#64. ΔS for Isothermal Mixing of Ideal Gases

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1 #64 Carnot Heat Engne ΔS for Isothermal Mxng of Ideal Gases ds = S dt + S T V V S = P V T T V PV = nrt, P T ds = v T = nr V dv V nr V V = nrln V V = - nrln V V

2 ΔS ΔS ΔS for Isothermal Mxng for Ideal Gases Dagram of Process ΔS V n = -nrln = -nrln V n + n mx = -n = ΔS V Rln V + ΔS = -n Rlny -n n = -n Rln n + n = -n Rlny = -n Rlny Rlny ΔS y ΔS Most ΔS ΔS for Isothermal Mxng for Ideal Gases mx.0 mx mx = - R s always deal & lny gas n 0 s dfferent lny postve mxng from = - n total R propertes are U, H, V ~, etc. y lny zero,

3 Carnot Heat Engne Electrcal Mechancal Potental (water behnd a dam) Heat Carnot heat engne converts Q to mechancal work or transports Q from a low T to a hgh T wth the nput of work 3

4 Carnot Heat Engne Dagram process wth T H & T C and work Process s cyclc usng a workng flud (show cycle on P-V dagram Workng Fluds Water - steam power plant Freon - refrgeraton Combuston Gases - car engne 4

5 Carnot Heat Engne We use an deal gas for a workng flud (a very poor choce) Why? An deal gas has a low heat capacty and there s no heat of vaporzaton, a very large process s requred to produce sgnfcant energy Carnot s an dealzed reversble process - obtans maxmum work and sets a thermodynamc lmt on the work obtanable from a gven amount of Q at T H and a specfed value of T C 5

6 Carnot Heat Engne Step A B rev. Isothermal expanson, Q H s absorbed and drves the expanson w = RT H ln(v B /V A ), ΔU = 0 Step B C the gas expands adabatcally and reversbly from V B to V C and T drops from T H to T C, Q = 0 w = ΔU = T C T H C v dt Carnot Heat Engne Step 3 C D Gas s compressed rev. sothermally from V C V D, heat s transferred to the low T reservor at T C ΔU = 0, sothermal deal gas w = RT C ln(v D /V C ) Step 4 D A Gas s compressed adabatcally and rev back to startng pont, compresson heats gas from T C to T H. Q = 0 adabatc w = ΔU = T H T C C V dt 6

7 Carnot Heat Engne Q cyc = Q H +Q C w cyc = RT C ln V D V C ΔU cyc = 0 = + RT H ln V B V A T H T C V dt + C C V dt T C w cyc = Q H - Q C, T H Carnot Heat Engne Q H = RT H ln V B V A Q C = RT C ln V D V C η efft w cyc Q H = Q H - Q C Q H = - Q C Q H η efft = - T C T H = T H - T C T H 7

8 Thermodynamc Effcency 0 < η efft <.0 Ths s the thermodynamc effcency not to be confused wth the mechancal effcency To maxmze η efft ether Q H or Q C 0 The nd law puts lmts on the amount of heat that can be converted to work Thermodynamc Temperature Scale Carnot s theorem: All reversble engnes operatng between the same heat reservors are equally effcent η of a Carnot engne s related to Q H /Q C, ths rato must be a functon of the reservor temperatures - the rato of the heats must be the rato of the temperatures 8

9 Carnot Theorems Theorem : All reversble heat engnes operatng between the same T s, T & T must have the same thermodynamc η Illustrate wth dagram Carnot Theorems Theorem : Reversble heat engnes have hghest η efft between any two T s Illustrate wth dagram and example 9

10 Carnot Theorems Theorem 3: For the same hgh temperature, T H, the engne that has the larger ΔT has the hgher effcency and produces more work Illustrate wth dagram Example One Carnot engne drves another n seres, as shown n the followng fgure. The heat released from the frst engne s absorbed by the second. Fnd the overall effcency, whch s determned by the total work produced dvded by the heat nput to the frst engne. 0

11 Fgure Soluton We call the ntermedate temperature T 0. From energy conservaton for the frst engne, we have W = Q Q. The effcency of the frst engne s η = W /Q = Q /Q = T 0 /T, whch gves Q /Q = T 0 /T. From energy conservaton for the second engne, we have

12 Soluton W = Q Q 3. The effcency of the second engne s η = W /Q = Q 3 /Q = T /T 0, whch gves Q 3 /Q = T /T 0. The overall effcency s η = (W + W )/Q = W /Q + W /Q = W /Q + (W /Q )(Q /Q ) = ( T 0 /T ) + ( T /T 0 )(T 0 /T ); η = T /T, whch s the effcency of a Carnot engne operatng between T and T. Example A power plant generates 800 MW of electrc power. At what rate does the plant generate waste heat f ts effcency s 8%? Assumng that the plant operates between 480 C and 80 C, what s the maxmum effcency possble?

13 Soluton Example An nventor clams to have bult an engne that takes n 3.0x0 8 J of thermal energy at 450K, rejects.4x0 8 J of thermal energy at 50K, and delvers.0x0 8 J of work n h of cyclc operaton. Is there anythng wrong wth ths clam? 3

14 Soluton 4