Thermodynamic Potentials

Thermodynamic Potentials We work with four key potentials in thermodynamics: the internal energy U , the enthalpy H ≡ U + P V , the Helmholtz potential F ≡ U − T S, and the Gibbs potential G ≡ U + P V − T S = H − T S. dU = T dS − P dV dH = dU + P dV − V dP = T dS − P dV + P dV − V dP = T dS − V dP dF = dU − T dS − SdT = T dS − P dV − T dS − SdT = −SdT − P dV dG = dH − T dS − SdT = T dS − V dP − T dS − SdT = −SdT − V dP (We are most interested in changes in these potentials—the zeroes of these potentials are not defined, and absolute values for these quantities are thus of no significance.) We only see T being paired with S and V being paired with P , suggesting some relationship between these linked variables, which we call conjugate variables. Conjugate variables represent generalized-force–generalized-displacement pairs, the product of each pair having units of energy. One member of each pair is intensive; the other is extensive. The most common of these pairs are P and V , and T and S, but other pairs include f and L, and γ and A. Where relevant, they appear in the expression for dU and the differentials of other thermodynamic potentials: elastic rod: dU = T dS + f dL liquid film: dU = T dS + γdA These potentials identify the availability of free energy in different conditions. Consider a system in contact with its surroundings at a temperature T0 and a pressure P0 . From the first and second laws, dU = δq + δw = δq + δwnon-exp − P0 dV dS ≥ δq/T0 ⇐⇒ δq ≤ T0 dS where we have distinguished expansive from non-expansive work. Now putting both laws together, dU ≤ T0 dS + δwnon-exp − P0 dV ⇐⇒ δwnon-exp ≥ dU − T0 dS + P0 dV This suggests the definition of a new quantity, the availability A, by A ≡ U − T0 S + P0 V , the differential of which can be substituted into the previous inequality to yield δwnon-exp ≥ dA. If no non-expansion work is done on the system, then δwnon-exp = 0 and dA is likewise 0. Hence all processes in such a system lower A (lower the availability of free energy), and equilibrium occurs at a state of minimized A. Under conditions of fixed volume and temperature dA = dF ; under conditions of fixed pressure and temperature dA = dG. The conditions of fixed pressure and temperature are most easily enforced; hence we frequently associate the Gibbs potential G with free energy, and speak of reaching equilibrium by minimizing the Gibbs free energy. (Less commonly, we also speak of the Helmholtz free energy.) (Question: Under conditions of constant entropy and constant volume, dA = dU ; under conditions of constant entropy and constant pressure, dA = dH. Why do we not frequently associate U or H with free energy?) These differentials for the thermodynamic potentials also allow us to derive useful relations between thermodynamic quantities. Since dU = T dS − P dV and (from calculus) dU = (∂U/∂S)V dS + (∂U/∂V )S dV (since U is a function of S and V alone; i.e. U = U (S, V )), by inspection,     ∂U ∂U = T and = −P ∂S V ∂V S Considering the other thermodynamic potentials, we also obtain     ∂H ∂H = T and = −V ∂S P ∂P S     ∂F ∂F = −S and = −P ∂T V ∂V T     ∂G ∂G = −S and = −V ∂T P ∂P T Note that these relations involve both the thermodynamic potentials and thermodynamic variables. We may also establish similar relations (called Maxwell relations) between partial derivatives of thermodynamic quantities. Since dU = T dS −P dV and dU = (∂U/∂S)V dS + (∂U/∂V )S dV , and since (from calculus) (∂/∂V (∂U/∂S)V )S = (∂/∂S (∂U/∂V )S )V , we have that     ∂P ∂T =− ∂V S ∂S V making use of our prior relations ((∂U/∂S)V = T and (∂U/∂V )S = −P ). Considering the other thermodynamic potentials, we also obtain     ∂T ∂V = ∂P S ∂S P     ∂S ∂P = ∂V T ∂T V     ∂S ∂V =− ∂P T ∂T P Note that these relations do not involve thermodynamic potentials. If necessary, they can be easily remembered via a suitable mnemonic (for example, “no pupil need study very thorougly”) and the subsequent construction of a “thermodynamic square:” −S V −P T Every set of three variables is of the shape x or y, representing a partial derivative, the first variable (starting from the top) being the numerator, the second the denominator, and the third the variable kept constant. The four Maxwell relations can be recreated by considering the four ts that can be formed by combining a x and a y and rotating around the thermodynamic square; then the Maxwell relations follow from equating the particular x and y that make up the t. Beyond the relations already discussed, calculus relationships between partial derivatives also dictate that         1 ∂y ∂z ∂x ∂x =  = −1 and that ∂x ∂y z ∂y z ∂z x ∂x y ∂y z Combining the two, we also get       ∂x ∂x ∂z =− ∂y z ∂z y ∂y x We also define various expansivities βx and compressibilities κx such that     1 ∂V ∂V ⇐⇒ βx = − −βx V = ∂T x V ∂T x     ∂V 1 ∂V −κx V = ⇐⇒ κx = − ∂p x V ∂p x where x is a constant thermodynamic variable. (βP for example, is the isobaric expansivity, and κS the adiabatic compressibility.) We can treat βx and κx as multipliers that describe the changes in volume with respect to changes in temperature and pressure respectively while a thermodynamic variable is kept constant.