Sc=kin( 2. 22g)=king,+kIn Br=s,+s (D43) Equation(D 4.2)is sometimes taken as the basic definition of entropy, but it should be remembered that it is only appropriate when each quantum state is equally likely. Equation (D3. 8)is more general and applies equally for equilibrium and non-equilibrium situations A simple numerical example shows trends in entropy changes and randomness for a system which can exist in three states. Consider the five probability distributions i)p1=1.0,P2=0,3=0;S=-k(mnl+0m0+0ln0)=0 i)P=0.P2=0.2,P2=0;S=-k{0.8M(08)+0.2b(02)+01m(0)]=0.00 i)n=08P2=0.1,P3=01;S=-k08m(0.8)+0.m0)+0.m0.=0639k iy)n=05P2=03,乃2=02;S=-40.5m(05)+03M(03)+0.2m(0.2)=1.03k 1-1-1-3-109米 The first distribution has no randomness For the second we know that state 3 is never found States (iii) and (iv) have thus higher randomI largest entropy. 1. 5. Numerical Example of the Approach to the equilibrium distribution Reynolds and Perkins give a numerical example which illustrates the above concepts and also the tendency of a closed isolated system to tend to equilibrium Reynolds and Perkins, Engineering Thermodynamics, McGraw-Hill, 1977. Sec. 6.7. pp 177-183 1D-61D-6 Sk k k SS C AB A B A B = ⋅ ln ln ln ( ) ΩΩ Ω Ω = + =+ (D.4.3) Equation (D.4.2) is sometimes taken as the basic definition of entropy, but it should be remembered that it is only appropriate when each quantum state is equally likely. Equation (D.3.8) is more general and applies equally for equilibrium and non-equilibrium situations. A simple numerical example shows trends in entropy changes and randomness for a system which can exist in three states. Consider the five probability distributions i) p pp 1 23 = == 10 0 0 ., , ; S k =− + + ( ) 110 0 ln ln ln 00 0 = ii) pp p 123 === 08 02 0 ., ., ; Sk k = − [ ] 0 8 0 8 0 2 0 2 0 0 0 500 . ln . . l ( ) + n . ln ( ) + ( ) = . iii) ppp 123 === 08 01 01 ., ., . ; Sk k = − [ ] 0 8 0 8 0 1 0 1 0 1 0 1 0 639 . ln . . l ( ) + n . . ln . . ( ) + ( ) = iv) pp p 123 === 05 03 02 ., ., . ; Sk k = − [ ] 0 5 0 5 0 3 0 3 0 2 0 2 1 03 . ln . . l ( ) + n . . ln . . ( ) + ( ) = v) pp p 123 === 13 13 13 /, /, / ; Sk k = −           3 = 1 3 1 3 ln . 1 099 . The first distribution has no randomness. For the second, we know that state 3 is never found. States (iii) and (iv) have progressively greater uncertainty about the distribution of states and thus higher randomness. State (v) has the greatest randomness and uncertainty and also the largest entropy. 1.D.5. Numerical Example of the Approach to the Equilibrium Distribution Reynolds and Perkins give a numerical example which illustrates the above concepts and also the tendency of a closed isolated system to tend to equilibrium. Reynolds and Perkins,Engineering Thermodynamics, McGraw-Hill, 1977. Sec. 6.7. pp.177-183