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
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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
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• 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
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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
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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
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= 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 ...
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6 Property Relations
• For any function with a continuous derivative like our state functions
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• 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
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• Rearranging and integrating
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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
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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
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10 Other Examples for Ideal Gases
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11 Other Examples for General Case
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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
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14 Entropy of Mixing
• From the definition of G:
∆ ∆ • We know Gmix and Hmix, so:
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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.
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16 Variation of Heat Capacity
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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
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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
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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
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20 Compressibility
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21 Other Effects
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