Mathematical Methods

Mathematical Methods Physics 313 Professor Lee Carkner Lecture 22 1

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 2

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 3

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 4

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

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 6

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

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

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

A König - Born Diagram V T U G P S H 10

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: (DS/DP)T = -(DV/DT)P

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

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

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

Heat Capacity Equations CP - CV = -T(dV/dT)P2 (dP/dV)T cP - cV = Tvb2/k • Examples: • Heat capacities are equal when: • T = 0 (absolute zero) • (dV/dT)P = 0 (when volume is at minima or maxima) 15

Mathematical Methods