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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 \( V_0 \), 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

\( B_0 = - V \left( \frac{\partial P}{\partial V} \right)_T. \)

Integrating gives

\( P = B_0 \ln(V_0/V). \, \)

\( V = V_0 \exp(-P/B_0). \, \)

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

Murnaghan equation of state

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

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

and the bulk modulus

\( B = - V \left( \frac{\partial P}{\partial V} \right)_T. \qquad (2) \)

Experimentally, the bulk modulus pressure derivative

\( B' = \left( \frac{\partial B}{\partial P} \right)_T \qquad (3) \)

is found to change little with pressure. If we take \( B' = B'_0 \) to be a constant, then

B = B_0 + B'_0 P \qquad(4) \)

where B_0 is the value of B when P = 0. We may equate this with (2) and rearrange as

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

Integrating this results in

\( P(V) = \frac{B_0}{B'_0} \left(\left(\frac{V_0}{V}\right)^{B'_0} - 1\right) \qquad (6) \)

or equivalently

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

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

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

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[2], is given by:

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

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

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