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
T_{1} = first transition temperature
ß_{0} = initial value of thercnal expansion coefficient at absolute zero temperature
V_{0} = initial volume at absolute zero temperature
V_{1} = volume at tranSition temperature
ß_{0}’ = initial vatue of thermal expansion coefficient based on second segment of curve
V_{0}’ = initial volume based on second segment of curve
With the initial votume of the first segment of the curve included, eq. (84) of Ref. 1 becomes
V = V_{0} + K/n³ T² 
(1)

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_{1}, ß_{1}, end ß_{0} — all of which can be determined from theory. The thermal coefficient of expansion at temperature T is
ß = 1/V dV/dT + ß_{0}  
= 2KT/n³ [1/V_{0} + KT²/n³ + ß_{0} 
(2)

= 2KT/n³V_{0} + KT² + ß_{0} 
At T_{1},
�_{1} = 2KT_{1}/n³V_{0} + KT_{1}² + ß_{0} 
Then,
ß_{1}�_{0}/2T_{1} = K/n砎_{0} + T_{1}� 
So,
2T_{1}/�_{1}�_{0} = n砎_{0}/K + t_{1}� 
K/n³ = V_{0}/2T_{1}/�_{1}�_{0}  T_{1}� 
(3)

Therefore,
V = V_{0} + V_{0}T²/2T_{1}/�_{1}�_{0}  T_{1}� 
(4)

This equation holds from T = 0 to T = T_{1}. Larson has deduced the following values of ß_{1}, ß_{0}, and T_{1}:
ß_{0} = 5.17 * 10^{6}/癒 for one unit
= 10.3 * 10^{6}/癒 for two units
= 15.5 * 10^{6}/癒 for three units
= 20.7 * 10^{6}/癒 for four units units
= units = [3576/T_{M}]
ß_{0} = 2/7 * ß_{1 }for electropositive elements
= 1/7 * ß_{1} 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 T_{1}) can be easily determined.
The equation for the volume for temperatures above T_{1} has the same form as eq. (4):
V_{0} T�


V = V_{0}^{1} +  ———  
2T_{1}

(5)


———   T�  
�_{1}�_{0}


For this equation to be of use,V_{0}’ and _{0}’ must be expressed in terms of known quantities such as V_{M} and T_{M}. Now,
V_{0}� T_{M}�  
V_{M} = V_{0}� +  ———–  
2T_{1}


———–   T_{1}�  
�_{1}�_{0}�

V_{0}� T_{1}�


V_{1} = V_{0}� +  ———–  
2T_{1}


———–   T_{1}�  
�_{1}�_{0}�

ln the equation for V_{1}, solve for V_{0}’ and put in equation for V_{M}:
V_{1}

( 
T_{M}�

)  
V_{M} =  ——————  1 +  —————  
1 + T_{1}�  2T_{1}  
——––—–

——–  T_{1}�  
2T_{1 }

�_{1}�_{0}�  
———–

 T_{1}�  
�_{1}�_{0}�

Or,
T_{M}�


1 +  —————–  
2T_{1}  
——––  T_{1}�  
�_{1}�_{0}�  
V_{M}/V_{1} =  ———————  
T_{1}�


1 +  —————–  
2T_{1}  
——––  T_{1}�  
�_{1}�_{0}� 
Let,
1


C_{2} =  —————  
2T_{1} 
(6)


——–  T_{1}�  
�_{1}�_{0} 
Then,
V_{M}  1 + C_{2}T_{M}�  
—– =  ————–  
V_{1} 
1  C_{2}T_{1}�

Solve for C_{2}:
V_{M}  V_{1}


C_{2} =  ———————  (7)  
V_{1}T_{M}�  V_{M}T_{1}� 

From eqs. (6) and (7), ß_{0}’, can be found:
2T_{1}


ß_{0}’ = ß_{1}   
V_{1}T_{M}²  V_{M}T_{1}² 
(8)


+ T_{1}²  
V_{M}  V_{1}

This value of ß_{0}’ can then be substituted into the equation for V_{0}’:
V1


VO’ =  ———————–  
T1²


1 +  —————— 
(9)


2T1  
———–  T1²  
ß1  ßo’ 
With ß_{0}’ and V_{0}’ known, eq. (5) is ready for use. Larson has deduced the following values of T_{M} and V_{M}:
T_{M }= 1.80 * T_{1} for one rot. unit vibrating  
= 4.56 * T_{1} for two rot. units  
= 9.32 * T_{1} for three rot. units  
= 17.87 * T_{1} for four rot. units  
G_{M}  
V_{M} = 1.0625 V_{0}  —– 
G_{0} 
where G_{0} 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}/V_{1} can be generalized to any pair of final to initial volumes:
V_{f}

1 + C_{2}T_{f}²  
— =

———— 
(10)


V_{i}

1 + C_{2}T_{i}² 
Compare this with the equation for a gas:
V_{f}  T_{f}  
— =  — 
(11)

V_{i}  T_{i} 
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_{1}, P_{2}, P_{3}, P_{4} etc. At these pressures the internal pressure P_{0} can change to P_{01}, P_{02}, P_{03}, P_{04}, 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:
V  P_{0} + P_{ref}  
—— =  ———— 
(12)


V_{ref} 
P_{0} + P

For the given value of P, the vatues of P_{ref}, V_{ref}, and P_{0} must be detennined by theory (or empiricnlly if necessary) before V can be found. For a four transition solid we have the following:
0 < P < P_{1}

V_{ref} = V_{0}

P_{ref} = _{0}

P_{0} = P_{0}

P_{1} < P < P_{2}

V_{ref} = V_{1}

P_{ref} = P_{1}

P_{0} P_{01}

P_{2} < P < P_{3}

V_{ref} = V_{2}

P_{ref} = P_{2}

P_{0} = P_{02}

P_{3 }< P < P_{4}

V_{ref} = V_{3}

P_{ref} = P_{3}

P_{0} = P_{03}

Now, in the MKS system,
aZy  KN  
P_{0} = 4.177 * 10^{23}  —–  —– 
(13)

S_{0}³  m² 
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_{0}), thus causing P_{0} 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:
B = 2 * P_{0} 
(14)

(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_{0} (for the moment) to s. Then the initial internal stress is
s = P = Js^{3} 
(15)

By definition,
ds  
E =  —– 
de 
And,
ds  ds  ds 
— =  —  — 
de  ds  de 
where e is the strain:
s  s_{0}  
e =  ——– 
s_{0} 
So,
s = s_{0e} + s_{0} 
thus,
ds 
— = s_{0} 
de 
Since
ds 
— = 3Js^{4} 
ds 
then
ds 
— = 3Js^{4 }s_{0} 
de 
and when s = s_{0} and T = 0 癒 for a pure substance,
ds 
— s_{0},T = 0 = 3Js_{03 = 3P0} 
de 
E = 3P_{0} 
(16)

(stress and pressure are in opposite directions)
Poisson’s ratio can be determined from the wellknown equation
3P_{0}  
n = .5   ——– 
(17)

6 2P_{0} 
Thus at zero temperature and pressure for a pure substance,
3P_{0}  
n = .5   ——– = .25 
(18)

6 2P_{0} 
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:
V_{f}  P_{0} + P_{i}  
— =  ———– 
(19)

V_{i}  P_{0} + P_{f} 
This compares with the equation for a gas:
V_{f}  P_{i}  
— =  — 
(20)

V_{i}  P_{f} 
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
P_{0}  
V_{0NEW} = V_{0 }  ——–– 
P_{0} + P 
Let T < T_{1}. Then,
V_{0NEW} T²  
V = V_{0NEW} +  ————————  
2T_{1NEW} 
(21)


————  T_{1NEW}2  
ß_{1}  ß_{0} 
The value of T_{1} is not the same as before. To get to the original value of V_{1 }the new value of T_{1} must be higher
2  
T_{1NEW} =  ———————  
ß_{1}  ß_{0}  
——— + ß_{1}  ß_{0} 
(22)


V_{1}  
—————–  
V_{0}  (P_{0}/ P_{0} + P)½ 
where V_{1} is calculated from the original T_{1}. I am assuming here that. is as before.
If P < P_{1} and T > T_{1}, then the value of V_{0}’ has to be modified, since T_{1} and T_{M} are different. I assume that ß_{0}’ 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:
( 
{

T_{1NEW}  [  V_{1}T_{M}2  V_{M}T_{1}2  ]  }  V_{M}V_{1}  +V_{M}  )  ½  
T_{MNEW} =  ———–  ——————— +  T_{1}2   T_{1NEW}²  — T_{1NEW}  
T_{1}  V_{M}  V_{1}  V_{1} 
Eq. (5) becomes
V_{0NEW}T²  
V = V_{0NEW + }  ———–  
2T_{1NEW} 
(24)


———–  T_{1NEW}2  
ß_{1}  ß_{0} 
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, aZy, from Tabte 22 of Ref, 1 are found; they are 43l. With these, the temperature of the first transition point, T_{1}, can be calcutated:
T_{1} = 8.98 (a+Z+y) = 8.98 (8) = 71.84^{o}K 
Silver has a maximum of four magnetit rotational units vibrating, so the solid end point is
T_{M} = 17.87 * T_{1} = 17.87 * 71.84 = 1283.78^{o}K 
In this case the endgoint appears to be somewhat higher than the empirical melting point, 1234 ^{o}K Thus it would seem that the thermal factors at the end point are towered by one to 331, so that
T_{M} = 17.87 * 8.98 (3+3+1) = 1123.31^{o}K 
Now the number of units to use in selecting ß_{1} is
[3576 / 1123.31] = 3 
and therefore
ß_{1} = 15.5 * 10^{6} / ^{o}K 
Since siiver is etectronegative,
s_{0} =  1/7 ß_{1} = 1/7 * 15.5 * 10^{6} = 2.214 * 10^{6} 
Then from eq. (4),
V  T²  
— = 1 + ————————  
V_{0}  2 * 71.84  
———————–   71.84²  
(15.5 + 2.214) * 10^{6} 
V  
— =  1 + 1.234 * 10^{7} T�  T < T_{1}, P = 0 
V_{0} 
This equation holds good up to T = T_{1} at which point
V_{1}  
— =  1 + 1.234 * 10^{7} T_{1}² = 1.0006369 
V_{0} 
For temperatures above T_{1}, the values of V_{0}’ and ß_{0}’ are needed. To calculate ß_{0}’ I am going to use the empirical value of T_{M} pending theoretical clarification.
From eqs. (8) and (9),
2 * 71.84  
ß_{0}’ = 15.5 * 10^{6  }  ———————————— 
1.0006369 * 1234²  1.0625 * 71.84²  
——————————————— + 7184²  
1.0625  1.0006369 
= 9.647 * 10^{6} 
1.0006369  
V_{0}’ =  —————–  
1 + 71.84²  
————  = 1.0004265  
2 * 71.84²  
———————–  71.84²  
(15.5  9.647) * 10^{6} 
Thus from eq. (5),
1.0004265 T²  
V = 1.0004265 +  —————— 
2 * 7184  
————————  71.84²  
(15.5  9.647) * 10^{6} 
V = 1.0004265 + 4.07623 * 10^{8} T²  T > T_{1}, P = 0, V_{0} = 1 
(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_{1} (which is approx. 10^{7} KN/m² ) so that the initial compressibility factors from Table 14 of Ref. 1 can be used: aZy = 481. From Table 4 of Ref. 1,
s_{0} = 2.87 x 10^{10} m. Then from eq. (13),
P_{0} = 4.177 * 10^{23} (4 * 8 * 1) / (2.87 * 10^{10})³  
= 5.654 * 10^{7} KN/m² 
Since P c p , P = 0, dnd V = V . Then eq. (12) is
V  (  5.554 * 10^{7}  )  
— =  ——————–  ½ P < 1.0 * 10^{7} KN/m²  
V^{0}  5.554 * 10^{7} + P  T = 0 
If P = .001 P ,
V/V^{0} = (P^{0}/1.001 P^{0})½ = .999500 
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:
B = 2 * P^{0} = 2 * 5.654 * 10^{7} = 1.1305 * 10^{8} KN/m² 
E = 13 * P^{0}1 = 3 * 5.654 * 10^{7} = 1.1692 * 10^{8} KN/m² 
V = .25 
For the combined pressure and temperature loading, eq. (21) yields
V_{0 }5.654 * 10^{7}  
——————–  T²  
V = V_{0}  5.654 * 10^{7}  5.654 * 10^{7} + P  
——————– +  ————————  
5.654 * 10^{7} + P  
2 T_{1}  
———————   T_{1}2  
(15.5 + 2.214) * 10^{6} 
The value of T_{1} to be used here comes from eq. (22):
T_{1} =  2 
————————————————————  
15.5 * 10^{6} + 2.214 * 10^{6} + (15.5 + 2.214) * 10^{6}  
———————————  
1.0006369  
————– 
If P = .001 P_{0}, then
5.654 * 10^{T  1} 
——————– 
5.65 * 10^{7} 1P 
T_{1} = 128.24 癒 
Putting this value of T_{1} into the above gives:
V  
— =  .99950 + 6.91095 * 10^{8} T²  T < T_{1} 
V_{0}  P = .001 P_{0} 
The new value of T_{1 }gives the new value of V_{0}’:
V_{0}’ =  1.0006369  
————–  
1 + 128.224  
—————————— = 1.0002614  
2 * 128.24  
——————   128.242  
(15.5  9.647) * 10^{6} 
Thus, for temperatures above T_{1}
V = 1.0002614 +  1,0002614 T²  
——————––  
2 * 128.24  
————–—   128.24²  
(15.5  9.647) * 10^{6} 

V = 1.0002614 + 2.28350 * 10^{8} T²  T > T_{1}, P = .001P_{0}  
V_{0} = 1 
Finally, from eq. (23) I find that the new melting temperature is:
T_{MNEW} = 1650.88 癒 
(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.
**********************
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).