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1 Mathematical Methods Physics 313 Professor Lee Carkner Lecture 22

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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

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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

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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

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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

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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

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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

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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

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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

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

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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: ( S/ P) T = -( V/ T) P

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

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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

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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

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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)

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1 The Second Law of Thermodynamics (II). 2 The Fundamental Equation We have shown that: dU = dq + dw plus dw rev = -pdV and dq rev = TdS We may write:

1 The Second Law of Thermodynamics (II). 2 The Fundamental Equation We have shown that: dU = dq + dw plus dw rev = -pdV and dq rev = TdS We may write:

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