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The Carnot Cycle Part 2

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Isochor. Curved line with slope: (dT/dS)V = T/CV ... Isochor. Entropy and Isotherms. We write change in entropy as: DS = dQ/T. If T is constant ... – PowerPoint PPT presentation

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Title: The Carnot Cycle Part 2


1
The Carnot Cycle - Part 2
  • Physics 313
  • Professor Lee Carkner
  • Lecture 17

2
Carnot and Temperature
  • How are the heat exchanges related to the
    temperature?
  • For isothermal processes heat work
  • For ideal gas
  • Q W nRT ln (Vf/Vi)
  • QH/QL TH/TL (ln V2/V3)/ln V4/V1)
  • The volume term equals 1 from PVg const
  • QH/QL TH/TL

3
Temperature Scale
  • Temperature can be related to the heat transfers
    of a Carnot engine
  • This can be used to form a temperature scale
  • Using the triple point of water
  • T 273.16 Q/QTP
  • Called the thermodynamic temperature
  • Equal to the ideal gas temperature

4
Efficiency
  • Can write the efficiency of a Carnot engine as
  • h 1 - (QL/QH)
  • h 1 - (TL/TH)
  • Increase the efficiency by increasing TH and
    decreasing TL
  • Efficiency is 1 for TL equal to absolute zero
  • For a Carnot refrigerator the coefficient of
    performance is
  • K TL / (TH - TL)

5
Entropy
  • The limits on efficiency for engines and
    refrigerators are expressions of entropy
  • Entropy is a measure of the randomness of a
    system
  • Entropy must always increase
  • Entropy represents a preferred direction for
    processes
  • You can violate the second law locally but not
    globally

6
Heat and Temperature
  • We saw that
  • QH/QC TH/TC
  • If we include the signs of the heat
  • QH/TH QC/TC 0
  • This is true for any Carnot cycle
  • Any curve can be represented as the sum of many
    Carnot cycles
  • S (Q/T) 0

7
Entropy Defined
  • For any reversible cycle
  • ? (dQ/T) 0
  • For the integral over the complete cycle
  • The integral along any reversible (non-closed)
    path represents the change in entropy
  • Sf - Si ? (dQ/T)
  • dS dQ/T

8
Ideal Gas Entropy
  • To calculate entropy need expression for dQ
  • dQ CVdT PdV
  • dS CV (dT/T) P/T dV
  • DS ? CV (dT/T) ? nR (dV/V)
  • DS n ?cV (dT/T) nR ln (Vf/Vi)
  • Similarly for
  • dQ CP dT -VdP
  • DS n ? cP (dT/T) - nR ln (Pf/Pi)

9
T and S
  • Heat can be expressed as
  • Q ? T dS
  • Heat is the area under the curve on a TS diagram
  • Only for reversible processes
  • A reversible cycle is a closed loop

10
The TS Diagram
  • How are standard processes plotted on a TS
    diagram?
  • Isotherm
  • horizontal line
  • Adiabatic
  • Since dQ 0, dS 0
  • No entropy change, so vertical line
  • Called an isentrope

11
Other Processes
  • Isobar
  • Curved line with slope
  • (dT/dS)P T/CP
  • Isochor
  • Curved line with slope
  • (dT/dS)V T/CV
  • Isobar has smaller slope than isochor since CP gt
    CV

12
TS Diagram
Isentrope
T
Isochor
Isobar
Isotherm
S
13
Entropy and Isotherms
  • We write change in entropy as
  • DS ? dQ/T
  • If T is constant
  • DS Q/T
  • The change in entropy for an isothermal process
    depends only on the temperature and the total
    heat exchange
  • True for a system in contact with a reservoir
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