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Mathematical Methods

Mathematical Methods. Physics 313 Professor Lee Carkner Lecture 20. Mathematical Thermodynamics. Experiment or theory often produce relationships in forms that are inconvenient for the problem at hand Many differential equations are hard to compute. Legendre Differential Transformation.

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Mathematical Methods

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  1. Mathematical Methods Physics 313 Professor Lee Carkner Lecture 20

  2. Mathematical Thermodynamics • Experiment or theory often produce relationships in forms that are inconvenient for the problem at hand • Many differential equations are hard to compute

  3. Legendre Differential Transformation • For an equation of the form: • we can define, • and get: • We use a Legendre transform when f is not a convenient variable and we want xdu instead of udx • e.g. replace PdV with -VdP

  4. Characteristic Functions • The internal energy can be written: dU = -PdV +T dS • We can use the Legendre transformation to find other expressions relating P, V, T and S • We will deal specifically with the hydrostatic thermodynamic potential functions, which are all energies

  5. Enthalpy • From dU = -PdV + T dS we can define: • H is the enthalpy • Since work is done in an isobaric process, enthalpy measures the energy needed to do work in this case • Change in internal energy is a measure of the energy needed for a temperature change

  6. Using Enthalpy • Enthalpy can be written: • For isobaric reactions DH = Q which is ~CDT • For an adiabatic process, DH =  V dP • Energy carried by flowing fluid • Can look up the flow work or heat needed to do isobaric work, etc.

  7. Helmholtz Function • From dU = T dS - PdV we can define: • A is called the Helmholtz function • Used when T and V are convenient variables • Can be related to the partition function

  8. Gibbs Function • If we start with the enthalpy, dH = T dS +V dP, we can define: • G is called the Gibbs function • For isothermal and isobaric processes, G remains constant • chemical reactions

  9. A PDE Theorem • The characteristic functions are all equations of the form: • or dz = M dx + N dy (M/y)x = (N/x)y

  10. Maxwell’s Relations • We can apply the previous theorem to the four characteristic equations to get: (T/V)S = (T/ P)S = ( S/V)T = (S/P)T = • We can also replace V and S (the extensive coordinates) with v and s • per unit mass

  11. Example: What is expression for dU? dU = TdS-PdV (T/V)S=-(P/S)V König - Born Diagram A V T U G P S H

  12. Using Maxwell’s Relations • Example: finding entropy • Using the last two Maxwell relations we can find the change in S by taking the derivative of P or V • Example: • Can read off “straddling” values on a table

  13. Key Equations • We can use the characteristic equations and Maxwell’s relations to find key relations involving:

  14. Entropy Equations T dS = CV dT + T (P/T)V dV T dS = CP dT - T(V/T)P dP • Examples: • Since b = (1/V) (V/T)P, the second equation can be integrated to find the heat

  15. Internal Energy Equations (U/P)T = -T (V/T)P - P(V/P)T • Example:

  16. Heat Capacity Equations CP - CV = -T(V/T)P2 (P/V)T • Examples: • Volume always increases with T and pressure always decreases with V • (V/T)P = 0 (when volume is at minima or maxima)

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