The Thermodynamic Potentials

1. The Thermodynamic Potentials Four Fundamental Thermodynamic Potentials Equilibrium dU = TdS - pdV dH = TdS + Vdp dG= Vdp - SdT dA = -pdV - SdT fixed V,S fixed S,P fixed P,T fixed T,V The appropriate thermodynamic potential to use is determined by the constraints imposed on the system. For example, since entropy is hard to control (adiabatic conditions are difficult to impose) G and A are more useful. Also in the case of solids p is a lot easier to control than V so G is the most useful of all potentials for solids.

2. Our discussion of these thermodynamic potentials has considered only “closed” (fixed size and composition) systems to this point. In this case two independent variables uniquely defines the state of the system. For example for a system at constant P and T the condition, dG = 0 defines equilibrium, i.e., equilibrium is attained when the Gibbs potential or Gibbs Free Energy reaches a minimum value. If the composition of the system is variable in that the number of moles of the various species present changes (e.g., as a consequence of a chemical reaction) then minimization of G at fixed P and T occurs when the system has a unique composition. For example, for a system containing CO, CO2, H2 and H2O at fixed P and T, minimization of G occurs when the following reaction reaches equilibrium.

3. If the number of moles of each of the individual species remain fixed we know that Since G is an EXTENSIVE property, for multi-component or open system it is necessary that the number of moles of each component be specified. i.e., Then

4. Chemical potential: The quantity is called the chemical potential of component i. It correspond to the rate of change of G with ni when the component i is added to the system at fixed P,T and number of moles of all other species. One can add the same open system term, for the other thermodynamic potentials, i.e., U, H and A.

5. The chemical potential is the partial molal Gibbs Free Energy (or U,H, A) of component i. Similar equations can be written for other extensive variables, e.g., Physically this corresponds to how the volume in the system changes upon addition of 1 mole of component ni at fixed P,T and mole numbers of other components.

6. Consider the relation for the Gibbs Free Energy: at fixed T at fixed P Now take the derivative of these quantities at fixed P and T respectively, Maxwell relations: These mathematical relations are used to connect experimentally measurable quantities to those that are not easily accessible

7. So, the RHS of each of these equations must be equal, If we compare the LHS of these equations, they must be equal since G is a state function and an exact differential and the order of differentiation is inconsequential,

8. Similarly we can develop a Maxwell relation from each of the other three potentials: A B C D

9. Let’s see how these Maxwell relations ca be useful. Consider the following for the entropy. Using the definition of the constant volume heat capacity and the definition of entropy for a reversible process

10. Dividing by dT, the entropy change with temperature at fixed P is Then, For the entropy change with volume at fixed T we can use the Maxwell relation B

11. Now from the ideal gas law, PV = nRT Integrating between states 1 and 2, This equation can be used to evaluate the entropy change at fixed T, problem 4.1.

12. Some important bits of information For a mechanically isolated system kept at constant temperature and volume the A = A(V, T) never increases. Equilibrium is determined by the state of minimum A and defined by the condition, dA = 0. For a mechanically isolated system kept at constant temperature and pressure the G = G(p, T) never increases. Equilibrium is determined by the state of minimum G and defined by the condition, dG = 0. Consider a system maintained at constant p. Then

13. Temperature dependence of H, S, andG Consider a phase undergoing a change in temp @ const P and @ Const. P

14. Temperature dependence of H, S, and G

15. Temperature Dependence of the Heat Capacity Cp Dulong and Petit value 1 T (K) Contributions to Specific Heat • Translational motion of free electrons ~ T1 • Lattice vibrations ~ T3 • Internal vib. within a molecule • Rotation of molecules • Excitation of upper energylevels • Anomalous effects

16. H H0 298K S G G0 Temp. dep. of H, S, and G S0≡ 0 pure elemental solids, Third Law T T ref. state for H is arbitrarily set@ H(298) = 0 and P = 1 atm for elemental substances H H slope = Cp T TS T slope = -S G G

17. T* Tm Thermodynamic Description of Phase Transitions 1. Component Solidification G @ T= Tm Gsolid Gl =Gs dGl = dGs ; G = 0 Gliquid T Where L is the enthalpy change of the transition or the heat of fusion (latent heat).

18. For a small undercooling to say T* ΔH and ΔS are constant (zeroth order approx.) Where ΔT = T m – T* * Note that L will be negative The location of the transition temp Tm will change with pressure

19. Clapeyron equation For a small change in melting point ΔT, we can assume that ΔS & ΔH are constant so The Clapeyron eq. governs the vapor pressure in any first order transition. Melting or vaporization transitions are called first-order transitions (Ehrenfestscheme) because there is a discontinuityin entropy, volume etc which are the 1st partial derivatives of G with respect to Xii.e., ;

20. There are phase transition for which ΔS = 0 and ΔV = 0 i.e., the first derivatives of G are continuous. Such a transition is not of first-order. According to Ehrenfest an nth order transition if at the transition point. Whereas all lower derivatives are equal. There are only two transitions known to fit this scheme gas – liquid 2nd order trans. in superconductivity Notable exceptions are; Curie pt. transin ferromag. Order-disorder trans. in binary alloys λ – transition in liquidhelium