Birch-Murnaghan equation of state
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
, 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
The next simplest reasonable model would be with a constant bulk modulus
Integrating gives
Francis D. Murnaghan
of Johns Hopkins University
in 1944. To begin with, we consider the pressure
and the bulk modulus
Experimentally, the bulk modulus pressure derivative
is found to change little with pressure. If we take
to be a constant, then
where
is the value of 
when 
We may equate this with (2) and rearrange as
Integrating this results in
or equivalently
Substituting (6) into
then results in the equation of state for energy.
Many substances have a fairly constant
of about 3.5.
of Harvard, is given by:
Again, E(V) is found by integration of the pressure:
Continuum mechanics
Continuum mechanics is a branch of mechanics that deals with the analysis of the kinematics and the mechanical behavior of materials modelled as a continuous mass rather than as discrete particles...
, an equation of state
Equation of state
In physics and thermodynamics, an equation of state is a relation between state variables. More specifically, an equation of state is a thermodynamic equation describing the state of matter under a given set of physical conditions...
suitable for modeling solids is naturally rather different from the ideal gas law
Ideal gas law
The ideal gas law is the equation of state of a hypothetical ideal gas. It is a good approximation to the behavior of many gases under many conditions, although it has several limitations. It was first stated by Émile Clapeyron in 1834 as a combination of Boyle's law and Charles's 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, withThe next simplest reasonable model would be with a constant bulk modulus
Bulk modulus
The bulk modulus of a substance measures the substance's resistance to uniform compression. It is defined as the pressure increase needed to decrease the volume by a factor of 1/e...
Integrating gives
Murnaghan equation of state
A more sophisticated equation of state was derived byFrancis D. Murnaghan
Francis Dominic Murnaghan (mathematician)
Francis Dominic Murnaghan was an Irish mathematician, former head of the mathematics department at Johns Hopkins University. His name is attached to developments in group theory and mathematics applied to continuum mechanics .He is the father of Francis Dominic Murnaghan, Jr., former U.S. federal...
of Johns Hopkins University
Johns Hopkins University
The Johns Hopkins University, commonly referred to as Johns Hopkins, JHU, or simply Hopkins, is a private research university based in Baltimore, Maryland, United States...
in 1944. To begin with, we consider the pressure
and the bulk modulus
Experimentally, the bulk modulus pressure derivative
is found to change little with pressure. If we take
to be a constant, thenwhere
is the value of when We may equate this with (2) and rearrange as
Integrating this results in
or equivalently
Substituting (6) into
then results in the equation of state for energy.Many substances have a fairly constant
of about 3.5.Birch–Murnaghan equation of state
The third-order Birch–Murnaghan isothermal equation of state, published in 1947 by Francis BirchFrancis Birch (geophysicist)
Albert Francis Birch was an American geophysicist best known for his experimental work on the properties of Earth-forming minerals at high pressure and temperature, in 1952 he published a well-known paper in the Journal of Geophysical Research ,where he demonstrated that the mantle is chiefly...
of Harvard, is given by:
Again, E(V) is found by integration of the pressure:
External links
- Equation of State Codes and Scripts This webpage provides a list of available codes and scripts used to fit energy and volume data from electronic structure calculations to equations of state such as the Birch–Murnaghan. These can be used to determine material properties such as equilibrium volume, minimum energy, and bulk modulus.