# Entropy Thermodynamics Professor Lee Carkner Lecture 13.

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Entropy Thermodynamics Professor Lee Carkner Lecture 13

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 =

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

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

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

Area Under TS Diagram

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)

Information From an hs Diagram

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 

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

Entropy of a Pure Substance

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

Isentropic Diagram

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

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

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

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

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 

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