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 First law Second law Multiply by T Subtract from the first law du = δq + δw δq δlw ds = T T TdS = δq δlw du TdS = δw + δlw 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 δwrev * = δwrev PdV = du TdS * δw = du TdS + PdV rev 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 = i n = n = n =µ n TPn,, TVn,, vsn,, PSn,, i n i n i n i n j i j i j i j i 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 U = U( S, V, n ) U U du = ds dv + S + V du = TdS PdV + H = H( S, P, n ) i U n U n Vn, Sn i i i, i SVn,, i n j i i SVn,, i n j H H dh = ds dp + S + P dh = TdS + VdP + F = F( T, V, n ) G = G( TP,, n ) i H n H n Pn, Sn i i i, i SPn,, i n j i i SPn,, i n j F F df = dt dv + T + V df = SdT PdV + G G dg = dt dp + T + P dg = SdT + VdP + i i G n G n dn i dn Pn, Tn i i i, i TPn,, i nj F n i i TPn,, i nj F n Vn, Tn i i i, i TVn,, i nj i i TVn,, i nj i dn dn i i 5
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,, b, nc... 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 a G = =µ n a TPn,, b, nc... a 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 For an ideal gas, PV = RT so: From the Maxwell relation So: V = RT P V R = T T P S V = P T T S R = P T T P Substituting and integrating WS2002 9 9
Property Relations For a solid or the general case, we need a relationship among the variables The thermal expansion coefficient (volumetric), α V, is defined as the rate of change of volume with temperature at constant pressure For a condensed phase, V and α 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 H mix = 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 G mix and H mix, so: WS2002 15 15
Heat Capacity Constant pressure heat capacity was defined as A more general definition comes from entropy H CP = T P 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 Q An adiabatic, reversible process is isentropic ( rev ) ds = δ 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 From the previous expression, the values of linear thermal expansion coeficient and applied force can be factored in to give S WS2002 20 20
Compressibility WS2002 21 21
Other Effects WS2002 22 22