Reading assignment - MSE 4100 - Fall 2011 5. Thermodynamics...

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5.3. Equilibrium conditions • For an isolated system , the ________________ has a maximum value. • For a system at constant temperature T , i.e . in contact with a thermostat, and with constant volume V the ___________________ becomes a minimum • For a system at constant temperature T and pressure p , the ______________ becomes a mimimum • In most alloys at atmospheric pressure, the term pV is quite small ( p 0) and it is possible to use the Helmholtz free energy instead of the Gibbs free energy • When material is exchanged with the environment, for example by diffusion across a phase boundary, the Gibbs energy changes (at constant p and T ) by μ i · dn i where dn i is the number of atoms of species i exchanged with the envi- ronment and μ i is the chemical potential or partial molar enthalpy of species i • Change of Gibbs energy dG with change in temperature dT , pressure dp and when material dn i is added • De±nition of the chemical potential as a function of the Gibbs energy G ( T , p , n i ) • Equilibrium condition in terms of the chemical potentials μ i F = E - T · S G = E - T · S + p · V dG = SdT + pdV - i μ i dn i μ i = @ G @ n i ± ± ± T,p,n i i μ i dn i =0 MSE 4100 - Fall 2011 5. Thermodynamics of alloys 4
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Example: Binary system of A and B atoms with phases α and β • A and B atoms can be exchanged between the phases α and β • At constant temperature and volume (or at pressures p 0), the equilibrium con- dition becomes • Number of A atoms leaving α phase and entering β phase has to be the same, hence • This results in the equilibrium condition for the chemical potentials of A and B atoms in the α and β phases • Written out in terms of the free energy of the α and β phases F α and F β • This is the mathematical representation of the well-known tangent rule • Generalizing these equations to a system of C components with P phases, results in a total of P ( C -1)+1 free variables (for constant pressure) - C -1 concentrations - P phases - 1 temperature T • We have C ( P -1) equilibrium conditions • The number of degrees of freedom for the system is simply the difference be- tween the number of free variables and the number of conditions i = A,B μ i dn i + i = A,B μ β i dn β i =0 dn i = - dn β i μ A = mu β A and μ B = μ β B @ F @ n A ± ± ± ± T,V,n B = @ F β @ n A ± ± ± ± B and @ F @ n B ± ± ± ± A = @ F β @ n B ± ± ± ± A MSE 4100 - Fall 2011 5. Thermodynamics of alloys 5
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• This is the same Gibbs phase rule as shown earlier except for constant pressure • If the pressure is allowed to vary, there is one more degree of freedom Statistical mechanics • Thermodynamic partition function Z where ω
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This note was uploaded on 12/05/2011 for the course MSE 4100 taught by Professor Hennig during the Fall '11 term at Cornell University (Engineering School).

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Reading assignment - MSE 4100 - Fall 2011 5. Thermodynamics...

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