MIT 3.00 Fall 2002 164 c W.C Carter ° Lecture 24 Implications of Equilibrium and Gibbs-Duhem Last Time Drawing Curves Correctly Stability, Global Stability, Metastability, Instability Equilibrium States With More Than One Variable For a system of fixed composition, δU (S, V ) can be expanded25 δU = ∂U ∂U dS + dV ∂S · ∂V ¸ 1 ∂ 2U ∂ 2U ∂ 2U 2 2 + (dS) + 2 dSdV + (dV ) + . . . 2 ∂S 2 ∂S∂V ∂V 2 For a local equilibrium ∂U = T◦ ∂S so that µ (dS, dV ) ∂2U ∂S 2 ∂2U ∂S∂V and ∂2U ∂S∂V ∂2U ∂V 2 ∂U = −P◦ ∂V ¶µ dS dV (24-1) (24-2) ¶ >0 (24-3) The matrix is called the Hessian of the system and for the inequality to be true it must be “positive definite” for a two-by-two matrix. 25 Assuming that U (S, V ) has continuous derivatives near the point (S, V ) that it is being expanded around. MIT 3.00 Fall 2002 165 c W.C Carter ° Necessary conditions for a local minimum are: ∂ 2U >0 ∂S 2 and ∂ 2U ∂ 2U − ∂S 2 ∂V 2 µ ∂ 2U ∂S∂V (24-4) ¶2 >0 (24-5) evaluated at the extrema. Therefore: µ ¶ ∂ 2U ∂T T = = >0 (24-6) 2 ∂S ∂S V CV CV > O for stability (If you add heat to a system, then its entropy must rise) The second part (Eq. 24-5) that must also positive can be written in terms of the Jacobian ¡ ¢ ¡ ∂U ¢ ∂( ∂U , ∂V S ) ∂(T, −P ) ∂S V = >0 (24-7) ∂(S, V ) ∂(S, V ) µ ¶ ∂P T <0 ∂V T CV µ ¶ ∂P <0 ∂V T (24-8) MIT 3.00 Fall 2002 166 c W.C Carter ° for a stable equilibrium. More Mathematical Thermodynamics: Homogeneous Functions Consider U (S, V, Ni ), if I scale all the extensive variables by multiplying each of the extensive variables with the same “scale factor” λ then U (λS, λV, λNi ) = λU (S, V, Ni ) (24-9) Functions that have the property of Equation 24-9, like U , are called “homogeneous degree one” (HD1) function of their variables. Notice that G is not a completely homogeneous function: G(λT, λP, λNi ) 6= λG(T, P, Ni ) (24-10) i.e., increasing the pressure is not like changing an extensive variable. However, G(T, P, λNi ) = λG(T, P, Ni ) (24-11) G is HD1 only in the Ni . Notice that (here lies a common mistake!) G(T, P, λXi ) 6= λG(T, P, Xi ) G is a different function than G. Consider carefully, what can be deduced from Equation 24-11. Taking the derivative with respect to λ (24-12) MIT 3.00 Fall 2002 167 c W.C Carter ° C X i=1 ∂G ∂(λNi ) = G(T, P, Ni ) ∂(λNi ) ∂λ (24-13) We get the following very important equation: C X µi Ni = G(T, P, Ni ) (24-14) i=1 This corresponds to what has been discussed about the relation of the Gibbs free energy. It corresponds to the internal degrees of freedom. The Gibbs-Duhem Relation Consider G= C X µ i Ni i=1 and compare it to our previous expression for dG: (24-15) MIT 3.00 Fall 2002 168 c W.C Carter ° It follows that (This is another important equation): 0 = −SdT + V dP − C X Ni dµi (24-16) i=1 This is the Gibbs-Duhem Equation. It will be used again and again. Notice that Equation 24-16 has the following form: ~ 0 = Y~ · dX (24-17) At equilibrium, a small virtual change in the system is normal to the size of the system.