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Birch-murnaghan Equation Of StateIn continuum mechanics, an equation of state suitable for modeling solids is naturally rather different from the ideal gas law. A solid has a certain equilibrium volume , and the energy increases quadratically as volume is increased or decreased a small amount from that value. The simplest plausible dependence of energy on volume would be a harmonic solid, with -
E = E_0 + \frac{1}{2} B_0 \frac{(V-V_0)^2}{V_0}. The next simplest reasonable model would be with a constant bulk modulus -
B = - V \left( \frac{\partial P}{\partial V} \right)_T. (2) -
E = E_0 + B_0 \left( V_0 - V + V \ln(V/V_0) \right). A more sophisticated equation of state was derived by F. D. Murnaghan. To begin with, we consider the pressure -
and the bulk modulus -
B = - V \left( \frac{\partial P}{\partial V} \right)_T. (2) Experimentally, the bulk modulus pressure derivative -
B' = \left( \frac{\partial B}{\partial P} \right)_T (3) is found to change little with pressure. If we take to be a constant, then -
B = B_0 + B'_0 P (4) where is the value of when We may equate this with (2) and rearrange as -
\frac{d V}{V} = -\frac{d P}{B_0 + B'_0 P}. (5) Integrating this results in -
P(V) = \frac{B_0}{B'_0} \left(\left(\frac{V_0}{V}\right)^{B'_0} - 1\right) (6) or equivalently -
V(P) = V_0 \left(1+B'_0 \frac{P}{B_0}\right)^{-1/B'_0}. (7) Substituting (6) into then results in the Birch--Murnaghan equation of state for energy. -
E(V) = E_0 + \frac{ B_0 V }{ B_0' } \left( \frac{ (V_0/V)^{B_0'} }{ B_0' - 1 } + 1 \right) - \frac{ B_0 V_0 }{ B_0' - 1 }. (8) Many substances have a fairly constant of about 3.5. References - F. D. Murnaghan, Proceedings of the National Academy of Sciences, vol. 30, p. 244, 1944.
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