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 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 P   − ext  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 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.

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