# Isothermal process on p-v, T-V, and p-t diagrams

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1 Isotherml roess on -, T-, nd -T digrms isotherml T = T 0 = onstnt = ( 1, 1, T 0 ) = ( 2, 2, T 0 ) = nrt 0 T 1 1 T 0 Q T 0 2 W T 0 T () = nrt 0 T() = T0 (T) = multivlued PHYS 1101, Winter 2009, Prof. Clrke 1

2 Isohori roess on -, T-, nd -T digrms isohori = 0 = onstnt = ( 1, 0, T 1 ) = ( 2, 0, T 2 ) 0 = nrt T 1 T T T 2 T 1 T () = multivlued T() = multivlued (T) = nrt 0 PHYS 1101, Winter 2009, Prof. Clrke 2

3 Isori roess on -, T-, nd -T digrms isori = 0 = onstnt = ( 0, 1, T 1 ) = ( 0, 2, T 2 ) 0 = nrt T Q T W T T 1 T 2 T () = 0 T() = 0 (T) = nr 0 PHYS 1101, Winter 2009, Prof. Clrke 3

4 Cliker question 1 Consider the - digrm elow in whih the system evolves from. If T 0 ~ 240K (nd thus RT 0 = 2,000 J mol 1 ), how mny moles of gs, n, re in the system? ) 5 (Nm 2 ) 10 5 T 0 isor isotherm isohor ) 10 5 ) 50 d) 1,000 e) Not enough informtion to tell 1 (m 3 ) first lw of thermodynmis: ΔE int = Q W ( = nc ΔT ) idel gs lw: = nrt PHYS 1101, Winter 2009, Prof. Clrke 4

5 Cliker question 1 Consider the - digrm elow in whih the system evolves from. If T 0 ~ 240K (nd thus RT 0 = 2,000 J mol 1 ), how mny moles of gs, n, re in the system? ) 5 (Nm 2 ) T 0 isor isotherm isohor (m 3 ) ) 10 5 ) 50 d) 1,000 e) Not enough informtion to tell 100,000 n = = = 50 RT 0 2,000 first lw of thermodynmis: ΔE int = Q W ( = nc ΔT ) idel gs lw: = nrt PHYS 1101, Winter 2009, Prof. Clrke 5

6 Cliker question 2 Consider the - digrm elow in whih the system evolves from. Wht is, the volume t stte? (Nm 2 ) 10 5 ) 0.5 m 3 ) 2.0 m 3 ) 4.0 m 3 T 0 d) 8.0 m 3 e) Not enough informtion to tell 1 (m 3 ) first lw of thermodynmis: ΔE int = Q W ( = nc ΔT ) idel gs lw: = nrt PHYS 1101, Winter 2009, Prof. Clrke 6

7 Cliker question 2 Consider the - digrm elow in whih the system evolves from. Wht is, the volume t stte? (Nm 2 ) 10 5 ) 0.5 m 3 ) 2.0 m 3 ) 4.0 m 3 T 0 d) 8.0 m 3 e) Not enough informtion to tell 1 (m 3 ) need to know first lw of thermodynmis: ΔE int = Q W ( = nc ΔT ) idel gs lw: = nrt PHYS 1101, Winter 2009, Prof. Clrke 7

8 Cliker question 3 Consider the - digrm elow in whih the system evolves from. Wht is, the volume t stte? (Nm 2 ) 10 5 ) 0.5 m 3 ) 2.0 m 3 ) 4.0 m T 0 d) 8.0 m 3 e) Not enough informtion to tell 1 (m 3 ) first lw of thermodynmis: ΔE int = Q W ( = nc ΔT ) idel gs lw: = nrt PHYS 1101, Winter 2009, Prof. Clrke 8

9 Cliker question 3 Consider the - digrm elow in whih the system evolves from. Wht is, the volume t stte? (Nm 2 ) 10 5 T 0 ) 0.5 m 3 ) 2.0 m 3 ) 4.0 m 3 d) 8.0 m 3 e) Not enough informtion to tell (m 3 ) = = = 2 m 3 first lw of thermodynmis: ΔE int = Q W ( = nc ΔT ) idel gs lw: = nrt PHYS 1101, Winter 2009, Prof. Clrke 9

10 Cliker question 4 Consider the - digrm elow in whih the system evolves from. Wht is the net hnge in internl energy, ΔE int? (Nm 2 ) 10 5 ) 0 J ) J ) out J T 0 d) 10 5 J e) Not enough informtion to tell 1 2 (m 3 ) first lw of thermodynmis: ΔE int = Q W ( = nc ΔT ) idel gs lw: = nrt PHYS 1101, Winter 2009, Prof. Clrke 10

11 Cliker question 4 Consider the - digrm elow in whih the system evolves from. Wht is the net hnge in internl energy, ΔE int? (Nm 2 ) 10 5 T 0 ) 0 J ) J ) out J d) 10 5 J e) Not enough informtion to tell ΔE int = nc ΔT 1 2 (m 3 ) first lw of thermodynmis: ΔE int = Q W ( = nc ΔT ) idel gs lw: = nrt PHYS 1101, Winter 2009, Prof. Clrke 11

12 Cliker question 5 Consider the - digrm elow in whih the system evolves from. Wht is the net work done y the system on its environment, W? (Nm 2 ) 10 5 ) 0 J ) J ) out J T 0 d) 10 5 J e) Not enough informtion to tell 1 2 (m 3 ) first lw of thermodynmis: ΔE int = Q W ( = nc ΔT ) idel gs lw: = nrt PHYS 1101, Winter 2009, Prof. Clrke 12

13 Cliker question 5 Consider the - digrm elow in whih the system evolves from. Wht is the net work done y the system on its environment, W? (Nm 2 ) 10 5 W ) 0 J ) J ) out J T 0 d) 10 5 J e) Not enough informtion to tell 1 2 (m 3 ) first lw of thermodynmis: ΔE int = Q W ( = nc ΔT ) idel gs lw: = nrt PHYS 1101, Winter 2009, Prof. Clrke 13

14 Cliker question 6 Consider the - digrm elow in whih the system evolves from. Wht is the net het trnsferred into the system, Q? (Nm 2 ) 10 5 ) J ) J ) 10 5 J T 0 d) 10 5 J e) Not enough informtion to tell 1 2 (m 3 ) first lw of thermodynmis: ΔE int = Q W ( = nc ΔT ) idel gs lw: = nrt PHYS 1101, Winter 2009, Prof. Clrke 14

15 Cliker question 6 Consider the - digrm elow in whih the system evolves from. Wht is the net het trnsferred into the system, Q? (Nm 2 ) 10 5 ) J ) J ) 10 5 J T 0 d) 10 5 J e) Not enough informtion to tell 1 2 (m 3 ) Q = ΔE int + W = J first lw of thermodynmis: ΔE int = Q W ( = nc ΔT ) idel gs lw: = nrt PHYS 1101, Winter 2009, Prof. Clrke 15

16 Internl Energy (revisited) f f E int = nc T = nrt = NkT C = C + R 2 2 n = numer of moles; 1 mole = rtiles (N A ) N = numer of rtiles R = gs onstnt = J mol 1 K 1 k = Boltzmnn s onstnt = J K 1 tye of gs degrees of freedom ( f ) seifi het t onstnt volume (C ) internl energy (E int ) seifi het t onstnt ressure (C ) montomi 3 R nrt R ditomi 5 R nrt R olytomi ( 3) ~6 3 R 3 nrt 4 R γ (C /C ) PHYS 1101, Winter 2009, Prof. Clrke 16

17 Aditi roesses reversile = ( 1, 1, T 1 ) = ( 2, 2, T 2 ) γ = onstnt irreversile = ( 1, 1, T 0 ) = ( 2, 2, T 0 ) 1 1 = dit 1 2 T 2 T 1 isotherms 2 T 0 isotherm PHYS 1101, Winter 2009, Prof. Clrke 17

18 Cliker question 7 Consider the - digrm elow in whih the system evolves reversily long the dit from stte to stte. This gs is (knm 2 ) 16 dit ) montomi (γ = 5/3) ) ditomi (γ = 7/5) ) olytomi (γ = 4/3) d) not enough informtion to tell (m 3 ) γ = onstnt PHYS 1101, Winter 2009, Prof. Clrke 18

19 Cliker question 7 Consider the - digrm elow in whih the system evolves reversily long the dit from stte to stte. This gs is (knm 2 ) 16 1 dit 1 8 γ = onstnt (m 3 ) ) montomi (γ = 5/3) ) ditomi (γ = 7/5) ) olytomi (γ = 4/3) d) not enough informtion to tell γ = 1(8) = 2 3 γ = γ = 16(1) = 2 4 γ = 4/3 olytomi γ γ PHYS 1101, Winter 2009, Prof. Clrke 19

20 Cliker question 8 Consider the - digrm elow in whih the system evolves reversily long the dit from stte to stte. How muh het is trnsferred to the system? ) 0 J (knm 2 ) 16 dit ) 8 kj ) 16 kj d) 128 kj e) not enough informtion to tell (m 3 ) γ = onstnt PHYS 1101, Winter 2009, Prof. Clrke 20

21 Cliker question 8 Consider the - digrm elow in whih the system evolves reversily long the dit from stte to stte. How muh het, Q, is trnsferred to the system? ) 0 J (knm 2 ) 16 dit ) 8 kj ) 16 kj d) 128 kj e) not enough informtion to tell γ = onstnt (m 3 ) By definition, Q = 0 for ll diti roesses. PHYS 1101, Winter 2009, Prof. Clrke 21

22 Cliker question 9 Consider the - digrm elow in whih the system evolves reversily long the dit from stte to stte. How muh work, W, does the system do on its environment? ) 20 kj ) 20 kj (knm 2 ) 16 dit ) 24 kj d) 24 kj e) 32 kj f) 32 kj g) not enough informtion to tell (m 3 ) E int = nc T = n3rt (for olytomi gs) first lw: ΔE int = Q W idel gs lw: = nrt PHYS 1101, Winter 2009, Prof. Clrke 22

23 Cliker question 9 Consider the - digrm elow in whih the system evolves reversily long the dit from stte to stte. How muh work, W, does the system do on its environment? ) 20 kj ) 20 kj (knm 2 ) 16 1 W dit 1 8 (m 3 ) ) 24 kj d) 24 kj e) 32 kj f) 32 kj g) not enough informtion to tell E int = nc T = n3rt (for olytomi gs) first lw: ΔE int = Q W ΔE int = 0 W = 3Δ(nRT) = 3Δ() = 3(16 8) = 24 W = 24 kj idel gs lw: = nrt PHYS 1101, Winter 2009, Prof. Clrke 23

24 Summry of Proesses 1 S 3 S 2 S 1 isentro: ΔS = 0 (reversile dit: Q = 0) S 4 isohor: Δ = 0 isor: Δ = 0 isotherm: ΔT = 0 = ( 1, 1, T 2, S 3 ) = ( 1, 2, T 1, S 1 ) = ( 2, 2, T 2, S 2 ) d = ( 4, 2, T 3, S 3 ) e = ( 3, 1, T 3, S 4 ) e 2 d T 2 T 1 T 3 free exnsion (irreversile dit: Q = 0) PHYS 1101, Winter 2009, Prof. Clrke 24

25 Summry of Proesses roess W Q ΔE int = nc ΔT ΔS isor ( 2 1 ) ( 2 1 ) ( 2 1 ) nc ln isohor 0 ( 2 1 ) ( 2 1 ) nc ln isotherm nrt ln nrt ln 0 nr ln isentro 0 0 free exnsion ( 2 ) 1 C R C R ( 2 ) 1 C R C R γ 1 γ nr ln ( 2 ) 1 ) ( 2 1 ) ( 2 1 ( 2 ) 1 PHYS 1101, Winter 2009, Prof. Clrke 25

26 PHYS 1101, Winter 2009, Prof. Clrke 26 All roesses on -, T-, nd T-S digrms T S T T S T T S S An isor (), isotherm (T), isentro (S), nd isohor () emnting from the sme initil stte () s mnifest on -, T-, nd T-S digrm.

27 Cliker question 10 Consider the - digrm elow in whih n = 1 mole of gs evolves reversily from stte to stte long the th shown. Wht is the net hnge in entroy? (Note, e = = Euler s numer, nd thus ln(e) = 1.) ) C ) C e ) C d) C e e) R f) R e g) no where ner enough informtion!! 1 e ( 2 ) ΔS = nc ln (isor) some formule, in se they hel 1 ΔS = nc ln ( 2 ) (isohor) 1 ΔS = nr ln ( 2 ) (isotherm) 1 PHYS 1101, Winter 2009, Prof. Clrke 27

28 Cliker question 10 Consider the - digrm elow in whih n = 1 mole of gs evolves revers -ily from stte to stte long the th shown. Wht is the net hnge in entroy? (Note, e = = Euler s numer, nd thus ln(e) = 1.) ) C ) C e ) C d) C e e) R f) R e g) Yes there is!! 1 e Sine S is stte vrile, it doesn t mtter whih th from to you hoose. Thus, hoose the isor. ( 2 ) ΔS = nc ln (isor) 1 ΔS = nc ln ( 2 ) (isohor) 1 ΔS = nr ln ( 2 ) (isotherm) 1 PHYS 1101, Winter 2009, Prof. Clrke 28

29 Cliker question 11 The evolution of system from stte to stte is shown on oth the - nd T-S digrms elow. Wht is the hnge in internl energy? (Nm 2 ) T (K) 2, ΔE int = Q W = nc ΔT 250 = nrt 1 2 (m 3 ) S (JK 1 ) ) 2000 J ) 2000 J ) 5000 J d) 5000 J e) 7000 J f) 7000 J PHYS 1101, Winter 2009, Prof. Clrke 29

30 Cliker question 11 The evolution of system from stte to stte is shown on oth the - nd T-S digrms elow. Wht is the hnge in internl energy? (Nm 2 ) 2,000 W = 2000 J T (K) Q > ~ 7500 J Q ~ 7000 J W = 2000 J ΔE int = Q W ~ 5000 J 1 2 (m 3 ) S (JK 1 ) ) 2000 J ) 2000 J ) 5000 J d) 5000 J e) 7000 J f) 7000 J PHYS 1101, Winter 2009, Prof. Clrke 30

31 Cliker question 12 The evolution of system from stte to stte is shown on oth the - nd T-S digrms elow. Aout how mny moles of gs re in the system? (Tke R = 8.) (Nm 2 ) T (K) 2, ΔE int = Q W = nc ΔT 250 = nrt 1 2 (m 3 ) S (JK 1 ) ) 0.5 ) 1 ) 1.5 d) 2 e) not enough informtion to tell PHYS 1101, Winter 2009, Prof. Clrke 31

32 Cliker question 12 The evolution of system from stte to stte is shown on oth the - nd T-S digrms elow. Aout how mny moles of gs re in the system? (Tke R = 8.) (Nm 2 ) 2,000 T (K) = nrt t stte : = 2000 RT ~ 2000 n ~ (m 3 ) S (JK 1 ) ) 0.5 ) 1 ) 1.5 d) 2 e) not enough informtion to tell PHYS 1101, Winter 2009, Prof. Clrke 32

33 Cliker question 13 The evolution of system from stte to stte is shown on oth the - nd T-S digrms elow. With n = 1 nd ΔE int = 5000 J, this gs is: (Nm 2 ) T (K) 2, ΔE int = nc ΔT C ~ 12 (montomi) 250 C ~ 21 (ditomi) C ~ 25 (olytomi) 1 2 (m 3 ) S (JK 1 ) ) montomi ) ditomi ) olytomi d) not enough informtion to tell PHYS 1101, Winter 2009, Prof. Clrke 33

34 Cliker question 13 The evolution of system from stte to stte is shown on oth the - nd T-S digrms elow. With n = 1 nd ΔE int = 5000 J, this gs is: (Nm 2 ) T (K) 2, ΔE int = nc ΔT C ~ 12 (montomi) 250 C ~ 21 (ditomi) C ~ 25 (olytomi) 1 2 (m 3 ) S (JK 1 ) ) montomi ) ditomi ) olytomi d) not enough informtion to tell PHYS 1101, Winter 2009, Prof. Clrke 34

35 Three itoril reresenttions of n engine Q in Q in T H Q out W > 0 In omlete thermodynmil yle, gs exnds t high ressure nd omresses t low ressure llowing work, W, to e extrted in eh yle. exnsion stroke T C omression stroke PHYS 1101, Winter 2009, Prof. Clrke 35 Q out Q H = Q in Q C = Q out W out = W

36 Mximum effiieny nd the Crnot yle T T H T Q in T H Q in W > 0 W > 0 S 1 Q in T C S2 dits isotherms T H T C S 1 Q out S 2 S T C d S 1 Q out S 2 S d Q out non-otiml thermo -dynmil yle for n engine otiml thermodyn -mil yle for n engine (Crnot yle) the Crnot engine yle on - digrm Note tht for n engine, the thermodynmil yle is lwys lokwise. PHYS 1101, Winter 2009, Prof. Clrke 36

37 Refrigertors (het ums) Q out S 1 Q in W < 0 For refrigertor, the yle is lwys ounterlokwise. Exnsion hens t low ressure, omression t high ressure nd this tkes work. Het is drwn in t T C nd exelled t T H. Q C = Q in Q H = Q out W in = W PHYS 1101, Winter 2009, Prof. Clrke 37 T C S2 Q out d dits isotherms T H Q in As for n engine, the most otiml thermodynmil yle for refrigertor is the Crnot yle trversed in the ounterlokwise diretion (oosite to the engine).

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