Irreversibility Physics 313 Professor Lee Carkner Lecture 16

Exercise #15 Carnot Engine  Power of engine   = 1 – Q H /Q L    = W/Q H   Source temp   = 1 – T L /T H   Max refrigerator COP  For a Carnot refrigerator operating between the same temperatures:   Since K < K C (8.2<9.9), refrigerator is possible

Entropy  Entropy (S) defined by heat and temperature  Total entropy around a closed reversible path is zero   Can write heat in terms of entropy: dQ = T dS 

General Irreversibility   Since  S = S f - S i S f > S i  This is true only for the sum of all entropies    Since only irreversible processes are possible,  Entropy always increases

Reversible Processes  Consider a heat exchange between a system and reservoir at temperature T    So: dS s = +dQ/T dS r = - dQ/T  For a reversible process the total entropy change of the universe is zero 

Irreversible Processes  How do you compute the entropy change for an irreversible process?    What is the change in entropy for specific irreversible processes?

Isothermal W to U  Friction or stirring of a system in contact with a heat reservoir   The only change of entropy is heat Q (=W) absorbed by the reservoir   S = W/T

Adiabatic W to U  Friction or stirring of insulated substance   System will increase in temperature   S =  dQ/T =  C P dT/T = C P ln (T f /T i )

Heat Transfer  Transferring heat from high to low T reservoir   For any heat reservoir  S = Q/T    S for cool reservoir = + Q/T C   Assumes no other changes in any other system 

Free Expansion  Gas released into a vacuum   Replace with a reversible isothermal expansion   Thus,  (dQ/T) =  (nRdV/V)   Note:   Entropy increases even though temperature does not change

Entropy Change of Solids  Solids (and most liquids) are incompressible   We can thus write dQ as CdT and dS as  (C/T)dT  If we approximate C as being constant with T   Note:   If C is not constant with T, need to know (and be able to integrate) C(T)

General Entropy Changes  For fluids that under go a change in T, P or V we can find the entropy change of the system by finding dQ   For example ideal gas:  dQ = C P dT – VdP   dQ = C V dT + PdV   These hold true for any continuous process involving an ideal gas with constant C 

Notes on Entropy  Processes can only occur such that S increases   Entropy is not conserved   The degree of entropy increase indicates the degree of departure from the reversible state 

Use of Entropy  How can the second law be used?   Example: total entropy for a refrigerator     S (reservoir) = (Q + W) /T H  The sum of all the entropy changes must be greater than zero:

Use of Entropy (cont.)  We can now find an expression for the work:  Thus the smallest value for the work is:  Thus for any substance we can look up S 1 -S 2 for a given Q and find out the minimum amount of work needed to cool it 