Equation of State First, density, ρ (g/cm3) is essentially the reciprocal of volume V (cm3) times molecular weight, M: ρ = MV/ Effect of Temperature: Expansivity The thermal expansivity, α, is 1 ∂V α = V ∂T P Integration gives the total effect on volume with changing temperature T ∫ α()TdT= ln(()/) VT Vo ≡Φ To Holland and Powell [1998] advocated a relationship between expansivity and temperature T(K), defined by a single constant a° for each mineral: α()Ta=° (110− T ) which gives ∂α// ∂TaT= 5 o 32/ and T o Φ≡ln(VT ( )/ Vo ) ==∫ α() TdT aTT{} (−−oo )20 ( T − T ) To where V(T) is the molar volume at temperature, Vo is the molar volume at STP, and To = 298K. The density at elevated temperature ρ(T) is related to the density at STP ρo by: −Φ ρρ()Te= o Effect of Pressure: Bulk Modulus The isothermal compressibility, βT, describes how volume changes as a function of pressure: 1 ∂V βT = − V ∂P T The isothermal bulk modulus, KT, (the reciprocal of the compressibility) is usually used instead: 11∂V = − KVT ∂P T Moduli at High Temperature The isothermal bulk modulus at elevated temperature KT(T) is related to the isothermal bulk modulus at STP KTo by: −δT Φ KTTT()= Keo where δT is the second Grüneisen parameter: ∂lnK 1 ∂K δ = T = T T ∂lnραP KT ∂T P The shear modulus at elevated temperature µ(T) follows in similar fashion from the shear modulus at STP µTo: −ΓΦ µµTT()Te= o where ∂lnµ 1 ∂µ Γ = = − ∂lnρP µα ∂T P Moduli at High Pressure The relationship between strain ε and volume or density is V ρ 32// 32 o = = ()(12− ε =+ƒ 12 ) V ρo where ƒ=−ε is the compression. The compression can be calculated recursively from 5/2 2 PK/')('T =ƒ3(1+2) ƒ{}1–2+ζζ ƒ ƒ/6[4 (43− K+ 53 K –5)] where ζ = 0754.(− K ') and KdKdP'(= TT / ) typically evaluated at To. The density at elevated pressure ρ(P) is then 32/ ρρ()P =+ƒo (12 ) Properties at High Temperature and Pressure The bulk modulus at elevated pressure and temperature KT(T,P) is 252/ KTPKTTT(, )= (){}153−− ( K ')(ƒ 3 K '−− 73512 )(' K )(+ƒ ) The expansivity at elevated pressure and temperature α(T,P) is −δT ααρρ(,TP )= () T[] ()/ P o The isentropic or adiabatic bulk modulus KS is KKTPTST=+(, )[]1 γα th (, TP ) where γth is the first Grüneisen parameter: VKTα γth = CV The shear modulus at elevated pressure and temperature µ(T,P) is 52/ 2 µµ(,TP )=+ƒ ()( T12 ){} 1− ƒ[] 53− µµ ' KTT ()/( T T).+ƒ 059[] (')'()/()KKTT− 4µµ+ 35 The density at elevated pressure and temperature ρ(P,T) is ρρρρ(,TP )= [()/]() Po T From this it is possible to calculate the P-wave velocity VP, shear wave velocity VS, and Poisson’s ratio ν: VKPS=+43//µ ρ VS = µ /ρ ν = ()/(3262KKSS− µµ+ ) Physical Properties of Rocks The physical property Ψ of a mineral aggregate can be calculated from the physical property Ψi of n constituent minerals using a a Reuss average (uniform stress): n ΨΨRii= ∑[]ν i=1 a Voigt average (uniform strain): n ΨΨVii=1/[/∑ ν ] i=1 or a Voigt–Reuss–Hill average: n n ΨΨΨΨVRH= {} V+ R//[/]/21= ∑ iννiii+ ∑ Ψ 2 i==11i where νi is the volume proportion of each mineral. Because mass in aggregates is a simple sum of component masses, only ΨV is used in calculating ρ for aggregates. Holland, T.J.B., and R. Powell, An internally consistent thermodynamic data set for phases of petrological interest, Journal of Metamorphic Geology, 16, 309-343, 1998..