Thermodynamic Stability: Free Energy and ... - McGill University

Free Energy -12- Chemistry 223principle we expect that the reaction should shift to the left (i.e., less dissociation) as pressure isincreased. Our final equation for α shows this. For this reaction, as we shall see in the next section,K P (T )decreases as temperature increases, which when used with our expression for α isagain consistent with LeChatellier’s principle.5.3. Temperature Dependence of K PThe equilibrium constant, K P ,isonly a function of the temperature. From its definition,cf. Eq. (34),d ln(K P )dT=− d(∆G(0) /RT )dT=− 1 ⎛RT 2 ⎜T d(∆G(0) )−∆G (0)⎞ ⎟ = T ∆S(0) +∆G (0)=⎝dT ⎠RT 2∆H(0)RT 2 ,(39)where the second to last equality follows when Eqs. (27a) and (34) are used. This is known asthe Gibbs-Helmholtz equation. Thus, by integrating we find thatln ⎡ K P (T 2 ) ⎤⎢ ⎥ =⎣K P (T 1 ) ∫ T 2⎦(0)∆H ⎛ 1dT ≈ − 1 ⎞RT2R ⎝ T 1 T 2 ⎠ . (40)T 1∆H (0)Equation (40) follows if we assume that ∆H (0) is approximately constant with respect to temperature,or equivalently, that ∆C P ≈ 0(which also implies that ∆S (0) is constant. Why?). Indeed,with this approximation, Eq. (40) simply states thatK P (T 2 ) ⎛K P (T 1 ) = exp ⎜− ∆G(0) (T 2 )+ ∆G(0) (T 1 ) ⎞⎟, (41)⎝RT 2 RT 1 ⎠where ∆G (0) (T ) =∆H (0) − T ∆S o ,asusual. Finally, note, that as in our discussion of LeChatellier’sprinciple, the equilibrium will shift to the product side, i.e., K P increases, when the temperatureis raised if ∆H (0) >0.There is a simple graphical way in which to apply the Gibbs-Helmholtz equation. Byexpressing Eq. (39) as a differential, it follows thatd ln(K P ) =∆H(0)(0)dT =−∆HRT2Rd⎛ 1 ⎞⎝ T ⎠ ;hence, plotting ln(K P )versus 1/T will give a curve whose slope at any point is −∆H (0) /R, and tothe extent that ∆H (0) is independent of temperature, will give a straight line. This is a powerfulway todetermine enthalpy (and entropy) changes without having to do calorimetry.5.4. Free Energy and Entropy of MixingPerhaps the simplest process is one where two samples of different pure gases are isothermallymixed as depicted in the figure belowFall Term, 2014