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Entropy Thermodynamics Professor Lee Carkner Lecture 13

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PAL # 12 Carnot Engine powering a heat pump Find work needed for heat pump COP HP = 1 /(1-T L /T H ) = 1/(1 – 275/295) = COP HP = Q H /W in W = Q/COP = 62000/14.75 = Total engine work is twice that needed for heat pump W engine = 2W HP = th = 1 – (T L /T H ) = 1 – (293/1073) = 0.727 th = W/Q H Q H = W/ th = 8406/0.727 =

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Clausius Inequality ∫ Q/T = 0 ∫ Q/T ≤ 0 Valid for all cycles

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Entropy The integral is a state quantity called the entropy s = ∫ Q/T Entropy is a tabulated property of a system like v, h or u S = Q/T

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TS Diagram Can plot a Temperature-Entropy Diagram The total heat is the integral of TdS, or the area under the process line on a TS diagram Similar to W being the area in a P v diagram

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Area Under TS Diagram

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Mollier Diagram The vertical (enthalpy) distance gives a measure of the work The horizontal (entropy) distance gives a measure of the irreversibilities (and thus inefficiencies)

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Information From an hs Diagram

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Entropy Rules Processes must proceed in the direction that increases entropy Entropy is not conserved Rate of entropy generations tells us the degree of irreversibility and thus efficiency

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Finding Entropy Change Entropy in tables assign zero to some arbitrary point For mixtures of vapor and liquid: We will normally assume the entropy of a compressed liquid is the same as a saturated liquid

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Entropy of a Pure Substance

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Isentropic Process e.g. well-insulated and frictionless Can use to determine the properties of the initial and final states of the system Find s for state 1 and you know that state 2 must have a P and T to give the same s

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Isentropic Diagram

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Disorder The higher the entropy the more random the distribution of molecules High entropy → high disorder → low quality → low efficiency

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The 3 rd Law Molecule motions decrease with decreasing temperature Only one configuration means absolute order

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Gibbs Equation but we already have relationships for Q Solving for Tds TdS – Pd V = dU Called the Gibbs equation Substituting into the Gibbs equation: Tds = dh - v dP

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Entropy Relations We can know write integrable equations ds = dh/T + v dP/T We still need: e.g., du = c v dT e.g., P v = RT

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Incompressible Entropy For solids and liquids the volume does not change much d v = 0 We can integrate the entropy equation: S = ∫ c dT/T = c ln T 2 /T 1

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Next Time Read: 7.7-7.13 Homework: Ch 7, P: 46, 54, 66, 93

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