THE EQUATION OF STATE OF SOLID MATTER For many years scientists and engineers have had available an excellent equation of state for gaseous matter. Now, at last, the Reciprocal System of Dewey B. Larson is able to give us an exact equation of state for solid matter. This paper will present a unified treatment of the subject, with Reference 1 as the starting point. I. Volume of Solid as a Function of Temperature with Pressure Constant From the material presented in Chapter 8 of Ref. 1, I have drawn a generalized plot of thermal expansion coefficient versus temperature, Figure l. The symbols are defined as follows:
With the initial votume of the first segment of the curve included, eq. (8-4) of Ref. 1 becomes
where K is a constant and n is the number of rotational units that are
themally vibrating.This equation can be put into a more usable form involving
T, T
At T
Then,
So,
Therefore,
This equation holds from T = 0 to T = T
= units = [3576/T
Thus given the volume of the solid at zero temperature, che rotational
factcrs of the element, and the solid and point temperature, the volume
V at any other temperature, (up to T The equation for the volume for temperatures above T
For this equation to be of use,V
ln the equation for V
Or,
Let,
Then,
Solve for C
From eqs. (6) and (7), ß
This value of ß
With ß
where G The ratio V
Compare this with the equation for a gas:
II. Volume of Solid as a Function of Pressure with Temperature Constant The comoression of a solid by hydrostatic pressure is discontinuous at
certain aressures nere denoted as P
For the given value of P, the vatues of P
Now, in the MKS system,
where a, Z, y are the rotational compression values (simitar to the thermat
values) and so is the base interatomic spacing. At each transition a,
Z, y can change (and possible S Before continuing the discussion of the equation of state I will discuss some subsidiary properties of matter: the bulk modulus, the modulus of elasticity, and Poisson’s ratio. Larson has derived the equation for compressibility; the solid bulk modulus is the inverse of this:
(at zero external pressure and zero temperature for a pure substance).
I witl not derive the equation for the modulus of elasticity, E. In eq.
(13) let the constants of the equation be written as J and generalize
s
By definition,
And,
where e is the strain:
So,
thus,
Since
then
and when s = s
(stress and pressure are in opposite directions) Poisson’s ratio can be determined from the well-known equation
Thus at zero temperature and pressure for a pure substance,
This is in the “ball park” for most solids; however, most substances used in construction are impure and at other than zero temperature dnd, in addition, may contain a proportion of tiquid molecules — thus drastically changing the values of Poisson’s ratio and the modulus of elasticity. These considerations will be left to another paper. Going back to eq. (12) we can generalize to the ratio of final to initial volume within a segment:
This compares with the equation for a gas:
III.Volume of Solid as a Function of Both Temperature and Pressure The solid can be considered to undergo a aressure change at zero temperature
and then a temperature change from the new volume. Let P <
Let T < T
The value of T
where V If P < P
(23)
Eq. (5) becomes
Equations (21) and (24j (combined) represent the complete equation of the solid state. IV. Exampte Calculations As an example, consider one volume unit of silver at zero degrets K and zero external pressure. Whnt is the volume at temperature T and pressure P? First the thermal rotational factors, a-Z-y, from Tabte 22 of Ref, 1
are found; they are 4-3-l. With these, the temperature of the first transition
point, T
Silver has a maximum of four magnetit rotational units vibrating, so the solid end point is
In this case the endgoint appears to be somewhat higher than the empirical
melting point, 1234
Now the number of units to use in selecting ß
and therefore
Since siiver is etectronegative,
Then from eq. (4),
This equation holds good up to T = T
For temperatures above T
Thus from eq. (5),
(Note: no crystal change from FCC is assumed here). Now we’11 go on to look at the pressure relations. Assume that P
is less than the first transition pressure P
Since P c p , P = 0, dnd V = V . Then eq. (12) is
If P = .001 P ,
The bulk modulus B, modulus of elasticity E, and Poisson’s ratio can now be calculated for a pure sample of silver at zero temperature:
For the combined pressure and temperature loading, eq. (21) yields
The value of T
If P = .001 P
Putting this value of T
The new value of T
Thus, for temperatures above T
Finally, from eq. (23) I find that the new melting temperature is:
(I have assumed, however, that this does not affect the original value
of ß Basically the same procedure could be used with other elements, atloys, and compounds. Corresponding equations do not exist in quantum mechanics. A solution in “principle only” is not a true solution. A true solution is based on principle and works in practice. ********************** Reference 1. Dewey B. Larson, |