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 = ( d z/ d x) y dx + ( d z/ d y) x dy • or dz = M dx + N dy • For an equation of the form: ( d M/ d y) x = ( d N/ d x) y 8

Maxwell’s Relations

• We can apply the previous theorem to the four characteristic equations to get: ( d T/ d V) S ( d T/ d P) S ( d S/ d V) T ( d S/ d P) T = - ( d P/ d S) V = ( d V/ d S) P = ( = -( d d P/ V/ d d T) T) V P • We can also replace V and S (the extensive coordinates) with v and s – per unit mass 9

V A T U G S H P

König - Born Diagram

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

Internal Energy Equations

( d U/ ( d U/ d P) T d V) T = T ( d P/ d T) V = -T ( d V/ d T) P - P( d - P V/ d P) 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

C P - C V = -T( d V/ d T) P 2 c P - c V ( d P/ d V) T = Tv b 2 / k • Examples: • Heat capacities are equal when: • T = 0 (absolute zero) • ( d V/ d T) P maxima) = 0 (when volume is at minima or 15