1 Mathematical Methods Physics 313 Professor Lee Carkner Lecture 22

2 Mathematical Thermodynamics Experiment or theory often produces relationships in a form that is inconvenient for the problem at hand –We can use mathematics for a change of variables into forms that are more useful Many differential equations are hard to compute –Want to find an equivalent expression that is easier to solve

3 Legendre Differential Transformation For an equation of the form: df = udx + vdy we can define, g = f - ux and get: dg = -x du +v dy

4 Characteristic Functions The internal energy can be written: dU = dW + dQ dU = -PdV +T dS We can use the Legendre transformation to find other expressions relating P, V, T and S These expressions are called characteristic functions of the first law

Enthalpy From dU = -PdV + T dS we can define: H = U + PV dH = VdP +TdS H is the enthalpy –Enthalpy is the isobaric heat H functions much like internal energy in a constant volume process Used for problems involving heat

6 Helmholtz Function From dU = T dS - PdV we can define: A = U - TS dA = - SdT - PdV A is called the Helmholtz function –Change in A equals isothermal work Used when T and V are convenient variables –Used in statistical mechanics

Gibbs Function If we start with the enthalpy, dH = T dS +V dP, we can define: G = H -TS dG = V dP - S dT G is called the Gibbs function –Used when P and T are convenient variables For isothermal and isobaric processes (such as phase changes), G remains constant –used with chemical reactions

8 A PDE Theorem The characteristic functions are all equations of the form: dz = (  z/  x) y dx + (  z/  y) x dy or dz = M dx + N dy For an equation of the form: (  M/  y) x = (  N/  x) y

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

10 König - Born Diagram H G A U S P VT

Using Maxwell’s Relations Example: finding entropy –Equations of state normally written in terms of P, V and T –Using the last two Maxwell relations we can find the change in S by taken the derivative of P or V Maxwell’s relations can also be written as finite differences Example: (  S/  P) T = -(  V/  T) P

12 Key Equations We can use the characteristic equations and Maxwell’s relations to find key relations involving: –entropy –internal energy –heat capacity

13 Entropy Equations T dS = C V dT + T (  P/  T) V dV T dS = C P dT - T(  V/  T) P dP Examples: If you have equation of state, you can find (  P/  T) V and integrate T dS to find heat Since  = (1/V) (  V/  T) P, the second equation can be integrated to find the heat

14 Internal Energy Equations (  U/  V) T = T (  P/  T) V - P (  U/  P) T = -T (  V/  T) P - P(  V/  P) T Example: The change in U with V or P can be found from the derivative of the equation of state

15 Heat Capacity Equations C P - C V = -T(  V/  T) P 2 (  P/  V) T c P - c V = Tv  2 /  Examples: Heat capacities are equal when: T = 0 (absolute zero) (  V/  T) P = 0 (when volume is at minima or maxima)