂T=∂U/∂T+p∂V/∂T I know that U =H-pV so ∂U/∂T =∂H/∂T-∂(pV)/∂T.
I know that ∂H/∂T = Cp but cannot prove the next relation.Can anyone give me a guidance for the next step ? I do not know how to prove the -αVΔH/ΔV though αV=∂V/∂T at constant P. Thanks in advance !
 
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