10-mathematics manupulation [相容模式]

10-mathematics manupulation [相容模式]

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Unformatted text preview: Mathematical manupulation in Thermodynamics Exact differential and Maxwell relation ΔU, ΔH, ΔS calculations for van der Waals gases ΔU, ΔH, ΔS calculations for Berthelot gases Joule Thomson coefficient Three techniques techniques Conditions for exact differential Divide through rule Cyclic rule rule Conditions for exact differential Conditions for exact differential if M = M ( x, y ) only ⎛ ∂M ⎞ ⎛ ∂M ⎞ dM = ⎜ ⎟ dx + ⎜ ⎜ ∂y ⎟ dy is an exact differential. ⎟ ⎝ ∂x ⎠ y ⎝ ⎠x ∂ ⎡⎛ ∂M ⎞ ⎤ ∂ ⎡⎛ ∂M then ⎟ ⎥ = ⎢⎜ ⎢⎜ ∂y ⎣⎝ ∂x ⎠ y ⎥ ∂x ⎢⎜ ∂y ⎢ ⎣⎝ ⎦x ⎞⎤ ⎟ ⎟⎥ ⎥ ⎠x ⎦ y ⎛ ∂P ⎞ ⎛ ∂Q ⎞ if ⎜ ⎟ = ⎜ ⎟ ⎜ ∂y ⎟ ⎝ ⎠ x ⎝ ∂x ⎠ y th then dM = Pdx + Qdy is an exact differential. Qd Basic Equations of Thermodynamics Equations of Thermodynamics H ≡ U + PV A ≡ U − TS Q ≡ H − TS dU = TdS − PdV dH = TdS + VdP dA = − SdT − PdV dG = − SdT + VdP Maxwell Relations Maxwell’s Relations ⎛ ∂T ⎞ ⎛ ∂P ⎞ ⎜ ⎟ = −⎜ ⎟ ⎝ ∂V ⎠ S ⎝ ∂S ⎠V ⎛ ∂T ⎞ ⎛ ∂V ⎞ ⎜ ⎟ =⎜ ⎟ ⎝ ∂P ⎠ S ⎝ ∂S ⎠ P ⎛ ∂S ⎞ ⎛ ∂P ⎞ ⎜ ⎟ =⎜ ⎟ ⎝ ∂V ⎠T ⎝ ∂T ⎠V ⎛ ∂S ⎞ ⎛ ∂V ⎞ −⎜ ⎟ = ⎜ ⎟ ⎝ ∂P ⎠T ⎝ ∂T ⎠ P Divide through rule through rule if dM = Pdx + Qdy is an exact differential, ⎛ ∂M ⎜ ⎝ ∂t ⎛ ∂M ⎜ ⎝ ∂x ⎞ ⎛ ∂x ⎞ ⎛ ∂y ⎞ ⎟ = P⎜ ⎟ + Q⎜ ⎟ ⎠s ⎝ ∂t ⎠ s ⎝ ∂t ⎠ s ⎞ ⎛ ∂y ⎞ ⎟ = P + Q⎜ ⎟ ⎠s ⎝ ∂x ⎠ s ⎛ ∂M ⎞ ⎛ ∂x ⎞ ⎟ = P⎜ ⎟ ⎜ ⎝ ∂t ⎠ y ⎝ ∂t ⎠ y Applications of divide through rules Applications of divide through rules ⎛ ∂U ⎞ ⎛ ∂S ⎞ ⎛ ∂P ⎞ ⎜ ⎟ = T⎜ ⎟ − P = T⎜ ⎟ −P ⎝ ∂V ⎠T ⎝ ∂V ⎠T ⎝ ∂T ⎠V ⎛ ∂H ⎞ ⎛ ∂S ⎞ ⎛ ∂V ⎞ ⎜ ⎟ = T⎜ ⎟ +V = V −T⎜ ⎟ ⎝ ∂P ⎠T ⎝ ∂P ⎠T ⎝ ∂T ⎠ P ⎛ ∂U ⎞ ⎛ ∂S ⎞ ⎟ ⎜ ⎟ = CV = T ⎜ ⎝ ∂T ⎠V ⎝ ∂T ⎠V ⎛ ∂H ⎞ ⎛ ∂S ⎞ ⎜ ⎟ = CP = T ⎜ ⎟ ⎝ ∂T ⎠ P ⎝ ∂T ⎠ P Pressure and volume dependence of Heat and volume dependence of Heat Capacity ∂ ⎡⎛ ∂U ⎞ ⎤ ∂ ⎡⎛ ∂U ⎞ ⎤ ⎛ ∂CV ⎞ ⎜ ⎟= ⎟⎥ = ⎟⎥ ⎢⎜ ⎢⎜ ∂V ⎠T ∂V ⎣⎝ ∂T ⎠ P ⎦ T ∂T ⎣⎝ ∂V ⎠T ⎦V ⎝ ∂ = ∂T ⎡ ⎛ ∂P ⎞ ⎤ ⎛ ∂2P ⎞ ⎟ − P⎥ = T ⎜ 2 ⎟ ⎢T ⎜ ⎜ ∂T ⎟ ⎝ ⎠V ⎦ ⎣ ⎝ ∂T ⎠V ∂ ⎡⎛ ∂H ⎞ ⎤ ∂ ⎡⎛ ∂H ⎞ ⎤ ⎛ ∂C P ⎞ ⎜ ⎟= ⎟⎥ = ⎟⎥ ⎢⎜ ⎢⎜ ⎝ ∂P ⎠T ∂P ⎣⎝ ∂T ⎠ P ⎦ T ∂T ⎣⎝ ∂P ⎠T ⎦ P ∂ = ∂T ⎡ ⎛ ∂ 2V ⎞ ⎛ ∂V ⎞ ⎤ ⎟ ⎥ = −T ⎜ 2 ⎟ ⎢V − T ⎜ ⎜ ∂T ⎟ ⎝ ∂T ⎠ P ⎦ ⎝ ⎠P ⎣ A. A. Calculations for van der Waals gas for van der Waals gas nRT n2a P= −2 V − nb V C vm R ig = α + β T + γT 2 nR ⎛ ∂P ⎞ ⎜ ⎟= ⎝ ∂T ⎠V V − nb ⎛ ∂2P ⎞ ⎜ 2⎟ =0 ⎜ ∂T ⎟ ⎝ ⎠V n2a ⎛ ∂P ⎞ T⎜ ⎟ − P = 2 V ⎝ ∂T ⎠V ΔU, ΔS and CV and ⎡ ⎛ ∂P ⎞ ⎤ n2a dU = CV dT + ⎢T ⎜ ⎟ − P ⎥ dV = CV dT + 2 dV ∂T ⎠V V ⎣⎝ ⎦ dS = CV C nR ⎛ ∂P ⎞ dV dT + ⎜ ⎟ dV = V dT + T ∂T ⎠V T V − nb ⎝ CV − CV ig CVm = CVm 2 V ⎛∂ P⎞ ⎛ ∂CV ⎞ =∫ ⎜ ⎟ dV = ∫∞ T ⎜ 2 ⎟ dV = 0 ⎜ ∂T ⎟ ∞ ∂V ⎝ ⎠T ⎝ ⎠V V ig 1. Isothermal Process (P1,V1,T1)→(P2,V2,T1) n2a dU = 2 dV V n2a 1⎞ 2 ⎛1 ⎜−⎟ ΔU = ∫ dV = n a⎜ ⎟ V1 V 2 ⎝ V1 V2 ⎠ ⎛ ⎞ (P2V2 − P1V1 ) = n 2 a⎜ 1 − 1 ⎟ + (P2V2 − P1V1 ) ΔH = ΔU + ⎜V V ⎟ 2⎠ ⎝1 nR V2 dS = V − nb V2 ΔS = ∫ V1 dV nR V − nb dV = nR ln 2 V − nb V1 − nb 2. Constant V Process (P1,V1,T1)→(P2,V1,T2) dU = CV dT = CV dT = nCvm dT ig ig ig ig C C dT nC dS = V dT = V dT = vm dT T T T T2 T2 ( ) ΔU = ∫ nCvm dT = nR ∫ α + βT + γT 2 dT ig T1 T1 ( )( ) β2 γ3 ⎡ 2 3⎤ = nR ⎢α (T2 − T1 ) + T2 − T1 + T2 − T1 ⎥ 2 3 ⎣ ⎦ ΔH = ΔU + (P2V1 − P1V1 ) = ΔU + (P2 − P1 )V1 ΔS = ∫ T2 T1 ( ig ) T2 1 nCvm dT = nR ∫ α + βT + γT 2 dT T1 T T ( ) ⎡ T2 γ2 2⎤ = nR ⎢α ln + β (T2 − T1 ) + T2 − T1 ⎥ T1 2 ⎣ ⎦ 3. Any Process with known initial and Any Process with known initial and end states (P1,V1,T1)→(P2,V2,T2) n2a dU = C dT + 2 dV V ig V ( )( ) β2 γ3 ⎡ 2 3⎤ ΔU = nR ⎢α (T2 − T1 ) + T2 − T1 + T2 − T1 ⎥ 2 3 ⎢ ⎥ ⎛1 1⎞ + n 2 a⎜ − ⎟ ⎜V V ⎟ 2⎠ ⎝1 ΔH = ΔU + (P2V2 − PV1 ) 1 dS = ig CV T dT + nR dV dV V − nb ( ) ⎡ T2 γ2 2⎤ ΔS = nR ⎢α ln + β (T2 − T1 ) + T2 − T1 ⎥ T1 2 ⎢ ⎥ V2 − nb + nR ln V1 − nb 4. Isobaric Process (P1,V1,T1)→(P1,V2,T2) ( )( ) β γ ⎡ 2 2 3 3⎤ Δ U = nR ⎢α (T2 − T1 ) + T2 − T1 + T2 − T1 ⎥ 2 3 ⎢ ⎥ ⎛1 1⎞ + n 2a⎜ − ⎜V V ⎟ ⎟ 2⎠ ⎝1 Δ H = Δ U + (P2V 2 − P1V1 ) ( ) ⎡ T γ2 2⎤ ΔS = nR ⎢α ln 2 + β (T2 − T1 ) + T2 − T1 ⎥ T1 2 ⎢ ⎥ V − nb + nR ln 2 V1 − nb 5. Adiabatic Process (P1,V1,T1)→ (?,V2,?) 5.1 Reversible dS = δQrev T =0 ( ) ⎡ V2 − nb T2 γ2 2⎤ 0 = nR ln + nR ⎢α ln + β (T2 − T1 ) + T2 − T1 ⎥ V1 − nb T1 2 ⎢ ⎥ ⇒T2 ⇒ P2 ( )( β2 γ3 ⎡ 2 3 ΔU = nR ⎢α (T2 − T1 ) + T2 − T1 + T2 − T1 2 3 ⎢ 1 ⎜ ⎟ )⎤ + n a⎛ V − V1 ⎞ ⎜ ⎟ ⎥ 2 ⎥ ⎝ 1 2 ΔH = ΔU + (P2V2 − P1V1 ) 5.2 Irreversibly against a constant external pressure , Pext dU = δQ + δW = − Pext dV ( ΔU = − Pext (V2 − V1 ) )( β2 γ3 ⎡ 2 3 ΔU = nR ⎢α (T2 − T1 ) + T2 − T1 + T2 − T1 2 3 ⎢ 1 ⎜ ⎟ )⎤ + n a⎛ V − V1 ⎞ ⎜ ⎟ ⎥ 2 ⎥ ⎝ 1 2 = − Pext (V2 − V1 ) ⇒ T2 ⇒ P2 ΔH = ΔU + (P2V2 − P1V1 ) ( ) ⎡ V2 − nb T2 γ2 2⎤ ΔS = nR ln + nR ⎢α ln + β (T2 − T1 ) + T2 − T1 ⎥ V1 − nb T1 2 ⎢ ⎥ ⎠ ⎠ B. Calculations for Berthelot gas Berthelot gas nRT n2a P= + V − nb TV 2 nR n2a ⎛ ∂P ⎞ + 22 ⎜ ⎟= ⎝ ∂T ⎠V V − nb V T CV ,m R ig = α + βT + γT 2 ⎛ ∂2P ⎞ 2n 2 a ⎜ 2 2⎟ =− 2 3 ⎜∂ T ⎟ VT ⎝ ⎠V 2n 2 a ⎛ ∂P ⎞ T⎜ ⎟ −P= TV 2 ⎝ ∂T ⎠V Expression for dU, dS, and CV for and 2n2 a dU = CV dT + dV 2 TV ⎡ nR CV n2a ⎤ dT + ⎢ dS = + 2 2 ⎥ dV T ⎢V − nb V T ⎥ CV − CV ig 2 V − 2n a ⎛ ∂2P ⎞ 2n 2 a = ∫ T ⎜ 2 ⎟ dV = ∫ 2 2 dV = 2 ∞ ⎜ ∂T ⎟ ∞TV TV ⎝ ⎠V V 1. Isothermal Process (P1,V1,T1)→(P2,V2,T1) 2n 2 a dU = dV 2 T1V 2n 2 a V2 dV 2n 2 a ⎛ 1 1 ⎞ ΔU = ∫V1 V 2 = T1 ⎜ V1 − V2 ⎟ ⎜ ⎟ T1 ⎝ ⎠ ΔH = ΔU + (P2V2 − PV1 ) 1 ⎡ nR n2a ⎤ dS = ⎢ + 2 2 ⎥ dV ⎢V − nb V T1 ⎥ V2 − nb n 2 a ⎛ 1 1 ⎞ ΔS = nR ln + 2⎜ − ⎟ V1 − nb T1 ⎜ V1 V2 ⎟ ⎝ ⎠ 2. Any Process Any Process (P1,V1,T1)→(P2,V2,T2) 2n 2 a dU = CV dT + dV 2 TV ⎡ nR CV n2a ⎤ dT + ⎢ dS = + 2 2 ⎥ dV T ⎢V − nb V T ⎥ (a) (P1, V1, T1) → (P*,V*,T1) (b) (P*,V*,T1) → (P**,V*,T2) (c) (P**,V*,T2) → (P2 ,V2, T2) V*→∞ a. (P a. (P1,V1,T1) → (P*,V*,T1), V*→∞ − 2n 2 a 2n 2 a ΔU1 = ∫ dV = V1 T V 2 T1V1 1 ∞ V* ΔS1 = ∫ V1 nR n2a V * − nb n 2 a [ + + 2 2 ]dV =nR ln V − nb V T1 V1 − nb V1T12 b. (P*,V*,T1) → (P**,V*,T2) (P dU = CV dT ig ig C dS = V dT T β2 γ3 ⎡ 2 3⎤ ΔU 2 = nR ⎢α (T2 − T1 ) + T2 − T1 + T2 − T1 ⎥ 2 3 ⎢ ⎥ ( )( ( ) ) ⎡ T2 γ2 2⎤ ΔS 2 = nR ⎢α ln + β (T2 − T1 ) + T2 − T1 ⎥ T1 2 ⎢ ⎥ c. (P**,V*,T2) → (P2,V2,T2) (P − 2n 2 a ΔU 3 = T2V2 V2 − nb n2a ΔS 3 = nR ln * − V − nb V2T2 2 ΔU = ΔU 1 + ΔU 2 + ΔU 3 ( )( ) β2 γ3 ⎡ 2 3⎤ = nR ⎢α (T2 − T1 ) + T2 − T1 + T2 − T1 ⎥ 2 3 ⎢ ⎥ ⎛1 1⎞ + 2n 2 a⎜ − ⎜V T V T ⎟ ⎟ 2 2⎠ ⎝ 11 ΔS = ΔS1 + ΔS 2 + ΔS 3 ( ) ⎡ T2 γ 2 2⎤ = nR ⎢α ln + β (T2 − T1 ) + T2 − T1 ⎥ T1 2 ⎢ ⎥ ⎛1 1⎞ ⎟ + n 2 a⎜ − ⎜V T 2 V T 2 ⎟ 22⎠ ⎝ 11 Δ H = Δ U + (P2V 2 − P1V1 ) 3. Const V Process (P1,V1,T1)→ (P2,V1,T2) ΔU = nR[α (T2 − T1 ) + β (T22 − T12 ) + γ3 (T23 − T13 )] 2 2n 2 a 1 1 (−) + V1 T1 T2 2 ΔS = nR[α ln T1 + β (T2 − T1 ) + γ2 (T22 − T12 )] T n2a 1 1 + ( 2 − 2) V1 T1 T2 Δ H = Δ U + ( P2 − P1 )V 1 4. Const P Process (P1,V1,T1)→ (P1,V2,T2) ΔU = nR[α (T2 − T1 ) + β (T22 − T12 ) + γ3 (T23 − T13 )] 2 + 2n 2 a ( 1 1 − ) V1T1 V2T2 2 ΔS = nR[α ln T1 + β (T2 − T1 ) + γ2 (T22 − T12 )] T 1 1 + n a( 2 − ) 2 V1T1 V2T2 2 ΔH = ΔU + P (V2 −V1) 1 5. Adiabatic Process (P1,V1,T1)→ (?,V2,?) 5.1 Reversible 2 ΔS = nR[α ln T1 + β (T2 − T1 ) + γ2 (T22 − T12 )] T + n 2 a( 1 1 − )=0 2 2 V1T1 V2T2 ⇒ T2 5.2 Irreversibly against a constant external pressure .Pext ΔU = nR[α (T2 − T1 ) + β (T22 − T12 ) + γ3 (T23 − T13 )] )] 2 + 2n 2 a ( 1 1 − ) = − Pext (V2 − V1 ) V1T1 V2T2 ⇒ T2 C. Calculations for Other Equation of Other Equation of State Vm = RT a + − bP P RT a⎞ ⎛ ∂V ⎞ ⎛R = n⎜ − ⎜ ⎟ ⎟ ∂T ⎠ P P RT 2 ⎠ ⎝ ⎝ ⎛ ∂ 2V ⎞ 2na ⎜ 2⎟ = 3 ⎜ ⎟ ⎝ ∂T ⎠ P RT ⎛ ∂V ⎞ ⎛ 2a ⎞ V −T⎜ − bP ⎟ ⎟ = n⎜ ⎝ ∂T ⎠ P ⎝ RT ⎠ Cvm = α + βT + γT 2 ig ΔH ⎛ ∂H ⎞ ⎛ ∂H ⎞ dH = ⎜ dT + ⎜ ⎟ ⎟ dP ⎝ ∂T ⎠ P ⎝ ∂P ⎠T ⎡ ⎛ ∂V ⎞ ⎤ = C P dT + ⎢V − T ⎜ ⎟ ⎥ dP ⎝ ∂T ⎠ P ⎦ ⎣ ⎛ 2a ⎞ = C P dT + n⎜ − bP ⎟dP ⎝ RT ⎠ ΔS ⎛ ∂S ⎞ ⎛ ∂S ⎞ dS = ⎜ dT + ⎜ ⎟ dP ⎟ ⎝ ∂T ⎠ P ⎝ ∂P ⎠T CP ⎛ ∂V ⎞ = dT − ⎜ ⎟ dP T ⎝ ∂T ⎠ P a⎞ C ⎛R = P dT − n⎜ − dP 2⎟ T ⎝ P RT ⎠ CP P ⎛ 2na ⎞ 2na ⎛ ∂C P ⎞ CP − CP = ∫ ⎜ P ⎟ dP = ∫0 − ⎜ ⎟ dP = − 3 3 0 RT ⎝ ∂P ⎠T ⎝ RT ⎠ 2na ig CP = CP − P 3 RT 2a ig C pm = C pm − P 3 RT ig P 1. Isothermal Process (P1,V1, T1)→(P2,V2,T1) 2a dH = n ( RT − bP ) dP 2a ΔH = n[ RT1 ( P2 − P ) − b ( P22 − P 2 )] 1 1 2 ΔU = ΔH − ( P2V2 − PV1 ) 1 dS = − n ( R − P a RT 2 ΔS = − n[( R ln P1 − P 2 ) dp a RT12 ( P2 − P1 )] 2. Any Process 2. Any Process (P1,V1,T1)→(P2,V2,T2) 2 dH = CpdT+ n( Ra − bP)dP T dS = Cp T a R dT − n( P − RT 2 )dP (a) (P1, V1, T1) → (P*,V*,T1) (b) (P*,V**,T1) → (P*,V**,T2) (c) (P*,V**,T2) → (P2 ,V2, T2) (a) (P1,V1,T1) → (P*,V*,T1) ΔH 1 = P*→0 − 2 na nb 2 P1 + P1 RT1 2 Δ S1 = − nR ln P* na − P 21 P1 RT1 P*→ 0 (b) (P*,V*, T1) → (P*,V**,T2) T2 ΔH2 = ∫ CP dT ig T1 = nR[(α + 1)(T2 − T1 ) + β (T22 −T 2) + γ3 (T23 − T13 )] 1 2 ig CP ΔS2 = ∫ dT T1 T T2 = nR[(α + 1) ln + β (T2 − T1 ) + γ (T22 − T12 )] 2 T1 T2 (c) (P*,V** ,T1) → (P2, V2 ,T2) b2 2a P2 − P2 ] ΔH 3 = n[ RT2 2 P a ΔS3 = n[ R ln 2* − P] 22 P RT2 ΔH = ΔH1 + ΔH 2 + ΔH 3 = nR[(α + 1)(T2 − T1 ) + β (T22 − T12 ) + γ3 (T23 − T13 )] 2 + 2na P2 P nb 2 ( − 1 ) − ( P2 − P 2 ) 1 R T 2 T1 2 ΔS = nR[(α + 1) ln T2 γ + β (T2 − T1 ) + (T22 − T12 )] T1 2 ⎡ P2 a P2 P1 ⎤ − n ⎢ R ln − ( 2 − 2 )⎥ P R T2 T1 ⎦ 1 ⎣ ΔU = ΔH − (P2V2 − PV1) 1 3. Const V Process (P1,V1,T1)→(P2,V1,T2) ΔH = nR[(α + 1)(T2 − T1 ) + β (T22 − T12 ) + γ3 (T23 − T13 )] 2 + 2na P2 P nb 2 ( − 1 ) − ( P2 − P 2 ) 1 R T 2 T1 2 ΔS = nR[(α + 1) ln T2 γ + β (T2 − T1 ) + (T22 − T12 )] T1 2 ⎡ P aP P⎤ − n ⎢ R ln 2 − ( 2 − 12 )⎥ P R T22 T1 ⎦ 1 ⎣ ΔU = ΔH − (P − P)V1 2 1 4. Const P Process (P1,V1,T1)→(P1,V2,T2) CP dS = dT T dH = CPdT ΔH = nR[(α + 1)(T2 − T1 ) + β (T22 − T12 ) + γ3 (T23 − T13 )] 2 + 2na 1 1 ( − )P 1 R T2 T1 T2 γ22 ΔS = nR[(α + 1) ln + β (T2 − T1 ) + (T2 − T1 )] T1 2 − naP ( 1 1 1 − 2) T22 T1 ΔU = ΔH − ( P2V2 − PV1 ) 1 5.1. Adiabatic Reversible (P1,V1,T1)→(P2,?,?) T2 γ2 ΔS = nR[(α + 1) ln + β (T2 − T1 ) + (T2 − T12 )] T1 2 P2 a P2 P − n[R ln − ( 2 − 1 )] = 0 P R T2 T12 1 ⇒T2 5.2 Adiabatic irreversible against a constant external pressure,Pext ΔU = −Pext (V2 −V1 ) ΔH −(PV2 − PV1) = −P t(V2 −V1) 2 1 ex β γ 2na P P nb 2 2 [(α +1)(T2 −T1) + (T22 −T12 ) + (T23 −T13 ) + ( 2 − 1 ) − (P P ) 2 1 2 3 R T2 T1 2 = ( P2 − Pext )V 2 − ( P 1 − Pext )V1 ⇒ T2 Cyclic rule rule P(V , T ), V (T , P), T ( P,V ) ⎛ ∂P ⎞ ⎛ ∂V ⎞ ⎛ ∂T ⎞ ⎜ ⎟⎜ ⎟ ⎜ ⎟ = −1 ⎝ ∂V ⎠T ⎝ ∂T ⎠ P ⎝ ∂P ⎠V Joule Thomson Experiment Joule Thomson Experiment P1,V1 U 2 − U1 = PV1 − P2V2 1 H 2 = H1 P2, V2 liquefaction of a gas using an isenthalpic of gas using an isenthalpic Joule-Thompson expansion Joule Thomson Coefficient Joule Thomson Coefficient ⎛ ∂T ⎞ μ ≡⎜ ⎟ ⎝ ∂P ⎠ H ⎛ ∂T ⎞ ⎛ ∂P ⎞ ⎛ ∂H ⎞ ⎜ ⎟⎜ ⎟⎜ ⎟ = −1 ⎝ ∂P ⎠ H ⎝ ∂H ⎠T ⎝ ∂T ⎠ P 1 ⎛ ∂H ⎞ 1⎡ ⎛ ∂T ⎞ ⎛ ∂V ⎞ ⎤ μ ≡⎜ ⎟ =− ⎜ ⎟ =− ⎟⎥ ⎢V − T ⎜ C P ⎝ ∂P ⎠T CP ⎣ ⎝ ∂P ⎠ H ⎝ ∂T ⎠ P ⎦ ⎛ ∂V ⎞ μ = 0, when V = T ⎜ ⎟ ∂T ⎠ P ⎝ Vm = RT a + − bP P RT Cvm = α + βT + γT 2 ig a⎞ ⎛ ∂V ⎞ ⎛R ⎜ ⎟ = n⎜ − 2⎟ ⎝ ∂T ⎠ P ⎝ P RT ⎠ ⎛ ∂V ⎞ ⎛ 2a ⎞ V −T⎜ = n⎜ − bP ⎟ ⎟ ⎝ ∂T ⎠ P ⎝ RT ⎠ 1⎡ 1 ⎛ 2a ⎛ ∂T ⎞ ⎛ ∂V ⎞ ⎤ ⎞ V −T⎜ − bP ⎟ ⎜ ⎟ =− ⎟ ⎥=− ⎢ CP ⎣ C P ,m ⎝ RT ⎝ ∂P ⎠ H ⎝ ∂T ⎠ P ⎦ ⎠ μ =⎜ 2 nRT na P= −2 V − nb V nR − nRT 2n 2 a ⎛ ∂P ⎞ ⎛ ∂P ⎞ ; +3 ⎜ ⎟= ⎜ ⎟= 2 V ⎝ ∂T ⎠V V − nb ⎝ ∂V ⎠T (V − nb) ⎛ ∂P ⎞ nR ⎜ ⎟ ⎛ ∂V ⎞ ⎝ ∂T ⎠V V − nb =− ⎜ ⎟ =− − nRT 2n 2 a ⎛ ∂P ⎞ ⎝ ∂T ⎠ P +3 ⎜ ⎟ (V − nb) 2 V ⎝ ∂V ⎠T nRT ⎛ ∂V ⎞ V − nb V = T⎜ ⎟= 2n 2 a nRT ⎝ ∂T ⎠ P − (V − nb) 2 V 3 2n 2 a nRTV nRT − 2= (V − nb) 2 V V − nb nRT ⎛ nb ⎞ 2n 2 a ⎜ ⎟= (V − nb) ⎝ V − nb ⎠ V 2 2a ⎛ nb ⎞ T= ⎜1 − ⎟ bR ⎝ V ⎠ 2 Joule Thomson Coefficient Thomson Coefficient Joule Thomson inversion temperature Joule Thomson inversion temperature Figure 3.6 All along the curves in the fi ure, μJfig T =0, and μJ-T is positive to the left of the curves and negative to the right. To experience cooling upon expansion at 100 atm, T must lie between 50 and 150 K for H2. The corresponding temperatures for N2 are 199 and 650 K. Example Problem 3.11 Example Problem 3.11 Example Problem 3.11 Using equation (3.43), (∂H/∂P)T=[(∂U/∂V)T+P](∂V/∂P)T+V show that μJ-T = 0 for an ideal gas. Solution μJ-T = − 1 CP ⎤ 1 ⎡⎛ ∂U ⎞ ⎛ ∂V ⎞ ⎛ ∂H ⎞ ⎛ ∂V ⎞ =− + P⎜ +V ⎥ ⎢⎜ ⎜ ⎟ ⎟⎜ ⎟ ⎟ CP ⎣⎝ ∂V ⎠T ⎝ ∂P ⎠T ⎝ ∂P ⎠T ⎝ ∂P ⎠T ⎦ =− 1 CP ⎡ ⎤ ⎛ ∂V ⎞ +V ⎥ 0+ P⎜ ⎢ ⎟ ⎝ ∂P ⎠T ⎣ ⎦ ⎡ ⎛ ∂ [ nRT / P ] ⎞ ⎤ ⎢P ⎜ ⎟ +V ⎥ ∂P ⎢ ⎥ ⎠T ⎣⎝ ⎦ 1 ⎡ nRT ⎤ =− − +V ⎥ = 0 CP ⎢ P ⎣ ⎦ 1 =− CP P3.11) Obtain an expression for the isothermal compressibility an expression for the isothermal compressibility 1 ⎛ ∂V ⎞ κ =− ⎜ ⎟ V ⎝ ∂P ⎠T for a van der Waals gas. van der Waals gas 1 ⎛ ∂Vm ⎞ κ =− ⎜ ⎟ =− Vm ⎝ ∂P ⎠T κ =− 1 1 =− ⎛ ∂P ⎞ ⎛⎛ ∂ ⎡ RT a ⎤ ⎞⎞ Vm ⎜ Vm ⎜⎜ −2 ⎟ ⎜ ∂Vm ⎢Vm − b Vm ⎥⎟⎟ ∂Vm ⎠T ⎝ ⎣ ⎦⎠⎟T ⎝⎝ ⎠ 1 ⎡2a RT ⎤ Vm ⎢ 3 − 2⎥ ⎢Vm (Vm − b) ⎥ ⎣ ⎦ P3 P3.13) show that that β2 CP = CV + TV κ P3.21) Derive the Joule coefficient the Joule coefficient 1⎡ ⎛ ∂T ⎞ ⎛ ∂P ⎞ ⎤ = P −T ⎜ ⎟ ⎥ ⎜⎟ ⎢ ⎝ ∂V ⎠U CV ⎣ ⎝ ∂T ⎠V ⎦ and calculate the Joule coefficient for an ideal gas and for a van der Waals gas. ...
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