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Observables

Observables. 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. Partial derivatives of the state equation define other measurable quantities.

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Observables

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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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