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General Relations

General Relations. Thermodynamics Professor Lee Carkner Lecture 24. PAL #23 Maxwell. Determine a relation for (  s/  P) T for a gas whose equation of state is P( v -b) = RT (  s/  P) T = -(  v /  T) P P( v -b) = RT can be written, v = (RT/P) + b

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General Relations

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  1. General Relations Thermodynamics Professor Lee Carkner Lecture 24

  2. PAL #23 Maxwell • Determine a relation for (s/P)T for a gas whose equation of state is P(v-b) = RT • (s/P)T = -(v/T)P • P(v-b) = RT can be written, v = (RT/P) + b • (s/P)T =-(v/T)P = -R/P

  3. PAL #23 Maxwell • Verify the validity of (s/P)T = - (v/T)P for refrigerant 134a at 80 C and 1.2 MPa • Can write as (Ds/DP)80 C = -(Dv/DT)1.2 MPa • Use values of T and P above and below 80 C and 1.2 MPa • (s1400 kPa – s1000 kPa) / (1400-1000) = -(v100 C – v60 C) / (100-60) • -1.005 X 10-4 = -1.0095 X 10-4

  4. Key Equations • We can use the characteristic equations and Maxwell’s relations to find key relations involving: • enthalpy • specific heats • so we can use an equation of state

  5. Internal Energy Equations du = (u/T)v dT + (u/v)T dv • We can also write the entropy as a function of T and v ds = (s/T)v dT + (s/v)T dv • We can end up with du = cvdT + [T(P/ T)v – P]dv • This can be solved by using an equation of state to relate P, T and v and integrating

  6. Enthalpy dh = (h/T)v dT + (h/v)T dv • We can derive: dh = cpdT + [v - T(v/T)P]dP • If we know Du or Dh we can find the other from the definition of h Dh = Du + D(Pv)

  7. Entropy Equations • We can use the entropy equation to get equations that can be integrated with a equation of state: ds = (s/T)v dT + (s/v)T dv ds = (s/T)P dT + (s/P)T dP ds = (cV/T) dT + (P/T)V dv ds = (cP/T) dT - (v/T)P dP

  8. Heat Capacity Equations • We can use the entropy equations to find relations for the specific heats (cv/v)T = T(2P/T2)v (cp/P)T = -T(2v/T2)P cP - cV = -T(v/T)P2 (P/v)T

  9. Equations of State • Ideal gas law: Pv = RT • Van der Waals (P + (a/v2))(v - b) = RT

  10. Volume Expansivity • Need to find volume expansivity b = • For isotropic materials: b = • where L.E. is the linear expansivity: L.E. = • Note that some materials are non-isotropic • e.g.

  11. Volume Expansivity

  12. Variation of b with T • Rises sharply with T and then flattens out • Similar to variations in cP

  13. Compressibility • Need to find the isothermal compressibility • = • Unlike b, a approaches a constant at 0 K • Liquids generally have an exponential rise of a with T: a = a0eaT • The more you compress a liquid, the harder the compression becomes

  14. Mayer Relation cP - cV = Tvb2/a • Known as the Mayer relation

  15. Using Heat Capacity Equations cP - cV = -T(v/T)P2 (P/v)T cP - cV = Tvb2/a • Examples: • Squares are always positive and pressure always decreases with v • T = 0 (absolute zero)

  16. Next Time • Final Exam, Thursday May 18, 9am • Covers entire course • Including Chapter 12 • 2 hours long • Can use all three equation sheets plus tables

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