First order phase transition/constancy of Gibbs

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SUMMARY

This discussion focuses on the principles of Gibbs free energy (G) in relation to first-order phase transitions, specifically addressing the conditions under which G attains its minimum value in equilibrium systems. The Clausius-Clapeyron equation is referenced to explain the behavior of a substance undergoing a discontinuous phase change at temperature TC, where G remains constant across two distinct states. The analysis concludes that latent heat (L) must be positive during this transition and indicates that TC is expected to decrease with increasing pressure.

PREREQUISITES
  • Understanding of Gibbs free energy (G) and its equation G = U - TS + pV
  • Familiarity with the Clausius-Clapeyron equation
  • Knowledge of first-order phase transitions and latent heat
  • Basic principles of thermodynamics, including the first law of thermodynamics
NEXT STEPS
  • Study the Clausius-Clapeyron equation in detail to understand phase transitions
  • Explore the implications of Gibbs free energy in thermodynamic systems
  • Investigate the relationship between pressure and temperature in phase diagrams
  • Examine real-world applications of latent heat in phase transitions
USEFUL FOR

This discussion is beneficial for students and professionals in thermodynamics, particularly those studying phase transitions, chemical engineering, and physical chemistry. It is also useful for researchers exploring the thermodynamic properties of materials under varying pressure and temperature conditions.

CAF123
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Homework Statement


a) Starting from the statement that total entropy (Ssystem+Ssurr) can only increase, show that G = U - TS +pV will attain its minimum value for a system in equilibrium with a fixed pressure and temperature reservoir.

b)At atmospheric pressure, a particular substance is found to undergo a discontinuous change between two states at temperature TC when heated. Its volume increases by ΔV and it absorbs latent heat L as its temperature is changed from just below TC to just above TC. Explain why at TC, the value of G is the same for the two states with different volumes.

c)Explain why L must be positive and comment on whether TC is expected to increase or decrease with pressure.

Homework Equations


Clausius-Clapyeron Equation

The Attempt at a Solution


a)Taking the differential of the given equation, I get dG = dU - TdS -SdT +pdV + Vdp. Eliminate two terms because the system is in thermal equilibrium at constant pressure/temperature. This gives dG = dU - TdS + pdV = 0 using the first law. Hence G is mimimum when the boundary conditions of the system permit a constant pressure/temperature environment. I did not really use the fact that the total S ≥ 0 though, so is there another derivation?

b)So is this process occurring at constant pressure and is TC the value of T on the boundary line between the two phases at that particular pressure? If the case, then at TC the two phases instantaneously have the same pressure/temperature. dG = Vdp - SdT = 0, so G is constant over the boundary line.

c) V increases upon heating at constant pressure. So I would imagine this would correspond to an increase in entropy of the system. So l = T(S2-S1) > 0. dP/dT is usually +ve for most substances. So dT/dP is decreasing, so in most cases expect TC to decrease with pressure. Did I do this right?

Many thanks.
 
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Also, I thought a latent heat was associated with a change of phase transition at constant temperature. It says that the substance absorbs latent heat from just below Tc to just above Tc, so it is therefore changing temperature, albeit infinitesimally if that is what 'just above (below)' means.
 

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