Birch-murnaghan Equation Of State

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Birch-murnaghan Equation Of State

:

E = E_0 + rac{1}{2} B_0 rac{(V-V_0)^2}{V_0}.

The next simplest reasonable model would be with a constant bulk modulus
:

B = - V \left( rac{\partial P}{\partial V} 
ight)_T.	(2)

:

E = E_0 + B_0 \left( V_0 - V + V \ln(V/V_0) 
ight).

MURNAGHAN EQUATION OF STATE

A more sophisticated equation of state was derived by
F. D. Murnaghan of Johns Hopkins University in 1944. To begin with, we consider the pressure
:

P = - \left( rac{\partial E}{\partial V} 
ight)_S	(1)

and the bulk modulus
:

B = - V \left( rac{\partial P}{\partial V} 
ight)_T.	(2)

Experimentally, the bulk modulus pressure derivative
:

B' =  \left( rac{\partial B}{\partial P} 
ight)_T	(3)

is found to change little with pressure. If
we take

B' = B'_0

to be a constant, then
:

B = B_0 + B'_0 P	(4)

where

B_0

is the value of

B

when

P = 0.

We may equate this with (2) and rearrange as
:

rac{d V}{V} = -rac{d P}{B_0 + B'_0 P}.	(5)

Integrating this results in
:

P(V) = rac{B_0}{B'_0} \left(\left(rac{V_0}{V}
ight)^{B'_0}
    - 1
ight)	(6)

or equivalently
:

V(P) = V_0 \left(1+B'_0
    rac{P}{B_0}
ight)^{-1/B'_0}.		(7)

Substituting (6) into

E = E_0 - \int P
dV

then results in the equation of state
for energy.
:

E(V) = E_0
 + rac{ B_0 V }{ B_0' } \left( rac{ (V_0/V)^{B_0'} }{ B_0' - 1 } + 1 
ight)
 - rac{ B_0 V_0 }{ B_0' - 1 }. 	(8)

Many substances have a fairly constant

B'_0

of about 3.5.

BIRCH-MURNAGHAN EQUATION OF STATE

The third-order Birch-Murnaghan isothermal equation of state, published in 1947 by Francis Birch of Harvard , is given by:

:

P(V)=rac{3B_0}{2}
\left[\left(rac{V_0}{V}
ight)^rac{7}{3} - 
\left(rac{V_0}{V}
ight)^rac{5}{3}
ight]
\left\{1+rac{3}{4}\left(B_0^\prime-4
ight)
\left[\left(rac{V_0}{V}
ight)^rac{2}{3} - 1
ight]
ight\}

Again, E(V) is found by integration of the pressure:
:

E(V) = E_0 + rac{9V_0B_0}{16}
\left\{
\left[\left(rac{V_0}{V}
ight)^rac{2}{3}-1
ight]^3B_0^\prime + 
\left[\left(rac{V_0}{V}
ight)^rac{2}{3}-1
ight]^2
\left[6-4\left(rac{V_0}{V}
ight)^rac{2}{3}
ight]
ight\}

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