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5-section - 1 5 Statistical Thermodynamics The objective of...

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1 5. Statistical Thermodynamics The objective of this section (and Chapter 15 of the text) is to show how the macro- scopic properties of molecular systems which you learned about in your study of classical thermodynamic may be described in terms of the quantities and variables of statistical mechanics. In doing this, we will need to draw on the definitions of and relationships among the properties and variables of classical thermodynamics. Recall that U is the total internal energy of the system The First Law of Thermodynamics states that changes in U are defined as Δ U = q heat + w where q heat is the amount of heat energy which passively flows into (positive q heat ) or out of (negative q heat ) the system, and w is the amount of energy which flows into (positive w ) or out of (negative w ) the system due to some type of work (mechanical, PV expansion/compression, electrical, . . . etc.) being done by/on the system. While U is a function of state , q heat and w are not. The value of w associated with an infinitesimal expansion or compression of the system is dw = - P opp dV where P opp is the pressure which is opposing the volume change dV . Enthalpy is defined as H = U + PV is also a function of state. Entropy is a function of state defined by the fact that dS = q rev T
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2 The Second Law of Thermodynamics states that in any “spontaneous” (or “natural” or “irreversible”) process dS > dq T The Helmholtz free energy is defined as A = U - TS The Gibbs free energy is defined as G = H - TS For infinitesimal reversible changes in a closed system with only PV work: dU = T dS - P dV = ∂U ∂S V,N dS + ∂U ∂V S,N dV (1) dH = T dS + V dP = ∂H ∂S P,N dS + ∂H ∂P S,N dP (2) dA = - S dT - P dV = ∂A ∂T V,N dT + ∂A ∂V T,N dV (3) dG = - S dT + V dP = ∂G ∂T P,N dT + ∂G ∂P T,N dP (4) 5.1 Internal Energy U and Heat Capacity C V We recall from Unit 3 that over an ensemble of N equivalent systems, the population in state i is defined by the canonical distribution function a i = N q e - β ε i the fraction of systems in state- i is p i = a i N = 1 q e - β ε i and that the ensemble average of any property i if state- i is h f i = f = X states i p i f i = 1 q ( N, V, β ) X states i ( f i e - β ε i )
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