5 Machinery of the First Law

5 Machinery of the First Law - The First Law: Machinery...

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The First Law: Machinery 8th Edition: Sections 2.10-2.12 & Further Information in Ch. 2 State Functions Exact and inexact differentials Changes in internal energy The Joule experiment Changes in internal energy at constant p Temperature Dependence of Enthalpy Changes in enthalpy at constant volume Isothermal compressibility Joule-Thomson effect C V vs. C p Chapter 3 of Atkins, 6th, 7th Ed: The First Law: the machinery Last updated Oct 10, 2007: Minor edit, slide 9
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State vs. Path Functions state functions : Properties are independent of how the substance is prepared, and are functions of variables such as pressure and temperature (define the state of system) examples: U : internal energy H : enthalpy path functions : Properties that relate to the preparation of the state of the substance examples : w : work done preparing a state q : energy transferred as heat state functions: system possesses U and H path functions: states do not possess q and w
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State Functions Initially: state has internal energy U i Path 1 : adiabatic expansion to final state with internal energy U f • work done on system is w Path 2 : non-adiabatic expansion to final state with U f q & and w are both done on the system U : property of state (same value of ) U in both cases) w , q : property of path
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Exact & Inexact Differentials Exact Differential: System is taken along a path, with ) U = U f - U i , and the overall change is the sum of the infinitessimal changes along the path (i.e., an integral): Inexact Differential: System is heated, total energy transferred as heat is the sum of individual contributions along each point of the path: q ' m f i ,p a th dq Do not write ) q : q is not a state function, energy is not q f - q i # q depends upon the path of integration (e.g., adiabatic vs. non- adiabatic) - path indepdence is expressed by saying that dq is an inexact differential - infinitessimal quantity that depends upon the path ( dw is also an inexact differential ) ) U ' m f i dU ) U is independent of path - path independence is expressed by saying that dU is an exact differential - an infinitessimal quantity, which when integrated gives a path independent result
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Work, heat, internal energy and pathways Consider a perfect gas in a cylinder with a piston: Initial state T , V i Final state T , V f Change of state: Path 1 : free expansion against no external pressure Path 2 : reversible isothermal expansion Path 3 : irreversible isothermal expansion against p ext 0 Calculate q , w and ) U for each pathway All pathways: internal energy arises from kinetic energy of molecules, so since processes are isothermal, ) U = 0, so q = -w Path 1 : free expansion, w = 0, so q = 0 Path 2 : w = - nRT ln ( V f / V i ), so q = nRT ln ( V f / V i ) Path 3 : w = - p ext ) V , so q = p ext ) V (since ) U = 0)
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Changes in Internal Energy, 1 For a closed system of constant composition, U is function of V and T (possible to express p in terms of V and T , so p is not independent here) Say V makes a small change to V + dV at constant T : U ) ' U % M U M V T dV or T changes to T + dT at constant V : U ) ' U % M U M T V dT The coefficients ( M U / M V ) T and ( M U / M T ) V are partial derivatives of U w.r.t.
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This note was uploaded on 07/31/2011 for the course CHM 170 taught by Professor Lemtayo during the Spring '11 term at MIT.

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5 Machinery of the First Law - The First Law: Machinery...

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