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    ∂lnραP 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 =   = −    ∂lnρP 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.