Chapter 7: Thermodynamic Driving Forces

“Thermodynamics is Two Laws and a Little Calculus”

I. Definitions

• Thermodynamic system - what we study– Open: can exchange U, V, n– Closed: can exchange U, V, but not n– Isolated: cannot exchange U, V, n

• Surroundings - everything else• Boundaries

– Semipermeable: allows some atoms to pass– Adiabatic: allows no heat to pass

• Phase: homogeneous; uniform in p, T, [A]

More Definitions

• Property: measurable of a system– Extensive = function of n, N, V

• U, S, H, G

– Intensive ≠ function of n, N• T, P, ρ, [A]

ReviewDegree of Freedom

Observation

( max W)

Driving Force

Ex. 2.2

pressure

V As V increases, gas expands.

p

Ex 2.3

diffusion

particle exch α Nj

As {Nj} increases, gases mix and particle distrib more uniform

Chem potential, μj

Ex. 3.4 U Heat flows until T is uniform

T

II. Fundamental Thermodynamic Equations: Entropy

• S(U, V, N1, N2, …)

• dS = (δS/δU)V,NdU + (δS/δV)U,NdV + Σ(δS/δNj)V,U,Ni dNj Eqn 7.1

• dS = T-1 dU + pT-1 dV - Σ μj T-1 dNj Eqn 7.5

• Note: dV, dNj, dU are differences in the degrees of freedom (DegF). p, μj, T are the driving forces. As driving forces (DF) become more uniform, d(DegF) 0.

Fundamental Thermodynamic Equations: Energy

• U(S, V, N)

• dU = (δU/δS)V,NdS + (δU/δV)S,NdV + Σ(δU/δNj)V,S,Ni dNj Eqn 7.2

• dU = TdS - pdV + Σ μj dNj Eqn 7.4

• Note: (δU/δS)V,N = T means that the increase in energy per increase in entropy is positive; as S increases, so does U and in proportion to T.

III. Equilibrium: dS = 0

• Identify system, variables (DegF), constants

• Identify constraints, relationships

• Maximize total entropy

• Apply constraint

• Combine and rearrange to find requirement for equilibrium

Thermal Equilibrium (Ex. 7.2)

• System = isolated = Object A (SA, UA, TA) + Object B (with similar properties); variables = UA, UB; constant = V, N ST(U) = SA + SB = S(UA, UB)

• UT = UA + UB = constant constraint dU = dUA + dUB = 0 or dUA = - dUB

• To maximize entropy: dST= 0 = (δSA/δUA)V,NdUA + (δSB/δUB)V,NdUB

• (δSA/δUA)V,N = (δSB/δUB)V,N 1/TA = 1/TB

Thermal Equilibrium (2)

• What does this mean? 1/TA = 1/TB TA = TB

• In order to maximize entropy, energy or heat will transfer until the temperatures are equal.

• Will heat flow from hot to cold or vice versa? Check dST = (1/TA - 1/TB)dUA

Mechanical Equilibrium (Ex. 7.3)

• Complete

Chemical Equilibrium (Ex. 7.5)

• Complete

Two Laws of Thermodynamics

• First Law

dU = δq + δw

dU = T dS – p dV (for closed system)

• Second Law

dS = δq/T

More Definitions

• State variables (state functions)

• Process variables(path functions)

• Quasi-static process: such that properties ≠ f(time, process speed)

• Reversible process: special case of quasi-static such that can be reversed with no entropy change (ideal case)

• Thermodynamic cycle: initial = final state

IV. Applications of Fundamental Thermodynamic Equations

• Reversible and Irreversible

• Work δw = -pext dV (quasi-static process)

– ΔV = 0– Δp = 0 isobaric– ΔT = 0 isothermal– q = 0 adiabatic

• Entropy

• Cycles