Phase Equilibria (Part - 2)- Thermodynamic and Statistical Physics, CSIR-NET Physical Sciences Physics Notes | EduRev

Physics for IIT JAM, UGC - NET, CSIR NET

Created by: Akhilesh Thakur

Physics : Phase Equilibria (Part - 2)- Thermodynamic and Statistical Physics, CSIR-NET Physical Sciences Physics Notes | EduRev

The document Phase Equilibria (Part - 2)- Thermodynamic and Statistical Physics, CSIR-NET Physical Sciences Physics Notes | EduRev is a part of the Physics Course Physics for IIT JAM, UGC - NET, CSIR NET.
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II. A system at constant volume and temperature

Since we now know that at equilibrium there is only one temperature defined for the two phases let us now remove the system from isolation and place it in a heat bath at temperature, T. We will still hold the volume constant. Under conditions of constant temperature and volume we know that the criterion for equilibrium is that the Helmholtz free energy seeks a minimum. That is,

Phase Equilibria (Part - 2)- Thermodynamic and Statistical Physics, CSIR-NET Physical Sciences Physics Notes | EduRev Our system now looks like,

Phase Equilibria (Part - 2)- Thermodynamic and Statistical Physics, CSIR-NET Physical Sciences Physics Notes | EduRev 
but we do not know the relationship between the pressures in the two phases.

Let us now transfer a small amount of volume, dV > 0, from phase α to phase β .

We know that dA = − SdT − pdV = − pdV for a constant temperature system. So,

Phase Equilibria (Part - 2)- Thermodynamic and Statistical Physics, CSIR-NET Physical Sciences Physics Notes | EduRev

The change in Helmholtz free energy for the entire system is,

Phase Equilibria (Part - 2)- Thermodynamic and Statistical Physics, CSIR-NET Physical Sciences Physics Notes | EduRev

Since dV is positive we conclude that

Phase Equilibria (Part - 2)- Thermodynamic and Statistical Physics, CSIR-NET Physical Sciences Physics Notes | EduRev

or 
Phase Equilibria (Part - 2)- Thermodynamic and Statistical Physics, CSIR-NET Physical Sciences Physics Notes | EduRev
If the system is not at equilibrium then Phase Equilibria (Part - 2)- Thermodynamic and Statistical Physics, CSIR-NET Physical Sciences Physics Notes | EduRev which makes sense since the β phase expanded at the expense of the α phase.

If the system is at equilibrium then Phase Equilibria (Part - 2)- Thermodynamic and Statistical Physics, CSIR-NET Physical Sciences Physics Notes | EduRev so that there is only one pressure defined for the system.

 


III. The system at constant pressure and temperature

Since we now know that at equilibrium both phases are at the same temperature and pressure, let us remove the constant volume restriction and place our system in a heat bath at temperature, T, and a pressure bath at pressure, p. (The atmosphere is a good example of a pressure bath at approximately one atmosphere pressure. You can make chambers to hold different pressures if you wish.) Our system looks like,

Phase Equilibria (Part - 2)- Thermodynamic and Statistical Physics, CSIR-NET Physical Sciences Physics Notes | EduRev

but we do not know the relationship between the chemical potentials in the two phases.

At constant temperature and pressure we know that

Phase Equilibria (Part - 2)- Thermodynamic and Statistical Physics, CSIR-NET Physical Sciences Physics Notes | EduRev

We also know that,

Phase Equilibria (Part - 2)- Thermodynamic and Statistical Physics, CSIR-NET Physical Sciences Physics Notes | EduRev

which at constant temperature and pressure becomes,

Phase Equilibria (Part - 2)- Thermodynamic and Statistical Physics, CSIR-NET Physical Sciences Physics Notes | EduRev

Let us now move a small amount of material, dn, from phase α to phase β (with dn > 0). Then,

Phase Equilibria (Part - 2)- Thermodynamic and Statistical Physics, CSIR-NET Physical Sciences Physics Notes | EduRev

and

Phase Equilibria (Part - 2)- Thermodynamic and Statistical Physics, CSIR-NET Physical Sciences Physics Notes | EduRev

The change in Gibbs free energy for the entire system is then

Phase Equilibria (Part - 2)- Thermodynamic and Statistical Physics, CSIR-NET Physical Sciences Physics Notes | EduRev

Since dn is positive by construction we conclude that,

Phase Equilibria (Part - 2)- Thermodynamic and Statistical Physics, CSIR-NET Physical Sciences Physics Notes | EduRev

or

Phase Equilibria (Part - 2)- Thermodynamic and Statistical Physics, CSIR-NET Physical Sciences Physics Notes | EduRev

If the system is not in equilibrium then Phase Equilibria (Part - 2)- Thermodynamic and Statistical Physics, CSIR-NET Physical Sciences Physics Notes | EduRevwhich makes sense since material wants to move from a region of higher chemical potential to a region of lower chemical potential.

If the system is in equilibrium then,

Phase Equilibria (Part - 2)- Thermodynamic and Statistical Physics, CSIR-NET Physical Sciences Physics Notes | EduRev

so that there is only one chemical potential defined for the system.

Our conclusion, then, is that two phases in equilibrium must have the same temperature, pressure and chemical potential. The above sequence of derivations can easily be extended to include more phases and/or extended to include mixtures, where we would find that the temperature, pressure, and the chemical potential of each component must be the same in every phase. We will use these facts a little later in our derivation of the Gibbs phase rule. 

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