Birch-Murnaghan equation of state

From Wikipedia, the free encyclopedia

In 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 <math>V_0</math>, 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

<math>

E = E_0 + \frac{1}{2} B_0 \frac{(V-V_0)^2}{V_0}. </math>

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

<math>

B = - V \left( \frac{\partial P}{\partial V} \right)_T. (2) </math>

<math>

E = E_0 + B_0 \left( V_0 - V + V \ln(V/V_0) \right). </math>

[edit] Murnaghan equation of state

A more sophisticated equation of state was derived by F. D. Murnaghan of Johns Hopkins University in 1944^[1]. To begin with, we consider the pressure

<math> P = - \left( \frac{\partial E}{\partial V} \right)_S (1)

</math> and the bulk modulus

<math>

B = - V \left( \frac{\partial P}{\partial V} \right)_T. (2) </math> Experimentally, the bulk modulus pressure derivative

<math>

B' = \left( \frac{\partial B}{\partial P} \right)_T (3) </math> is found to change little with pressure. If we take <math>B' = B'_0</math> to be a constant, then

<math>

B = B_0 + B'_0 P (4) </math> where <math>B_0</math> is the value of <math>B</math> when <math>P = 0.</math> We may equate this with (2) and rearrange as

<math>

\frac{d V}{V} = -\frac{d P}{B_0 + B'_0 P}. (5) </math> Integrating this results in

<math>

P(V) = \frac{B_0}{B'_0} \left(\left(\frac{V_0}{V}\right)^{B'_0}

   - 1\right)	(6)

</math> or equivalently

<math>

V(P) = V_0 \left(1+B'_0

   \frac{P}{B_0}\right)^{-1/B'_0}.		(7)

</math> Substituting (6) into <math>E = E_0 - \int P dV</math> then results in the equation of state for energy.

<math>

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)

</math>

Many substances have a fairly constant <math>B'_0</math> of about 3.5.

[edit] Birch-Murnaghan equation of state

The third-order Birch-Murnaghan isothermal equation of state, published in 1947 by Francis Birch of Harvard^[2], is given by:

<math>

P(V)=\frac{3B_0}{2} \left[\left(\frac{V_0}{V}\right)^\frac{7}{3} - \left(\frac{V_0}{V}\right)^\frac{5}{3}\right] \left\{1+\frac{3}{4}\left(B_0^\prime-4\right) \left[\left(\frac{V_0}{V}\right)^\frac{2}{3} - 1\right]\right\} </math>

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

<math>

E(V) = E_0 + \frac{9V_0B_0}{16} \left\{ \left[\left(\frac{V_0}{V}\right)^\frac{2}{3}-1\right]^3B_0^\prime + \left[\left(\frac{V_0}{V}\right)^\frac{2}{3}-1\right]^2 \left[6-4\left(\frac{V_0}{V}\right)^\frac{2}{3}\right]\right\} </math>

[edit] References

↑ F.D. Murnaghan, 'The Compressibility of Media under Extreme Pressures', in Proceedings of the National Academy of Sciences, vol. 30, pp. 244-247, 1944.