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Chemical Potential, Μ Partial Molar Properties Entropy of Mixing Compressibility Thermoelastic Effect Magnetic Effects

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Chemical Potential, Μ Partial Molar Properties Entropy of Mixing Compressibility Thermoelastic Effect Magnetic Effects Chapter 3 • Property Relations The essence of macroscopic thermodynamics Dependence of U, H, S, G, and F on T, P, V, etc. • Concepts Energy functions F and G Chemical potential, µ Partial Molar properties Entropy of mixing Compressibility Thermoelastic effect Magnetic Effects WS2002 1 1 Property Relations dU=+δδ Q W • First law δδQ lw dS =− Second law T T • Multiply by T TdS=−δδ Q lw Subtract from the first law dU −=+ TdSδδ W lw • Energy funciton for Enthalpy WS2002 2 2 Helmholtz Free Energy • Also called the work function A measure of the work required to change a system from one state to another • From the previous discussion Integrating at constant temperature Rearranging By definition • Energy Function for Helmholtz Free Energy WS2002 3 3 Gibbs’ Free Energy • In the laboratory, we often change the state of a system at constant temperature and pressure exclusive of any P-V type work • Conditions arise for phase changes at constant pressure • Under these conditions, the work required to initiate a phase change is the reversible work minus the P-V work δδ=−=−* Wrev W rev PdV dU TdS δ * =− + Wrev dU TdS PdV • Integrating at constant temperature and pressure • Gibbs’ Free Energy WS2002 4 4 Chemical Potential • Up to now we have discussed changes in closed systems in homogeneous materials • Terms can be added to the definitions of U, H, F, and G to deal with changes in composition • From these equations, it can be shown that: ∂∂∂∂G F U H = = = =µ i ni ≠≠≠≠ni ni ni TPn, ,ji n TVn , , ji n vSn ,, ji n PSn ,, ji n • The chemical potential is a measure of the propensity of a constituent of a system to undergo change • At equilibrium, the chemical potentials of all the species in a system are equal WS2002 5 = UUSVn(, ,i ) ∂U ∂U ∂U dU = dS + dV + ∑ dn ∂ ∂ ∂ i S V i ni ≠ Vn,,ii Sn SVn,,ij n ∂U dU=−+ TdS PdV ∑ ∂ i ni ≠ SVn,,ij n = HHSPn(,,i ) ∂H ∂H ∂H dH = dS + dP + ∑ dn ∂ ∂ ∂ i S P i ni ≠ Pn,,ii Sn SPn,, ij n ∂H dH=+ TdS VdP +∑ ∂ i ni ≠ SPn,, ij n = FFTVn(, ,i ) ∂F ∂F ∂F dF = dT + dV + ∑ dn ∂ ∂ ∂ i T V i ni ≠ Vn,,ii Tn TVn,,ij n ∂F dF=− SdT − PdV +∑ ∂ i ni ≠ TVn,,ij n = GGTPn(,,i ) ∂G ∂G ∂G dG = dT + dP + ∑ dn ∂ ∂ ∂ i T P i ni ≠ Pn,,ii Tn TPn,, ij n ∂G dG=− SdT + VdP +∑ ∂ i ni ≠ 5 TPn,, ij n Partial Molar Quantities • Partial derivative of a quantity with respect to mass (n) at constant T, P The rate of change of a quantity as a component is added Designated by a bar over the variable ∂V Va = ∂ na TPn, ,bc , n ... • Very useful in the study of solutions For molar volume, it is the volume of that component in solution • For G only, partial molar free energy is equal to the chemical potential ∂G Ga = =µ ∂ a na TPn, ,bc , n ... WS2002 6 6 Property Relations • For any function with a continuous derivative like our state functions • • This analysis gives the Maxwell relations for the energy functions relationship among variables used to relate one property to other values can relate a calculated quantity (S) to measurable quantities (T,P) WS2002 7 7 Property Relations • For Gibbs’ free energy • • Rearranging and integrating WS2002 8 8 Property Relations RT • For an ideal gas, PV = RT so: V = P ∂V R = ∂ T P T ∂S ∂V − = ∂ ∂ From the Maxwell relation P TPT ∂S R So: =− ∂P T T • Substituting and integrating WS2002 9 9 Property Relations • For a solid or the general case, we need a relationship among the variables α • The thermal expansion coefficient (volumetric), V, is defined as the rate of change of volume with temperature at constant pressure α • For a condensed phase, V and V are constant over normal ranges of pressure, so WS2002 10 10 Other Examples for Ideal Gases WS2002 11 11 Other Examples for General Case WS2002 12 12 Enthalpy of Mixing • dH is given by • Rearranging the definition of dH Now, we need a relation between S and V Check the other energy function definitions • From the definition of G • Maxwell relation • Substituting back into the dH relation • From the ideal gas law • Giving the change in enthalpy with pressure at constant temperature ∆ • From this, it can be argued that Hmix = 0 since there is no interaction among molecules WS2002 13 13 Free Energy of Mixing • For each component of a gas before mixing • For each component of a gas mixture • The free energy of mixing is then WS2002 14 14 Entropy of Mixing • From the definition of G: ∆ ∆ • We know Gmix and Hmix, so: WS2002 15 15 Heat Capacity ∂H C = • Constant pressure heat capacity was defined as P ∂T P • A more general definition comes from entropy • Generically, then where I is any process variable • To check this result, go the definition of enthalpy at constant pressure substituting for dS • So for any process variable magnetic, electric, volume, etc. WS2002 16 16 Variation of Heat Capacity WS2002 17 17 Isentropic P-T Relationship • In Ch. 1, we derived P-T relationships for adiabatic, reversible processes • The same relationships can be derived for an isentropic process δQ An adiabatic, reversible process is isentropic ( dS = rev ) T WS2002 18 18 Isentropic Compression of Solids • The relationship derived previously is valid here • Giving the relationship for ∆T as a function of pressure • If the pressure on copper is increased from 1 to 10 atm., calculate ∆T WS2002 19 19 Thermoelastic Effect • Also called the adiabatic thermoeleastic effect • The previous relationship was for an isostatic pressure We can alter it to account for uniaxial stress application ∂T Numerically, we want to know the value of ∂σ S • From the previous expression, the values of linear thermal expansion coeficient and applied force can be factored in to give WS2002 20 20 Compressibility WS2002 21 21 Other Effects WS2002 22 22.
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