The Thermodynamic Potentials Four Fundamental Thermodynamic Potentials dU = TdS - pdV dH = TdS + Vdp dG = Vdp - SdT dA = -pdV - SdT 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. fixed V,S fixed S,P fixed P,T fixed T,V Equilibrium

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, CO 2, H 2 and H 2 O at fixed P and T, minimization of G occurs when the following reaction reaches equilibrium.

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 If the number of moles of each of the individual species remain fixed we know that

Chemical potential: The quantity is called the chemical potential of component i. It correspond to the rate of change of G with n i 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.

Physically this corresponds to how the volume in the system changes upon addition of 1 mole of component n i at fixed P,T and mole numbers of other components. 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.,

Maxwell relations: These mathematical relations are used to connect experimentally measurable quantities to those that are not easily accessible 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,

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, So, the RHS of each of these equations must be equal,

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

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

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

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.

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

Temperature dependence of H, S, and G Consider a phase undergoing a change in const P Const. P

Temperature dependence of H, S, and G

Temperature Dependence of the Heat Capacity CpCp 1 T (K) Contributions to Specific Heat 1.Translational motion of free electrons ~ T 1 2.Lattice vibrations ~ T 3 3.Internal vib. within a molecule 4.Rotation of molecules 5.Excitation of upper energy levels 6.Anomalous effects Dulong and Petit value

Temp. dep. of H, S, and G H H0H0 298K ref. state for H is arbitrarily H(298) = 0 and P = 1 atm for elemental substances T S 0 ≡ 0 pure elemental solids, Third Law S T T G G0G0 T G H TS G H slope = C p slope = -S

Thermodynamic Description of Phase Transitions 1. Component Solidification T*TmTm G solid G G liquid T= T m G l =G s dG l = dG s ;  G = 0 Where L is the enthalpy change of the transition or the heat of fusion (latent heat).

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

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 (Ehrenfest scheme) because there is a discontinuity in entropy, volume etc which are the 1 st partial derivatives of G with respect to X i i.e., ;

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 n th 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 2 nd order trans. in superconductivity Notable exceptions are; Curie pt. trans in ferromag. Order-disorder trans. in binary alloys λ – transition in liquid helium