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Understanding Molar Response Functions and Thermodynamic Potentials in Thermodynamics

This article explores key concepts in thermodynamics, focusing on molar response functions and the relations between various thermodynamic potentials. It discusses how extensive variables are independent of system size, introduces the notion of response functions under constant conditions, and clarifies the roles of heat capacity and elasticity. The text elaborates on Maxwell relations, Legendre transformations, and the interrelations among Helmholtz free energy, Gibbs free energy, and other potentials. By providing insights into derivatives and the Jacobian matrix, this work aims to elucidate complex thermodynamic interactions.

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Understanding Molar Response Functions and Thermodynamic Potentials in Thermodynamics

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  1. Observables

  2. Molar System • The ratio of two extensive variables is independent of the system size. • Denominator N as particle • Denominator N as mole • Entropy also scales by N. • Equation of state as well

  3. Partial derivatives of the state equation define other measurable quantities. Response functions Response functions can be limited to systems of constant number. Molar response functions Use definitions of heat and work Response Function

  4. Thermal Response • Heat capacity is a measurement of the thermal response of a system. • Since there are multiple variables, one selects certain ones to be constant. • Total or molar

  5. Mechanical Response • Mechanical response functions compare other extensive variables to conjugate forces or temperature. • Isothermal compressibility – volume vs pressure • Elastic constant – length vs force • Thermal expansivity – volume vs temperature A Dx A

  6. Derivatives can be reordered. Second order equivalence Forms a Maxwell relation Maxwell relations can be applied to response functions to create new equivalences. Example: integrable expression for temperature change. Maxwell Relations Robertson (1998)

  7. New Parameter • Consider a function A(x,y) with two independent variables. • Plot as curve • Take a line of variable slope f1. • Depends on new variable z • Maximize f2 = A(x) to find y*(z). • Function B of three variables • Variable x is passive

  8. Legendre Transformation • The change of variables is a Legendre transformation. • Swaps dependent and independent variables • Transforms Lagrangian to Hamiltonian

  9. The total energy U is a potential that can be transformed. The Helmholtz free energy swaps entropy and temperature. The enthalpy swaps volume and pressure. Thermodynamic Potentials

  10. The thermodynamic potentials can be transformed twice. The Gibbs free energy depends on temperature and pressure. The grand potential swaps chemical potential for particle number. Double Transforms

  11. Maxwell Diagram • The Maxwell relationships can be used to describe the relationships between potentials. • Diagram with potentials and variables • Example with P, V coordinates • In the diagram the potentials are functions of the adjacent corners. • Complementary variables at opposite corners • Transform to adjacent side V F T U G S H P

  12. Jacobian • The partial derivatives of a set of variables can be expressed as a Jacobian matrix. • The properties of the Jacobian can be used to simplify complicated expressions. • Example: specific heat

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