Entropy and Spontaneous Processes Pext dw = –Pext dV dV dq = C dT at constant V Tsurr v P, T dU = dq + dw closed system assume spontaneous process; irreversible path dq dq dS = rev reversible path, P = P T ext dq rev = T dS dw max = – P dV maximum |work| on expansion dU = T dS – P dV reversible path T dS – P dV = dq + dw dU independent of path Evaluate dU and dS for an irreversible process from a corresponding reversible process T dS = dq + P dV –Pext dV dq P P dq P P dS = + dV – ext dV = + − ext dV T T T T T T dq P - P dS = + ext dV P – P is the pressure gradient T T ext adiabatic: dq = 0 if P > P ext need a pressure gradient to do work P - P ext > 0 and the system expands so dV > 0 so that dS > 0 T adiabatic: dq = 0 if P < P ext P - P ext < 0 and the system contracts so dV < 0 so guarantees that dS > 0 T P - P for a reversible process P = P and dw = dw and then ext dV = 0 ext max T P - Pext for a spontaneous expansion |dw| < |dw | but dV > 0 max T P - P ext dV = "lost work" term energy wasted to increase entropy T dq "lost work" term always positive so dS ≥ Clausius inequality T dU P dS = + dV closed system, entropy production: dU = dq + dw T T Colby College now focus on the universe: dq surr dq dS univ = dS + dS surr dS surr = = − Tsurr Tsurr dq P - Pext dq dS univ = + dV − T T Tsurr 1 1 P Pext dS univ = − dq + − dV T Tsurr T T constant V: if T < T surr need a temperature gradient for heat transfer 1 1 − > 0 and the system absorbs energy so dq > 0 so that dS > 0 T Tsurr constant V: if T > T surr 1 1 − < 0 and the system liberates energy so dq < 0 so guarantees that dS > 0 T Tsurr heat transfer term is always positive for a spontaneous process, "lost work" term is always positive for a spontaneous process: Entropy of the universe always increases for spontaneous processes 1 1 − is the thermodynamic force for energy transfer in the form of heat T Tsurr P Pext − is the thermodynamic force for energy transfer in the form of work T T Tsurr − T P - Pext dS univ = dq + dV T T surr T Non-equilibrium, irreversible thermodynamics dS 1 1 dq P P dV univ = − + − ext rate of entropy production dt T Tsurr dt T T dt dq KA J = (T – T ) K= thermal conductivity = , A = contact area dt δ surr m K s KA – δ C t (T – Tsurr ) = (T o – Tsurr ) e P D. Kondepudi, I. Prigogine, Modern Thermodynamics , Wiley, Chichester, England, 1998, Chap. 3. K. S. Sanchez, R. A. Vergenz, "Understanding Entropy Using the Fundamental Stability Conditions," J. Chem. Ed. , 1994 , 71(7) , 562-566. Colby College .