Download Thermodynamics: Combining the First and Second Laws and more Lecture notes Chemistry in PDF only on Docsity! Combining the First and Second Laws The First Law is given by: dU = dq + dw (1) For a reversible change in a closed system and in the absence of non-pV work, dwrev = !pdV (2) and, from the Second Law, dqrev = TdS (3) Substitution of Eqs. (2) and (3) into Eq. (1), produces the so-called Fundamental Equation, which expresses U as a function of S and V: dU = TdS - pdV (4) Because, as a state function, dU is an exact differential (i.e., its integral is independent of path), an infinitesimal change dU can be expressed in terms of dS and dV by (5) Thus, comparing Eq. (5) to Eq. (4), the two partial differentials can be identified as The Maxwell Relations Mathematically, an infinitesimal change in a function f(x,y) can be expressed as df = gdx + hdy (6) where g and h are functions of x and y. If df is an exact differential, then (7) In the present context, df is dU, dx is dS, dy is dV, g is T, and h is -p. Therefore, replacing the variables in Eq. (7) with their equivalents yields (8) Eq. (8) is one of the four so-called Maxwell relations, each of which can be similarly derived from one of the expressions for the exact differentials dU, dH, dA, or dG, and which reveal some useful relationships among thermodynamic state variables: Variation of Gibbs Energy with Temperature Since G = H ! TS, (10) then (11) For changes in G (ÄG = Gf - Gi) brought about by changes in H, (12) Eq. (12) is the so-called Gibbs-Helmholtz equation, which shows how G/T varies with temperature. This equation is of great importance in chemistry because the equilibrium constant of a chemical reaction is directly related to G/T. Variation of Gibbs Energy with Pressure The effect of pressure on Gibbs energy at constant temperature can be determined from Eq. (9) above by setting dT = 0, giving dG = Vdp, and then integrating between the two pressures: (13) Because solids and liquids are essentially incompressible, their molar volumes and Gibbs energies are almost unaffected by pressure changes: For gases, both the molar volume and the Gibbs energy are strongly affected by pressure changes. For a perfect gas, we can replace Vm in Eq. (13) with RT/p and assuming that RT is a constant, (14) If pi = pE (i.e., 1 bar) and pf = p, then (15) For real gases, molecular interactions (attractions and repulsions) are not negligible: To account for non-ideal behavior in real gases, the perfect gas pressure, p, in Eq. (15) is replaced by an “effective” pressure called the fugacity, f, resulting in In effect, the fugacity is essentially the perfect gas pressure, p, multiplied by a fugacity coefficient, ö, i.e., f = öp. ö is related to the compression factor, Z, by
