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  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

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

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

Using Enthalpy  Enthalpy can be written:    For isobaric reactions  H = Q which is ~C  T  For an adiabatic process,  H =  V dP    Energy carried by flowing fluid   Can look up the flow work or heat needed to do isobaric work, etc.

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

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

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

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

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

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

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

Entropy Equations T dS = C V dT + T (  P/  T) V dV T dS = C P dT - T(  V/  T) P dP   Examples:   Since  = (1/V) (  V/  T) P, the second equation can be integrated to find the heat

Internal Energy Equations (  U/  P) T = -T (  V/  T) P - P(  V/  P) T  Example: 

Heat Capacity Equations C P - C V = -T(  V/  T) P 2 (  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)