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110c-lecture2 - Reversible Expansion of Ideal Gas and P-V...

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Reversible Expansion of Ideal Gas and P-V Work Volume Pressure Final Initial V 2 V 1 Reversible Expansion (n-steps) Volume Initial V 2 V 1 Final P ext One-step (const P) w 1-step = -2 bar*(1 - 0.5 dm 3 ) = -100 J w rev = -nRT*ln(V 2 /V 1 ) = -(2 bar·dm 3 )*ln(1/0.5) = -139 J = = 2 1 2 1 V V ext dV P w w δ W 1-step = -P ext Δ V P ext = nRT V w rev = dV V nRT V V 2 1 = -nRT*ln(V 2 /V 1 ) n nRT=P 2 V 2 2 bar =
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Thermodynamic State Functions Depend Only on Final and Initial States DEFN: State function —a property that depends only on the state of a system (e.g P, V or T of an ideal gas) and not how the system was brought to that state. Energy (U) is a state function and is a measure of average motion of gas atoms (depends only on T for ideal gas). U U U dU Δ = = 1 2 1 2 (U is a state function) Work (w) is NOT a state function of a system because it depends on the path of a process and NOT simply the difference between final and initial states (Remember the different work above to compress a gas). w or w w dV P w w V V ext Δ = = 1 2 2 1 2 1 δ (w is a path function: P ext depends on V) Heat (q) also is NOT a state function because it depends on details of the path (temperature) and NOT simply difference in heat between final and initial states. q or q q dT T q q q T T Δ = = 1 2 2 1 2 1 (q is a path function) δ w is an inexact differential δ q is an inexact differential dU is an exact differential
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First Law of Thermodynamics DEFN: w q U + = Δ is the First Law of Thermodynamics. Conservation of Energy: Total energy of universe remains constant. Δ E univ = Δ E sys + Δ E surr = 0 ⇒Δ E sys = - Δ E surr Δ U = U f –U i = q + w A B Initial Final The energy of a system (U) is changed by doing work (w) or exchanging heat (q) dU = δ q+ δ w (differential form for infinitesimally small changes) First Law allows us to calculate heat (q), work (w) or energy (U): Constant Volume: δ w = -PdV = 0 dU = q = dT Isothermal ( Δ T = 0, dU =0): dU = 0 ⇒δ w=- q w = -nRTln Adiabatic ( δ q=±0): q=0 dU = w dT T C U T T V ) ( 2 1 = Δ V T q = 2 1 V V V dV nRT w V 2 V 1 ( Δ V = 0, δ w=±0) Change in energy ( Δ U) is work (w) performed plus heat (q) transferred.
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Heat (q) of Ideal Gas Expansion is Path Dependent Path A (isothermal): Δ T = 0 ⇒δ w A = - δ q A w A = -PdV = -nRT 1 dV V w A = -q A = = -nRT 1 ln V 2 2 1 V V V dV nRT V 1 Path B + C : q A = 1 2 1 ln V V nRT q B+C = q B + q C = 0 + 1 2 ) ( T T dT T Cv Path B (adiabatic): q B = 0 Path C ( V = 0): dU = q C & U = q C dT T q dU T T ∫∫ = 1 2 q C = dT T C T T V ) ( 1 2 = q C
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110c-lecture2 - Reversible Expansion of Ideal Gas and P-V...

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