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# 240_l13 - The Second Law The Machinery Chapter 5 of Atkins...

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The Second Law: The Machinery Sections 5.1-5.6, 6th Ed.; 5.1 - 5.5, 7th Ed.; 3.7-3.9 8th Ed. Combining First and Second Laws Properties of the Internal Energy Properties of the Gibbs Energy The Chemical Potential of a Pure Substance Real Gases: The Fugacity Definition of Fugacity Standard States of Gases Relation Between Fugacity and Pressure Chapter 5 of Atkins: The Second Law: The Concepts Last updated: Oct. 30, 2006 - Slide 1 updated

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The Second Law: The Machinery The laws of thermodynamics we have learned thus far now enable us to begin finding relationships between properties that may have not been thought to be related with one another - there are many interesting “hidden” relationships that can be extracted - lets now combine the laws: The First Law of Thermodynamics: dU ' dq % dw For reversible change in constant composition system with no non- expansion work dw rev ' & pdV dq rev ' TdS Therefore: dU ' TdS & pdV dU is an exact differential, independent of path, so the same values for dU are obtained regardless of change being reversible or irreversible (closed system, no non-expansion work) Fundamental Equation
Properties of Internal Energy Reversible change : T dS same as dq , and -p dV same as dw Irreversible change : T dS > dq (Clausius inequality), and -p dV > dw For a system of constant composition: dw + dq = T dS + -p dV dU ' T dS & p dV When S and V are changed, dU % dS and dU % dV , from the fundamental equation, suggesting that dU should be written as a function of S and V dU ' M U M S V dS % M U M V S dV The equation above means that the change in U is proportional to changes in S and V , with the coefficients being slopes of plots of U against S at constant V , and U against V at constant S . For systems of constant composition: M U M S V ' T M U M V S ' & p Thermodynamic definition of temperature Thermodynamic definition of pressure

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Maxwell Relations We know that dU = T dS - p dV is exact, then M T M V S ' & M p M S V We have generated a relationship between quantities that would not seem to be related on first sight! In fact, the four Maxwell relations can be derived from the four state functions U , H , G and A : M T M V S ' & M p M S V M T M p S ' M V M S p M p M T V ' M S M V T M V M T p ' & M S M p T The state functions are exact differentials , meaning that they must pass the test that indicate their independence of path taken: df ' g dx % h dy is exact if M g M y x ' M h M x y
Variation of Internal Energy with Volume In Chapter 3, we defined a coefficient called the internal pressure B T ' M U M V T B T ' T M p M T V & p which can be written as a thermodynamic equation of state , in terms of p and T , and can therefore be applied to any substance This expression for B T is obtained by dividing by dV and substituting in: dU ' M U M S V dS % M U M V S dV M U M S V ' T M U M V S ' & p M U M V T ' M U M S V M S M V T % M U M V S ' T M S M V T & p M S M V T ' M p M T V a Maxwell relation

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Properties of the Gibbs Energy For a closed system doing no non-expansion work, dU can be replaced by the fundamental equation, dU = T dS - p dV dG ' ( TdS & pdV ) % pdV %
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