Isothermal process on p-V, T-V, and p-T diagrams isothermal ⇒ T = T0 = constant a = (p1, V1, T0) b = (p2, V2, T0) pV = nRT0 p T p a p1 p1 a a b T0 Q T0 b p2 p2 b W V1 V2 V V1 V2 V T0 T nRT p V 0 T V T p T ( ) = V ( ) = 0 ( ) = multivalued PHYS 1101, Winter 2009, Prof. Clarke 1 Isochoric process on p-V, T-V, and p-T diagrams isochoric ⇒ V = V0 = constant a = (p1, V0, T1) b = (p2, V0, T2) pV0 = nRT p T p p1 a T1 a p1 a p2 b T2 b p2 b V0 V V0 V T2 T1 T nRT p(V) = multivalued T(V) = multivalued p(T) = V0 PHYS 1101, Winter 2009, Prof. Clarke 2 Isobaric process on p-V, T-V, and p-T diagrams isobaric ⇒ p = p0 = constant a = (p0, V1, T1) b = (p0, V2, T2) p0V = nRT p T p T2 b Q a b a b p0 p0 W T1 a V1 V2 V V1 V2 V T1 T2 T p V p V p T V 0 p T p ( ) = 0 ( ) = nR ( ) = 0 PHYS 1101, Winter 2009, Prof. Clarke 3 Clicker question 1 Consider the p-V diagram below in which the system evolves from a → b –1 → c. If T0 ~ 240K (and thus RT0 = 2,000 J mol ), how many moles of gas, n, are in the system? p (Nm–2) a) 5 isobar b) 105 a b 105 isotherm c) 50 d) 1,000 T0 isochor e) Not enough information to tell pc c 3 1 Vc V (m ) first law of thermodynamics: ΔEint = Q – W ( = nCVΔT ) ideal gas law: pV = nRT PHYS 1101, Winter 2009, Prof. Clarke 4 Clicker question 1 Consider the p-V diagram below in which the system evolves from a → b –1 → c. If T0 ~ 240K (and thus RT0 = 2,000 J mol ), how many moles of gas, n, are in the system? p (Nm–2) a) 5 isobar b) 105 a b 105 isotherm c) 50 d) 1,000 T0 isochor e) Not enough information to tell pc c pV 100,000 3 n = = = 50 1 Vc V (m ) RT0 2,000 first law of thermodynamics: ΔEint = Q – W ( = nCVΔT ) ideal gas law: pV = nRT PHYS 1101, Winter 2009, Prof. Clarke 5 Clicker question 2 Consider the p-V diagram below in which the system evolves from a → b → c. What is Vc, the volume at state c? 3 p (Nm–2) a) 0.5 m 3 b) 2.0 m a b 105 c) 4.0 m3 d) 8.0 m3 T0 e) Not enough information to tell pc c 3 1 Vc V (m ) first law of thermodynamics: ΔEint = Q – W ( = nCVΔT ) ideal gas law: pV = nRT PHYS 1101, Winter 2009, Prof. Clarke 6 Clicker question 2 Consider the p-V diagram below in which the system evolves from a → b → c. What is Vc, the volume at state c? 3 p (Nm–2) a) 0.5 m 3 b) 2.0 m a b 105 c) 4.0 m3 d) 8.0 m3 T0 e) Not enough information to tell pc c 3 need to know pc 1 Vc V (m ) first law of thermodynamics: ΔEint = Q – W ( = nCVΔT ) ideal gas law: pV = nRT PHYS 1101, Winter 2009, Prof. Clarke 7 Clicker question 3 Consider the p-V diagram below in which the system evolves from a → b → c. What is Vc, the volume at state c? 3 p (Nm–2) a) 0.5 m 3 b) 2.0 m a b 105 c) 4.0 m3 d) 8.0 m3 T0 e) Not enough information to tell 4 5 ×10 c 3 1 Vc V (m ) first law of thermodynamics: ΔEint = Q – W ( = nCVΔT ) ideal gas law: pV = nRT PHYS 1101, Winter 2009, Prof. Clarke 8 Clicker question 3 Consider the p-V diagram below in which the system evolves from a → b → c. What is Vc, the volume at state c? 3 p (Nm–2) a) 0.5 m 3 b) 2.0 m a b 105 c) 4.0 m3 d) 8.0 m3 T0 e) Not enough information to tell 4 5 ×10 c p p V = p V ⇒ V = a V = 2 m3 3 c c a a c p a 1 Vc V (m ) c first law of thermodynamics: ΔEint = Q – W ( = nCVΔT ) ideal gas law: pV = nRT PHYS 1101, Winter 2009, Prof. Clarke 9 Clicker question 4 Consider the p-V diagram below in which the system evolves from a → b → c. What is the net change in internal energy, ΔEint? p (Nm–2) a) 0 J 4 b) 5.0 ×10 J a b 105 4 c) about 7.0 ×10 J 5 d) 10 J T0 e) Not enough information to tell 4 5 ×10 c 3 1 2 V (m ) first law of thermodynamics: ΔEint = Q – W ( = nCVΔT ) ideal gas law: pV = nRT PHYS 1101, Winter 2009, Prof. Clarke 10 Clicker question 4 Consider the p-V diagram below in which the system evolves from a → b → c. What is the net change in internal energy, ΔEint? p (Nm–2) a) 0 J 4 b) 5.0 ×10 J a b 105 4 c) about 7.0 ×10 J 5 d) 10 J T0 e) Not enough information to tell 4 5 ×10 c ΔE = nC ΔT 3 int V 1 2 V (m ) first law of thermodynamics: ΔEint = Q – W ( = nCVΔT ) ideal gas law: pV = nRT PHYS 1101, Winter 2009, Prof. Clarke 11 Clicker question 5 Consider the p-V diagram below in which the system evolves from a → b → c. What is the net work done by the system on its environment, W? p (Nm–2) a) 0 J 4 b) 5.0 ×10 J a b 105 4 c) about 7.0 ×10 J 5 d) 10 J T0 e) Not enough information to tell 4 5 ×10 c 3 1 2 V (m ) first law of thermodynamics: ΔEint = Q – W ( = nCVΔT ) ideal gas law: pV = nRT PHYS 1101, Winter 2009, Prof. Clarke 12 Clicker question 5 Consider the p-V diagram below in which the system evolves from a → b → c. What is the net work done by the system on its environment, W? p (Nm–2) a) 0 J W 4 b) 5.0 ×10 J a b 105 4 c) about 7.0 ×10 J 5 d) 10 J T0 e) Not enough information to tell 4 5 ×10 c 3 1 2 V (m ) first law of thermodynamics: ΔEint = Q – W ( = nCVΔT ) ideal gas law: pV = nRT PHYS 1101, Winter 2009, Prof. Clarke 13 Clicker question 6 Consider the p-V diagram below in which the system evolves from a → b → c. What is the net heat transferred into the system, Q? 4 p (Nm–2) a) –5.0 ×10 J 4 b) 5.0 ×10 J a b 105 5 c) –10 J 5 d) 10 J T0 e) Not enough information to tell 4 5 ×10 c 3 1 2 V (m ) first law of thermodynamics: ΔEint = Q – W ( = nCVΔT ) ideal gas law: pV = nRT PHYS 1101, Winter 2009, Prof. Clarke 14 Clicker question 6 Consider the p-V diagram below in which the system evolves from a → b → c. What is the net heat transferred into the system, Q? 4 p (Nm–2) a) –5.0 ×10 J 4 b) 5.0 ×10 J a b 105 5 c) –10 J 5 d) 10 J T0 e) Not enough information to tell 4 5 ×10 c Q E W 0 105 J 3 = Δ int + = + 1 2 V (m ) first law of thermodynamics: ΔEint = Q – W ( = nCVΔT ) ideal gas law: pV = nRT PHYS 1101, Winter 2009, Prof. Clarke 15 Internal Energy (revisited) f f E = nC T = nRT = NkT C = C + R int V 2 2 p V 22 n = number of moles; 1 mole = 6.0221 × 10 particles (NA) N = number of particles R = gas constant = 8.3147 J mol–1 K–1 –23 –1 k = Boltzmann’s constant = 1.3807 × 10 J K type of gas degrees of specific heat at internal specific heat at γ freedom constant energy constant (Cp/CV ) ( f ) volume (CV ) (Eint) pressure (Cp ) monatomic 3 3 R 3 nRT 5 R 5 2 2 2 3 diatomic 5 5 R 5 nRT 7 R 7 2 2 2 5 4 polyatomic ( 3) ~6 3 R 3 nRT 4 R ≥ 3 PHYS 1101, Winter 2009, Prof. Clarke 16 Adiabatic processes reversible irreversible a = (p1, V1, T1) a = (p1, V1, T0) b = (p2, V2, T2) b = (p2, V2, T0) γ pV = constant p1V1 = p2V2 p p a a p1 adiabat p1 isotherm T1 isotherms T0 T2 b b p2 p2 V1 V2 V V1 V2 V PHYS 1101, Winter 2009, Prof. Clarke 17 Clicker question 7 Consider the p-V diagram below in which the system evolves reversibly along the adiabat from state a to state b. This gas is… a) monatomic (γ = 5/3) p (kNm–2) b) diatomic (γ = 7/5) b 16 c) polyatomic (γ = 4/3) d) not enough information to tell adiabat a 1 3 1 8 V (m ) γ pV = constant PHYS 1101, Winter 2009, Prof. Clarke 18 Clicker question 7 Consider the p-V diagram below in which the system evolves reversibly along the adiabat from state a to state b. This gas is… a) monatomic (γ = 5/3) p (kNm–2) b) diatomic (γ = 7/5) b 16 c) polyatomic (γ = 4/3) d) not enough information to tell adiabat γ γ 3γ paVa = 1(8) = 2 = a 1 γ γ 4 3 pbVb = 16(1) = 2 1 8 V (m ) = 4/3 polyatomic γ ⇒ γ ⇒ pV = constant PHYS 1101, Winter 2009, Prof.