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Entropy

1. Temperature, volume, pressure and energy are all state variables: their values depend only on the current state of the system, not on how it got there. Neither heat nor work are state variables.
2. The change in entropy for any process is defined by the integral of

   dS = dQ / T.

   where T is the temperature at which dQ is transferred.

   S is a state variable, so we can evaluate the integral over any convenient path (ie., an isothermal path even if the process was not isothermal).

   S differs from other state variables in that for any real process, ΔS > 0. This is one statement of the second law of thermodynamics.

3. The first law of thermodynamics can then be written

   dE = T dS - P dV.

   Defining the enthalpy as H ≡ E + P V, it takes the form

   dH = T dS + V dP.

   Since chemical processes commonly take place at constant pressure, the enthalpy in a chemical reaction indicates whether the reaction is exothermic (releases heat) or endothermic (absorbs heat).

   The Helmholtz free energy F ≡ E - T S, giving

   dF = -S dT - P dV.

   In an isothermal process, the Helmholtz free energy is the maximum amount of energy available to do work.

   Finally, the Gibbs free energy G ≡ E - T S + P V yields

   dG = -S dT + V dP.

   The Gibbs free energy can also be written as H - T S or F + P V. The Gibbs free energy is constant during phase transitions in which both the temperature and pressure are constant.

   By holding one variable constant, we can obtain a whole host of relations, such as

   • T = dE/dS_{constant V} = dH/dS_{constant P}
   • P = -dE/dV_{constant S} = -dF/dV_{constant T}
   • V = dH/dP_{constant S} = dG/dP_{constant T}
   • S = -dF/dT_{constant V} = -dG/dT_{constant P}
4. Microscopically, if Ω is the number of accessible states,

   S = k ln Ω.

   A useful analogy: if order is not an observable, Ω would correspond to the number of permutations, and S would correspond to the number of combinations.

   Since entropy is associated with the transfer of heat, it is useful to think of an increase in entropy as the distribution of thermal energy into a larger number of degrees of freedom.

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©2011, Kenneth R. Koehler. All Rights Reserved. This document may be freely reproduced provided that this copyright notice is included.

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