## Murnaghan integrated linear equation of state

The isothermal bulk modulus, KT, is assumed to be independent from compression, namely pressure in the case of the simplest EOS. Here, we assume that the isothermal bulk modulus is a linear function of pressure. Namely,

(1)

where KT0 is the isothermal bulk modulus at zero pressure, and K0' is its first pressure derivative.

By generalizing Eq (2) in the simplest EOS page, we have:

(2)

(3)

To solve the differential equation of Eq (3),

(4)

By integrating the both sides of Eq (4),

(5)

The integral constant C is determined from the conditions of V = V0 at P = 0:

(6)

By substituting Eq. (6) to Eq. (5), we have:

(7)

By taking out the natural logarithms, we have:

(8)

Finally, we have Murnaghan's integrated linear EOS:

(9)

The Murnaghan's EOS has been much less often used than the 3rd-order Burch-Murnaghan EOS. The  3rd-order Burch-Murnaghan EOS seems to be considered more advanced than the Murnaghan's EOS. However, as is discussed in the EOS page, the Murnaghan's EOS also shows satisfactory results at least for MgSiO3 bridgmanite. Therefore, the Murnaghan's EOS is equivalently useful to other EOS for researchers in physics of the Earth and planetary interiors.