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:

ß = thermal expansion coefficient

T = temperature

T_M = temperature of solid end point (at or close to melting point)

V_M = volume at solid end point

T₁ = first transition temperature

ß₀ = initial value of thercnal expansion coefficient at absolute zero temperature

V₀ = initial volume at absolute zero temperature

V₁ = volume at tranSition temperature

ß₀’ = initial vatue of thermal expansion coefficient based on second segment of curve

V₀’ = initial volume based on second segment of curve

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, T₁, ß₁, end ß₀ — all of which can be determined from theory. The thermal coefficient of expansion at temperature T is

At T₁,

Then,

So,

Therefore,

This equation holds from T = 0 to T = T₁. Larson has deduced the following values of ß₁, ß₀, and T₁:

ß₀ = 5.17 * 10^-6/°K for one unit

     = 10.3 * 10^-6/°K for two units

     = 15.5 * 10^-6/°K for three units

     = 20.7 * 10^-6/°K for four units units

  = units = [3576/T_M]

ß₀ = -2/7 * ß₁ for electropositive elements

     = -1/7 * ß₁ for some 2lectronegative alements

 Tl = 8.98 (a + z + y)°K

 a, z, y are from Table 22 of Ref. 1

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 T₁) can be easily determined.

The equation for the volume for temperatures above T₁ has the same form as eq. (4):

For this equation to be of use,V₀’ and ₀’ must be expressed in terms of known quantities such as V_M and T_M. Now,

ln the equation for V₁, solve for V₀’ and put in equation for V_M:

Or,

Let,

Then,

Solve for C₂:

From eqs. (6) and (7), ß₀’, can be found:

This value of ß₀’ can then be substituted into the equation for V₀’:

With ß₀’ and V₀’ known, eq. (5) is ready for use. Larson has deduced the following values of T_M and V_M:

where G₀ is the initial crystal geometric constant and G_M is the final one (some solids cnange crystalline structure as they expand).

The ratio V_M/V₁ can be generalized to any pair of final to initial volumes:

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 P₁, P₂, P₃, P₄ etc. At these pressures the internal pressure P₀ can change to P₀₁, P₀₂, P₀₃, P₀₄, etc., thus altering the slope of the compression curve. Larson has shown that between the transitions, the volume naries as the inverse square root of the pressure. The most general way to express this is with the following aquation:

For the given value of P, the vatues of P_ref, V_ref, and P₀ must be detennined by theory (or empiricnlly if necessary) before V can be found. For a four transition solid we have the following:

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 S₀), thus causing P₀ to change.

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 s₀ (for the moment) to s. Then the initial internal stress is

By definition,

And,

where e is the strain:

So,

thus,

Since

then

and when s = s₀ and T = 0 °K for a pure substance,

(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 < _P1 Then

Let T < T₁. Then,

The value of T₁ is not the same as before. To get to the original value of V₁ the new value of T₁ must be higher

where V₁ is calculated from the original T₁. I am assuming here that. is as before.

If P < P₁ and T > T₁, then the value of V₀’ has to be modified, since T₁ and T_M are different. I assume that ß₀’ is the same. Then the second term on the right in eq. (8) is the same and the new value of T_M can be found:

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, T₁, can be calcutated:

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 ^oK Thus it would seem that the thermal factors at the end point are towered by one to 3-3-1, so that

Now the number of units to use in selecting ß₁ is

and therefore

Since siiver is etectronegative,

Then from eq. (4),

This equation holds good up to T = T₁ at which point

For temperatures above T₁, the values of V₀’ and ß₀’ are needed. To calculate ß₀’ I am going to use the empirical value of T_M pending theoretical clarification.
From eqs. (8) and (9),

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 P₁ (which is approx. 10⁷ KN/m² ) so that the initial compressibility factors from Table 14 of Ref. 1 can be used: a-Z-y = 4-8-1. From Table 4 of Ref. 1,
s₀ = 2.87 x 10^-10 m. Then from eq. (13),

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 T₁ to be used here comes from eq. (22):

If P = .001 P₀, then

Putting this value of T₁ into the above gives:

The new value of T₁ gives the new value of V₀’:

Thus, for temperatures above T₁

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^₁) .

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.

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Reference

1. Dewey B. Larson, Solid Matter, prepublication version of second volume of the revised edition of The Structure of the Physical Universe (Portland, Oregon: North Pacific Publishers, 1980).