K. R. SHARMA

848

Table 2. Materials that possess NTE.

# Material Journal Reference

1.0 ZrW2O8, Zirconium

Tungstate (cubic lattice) Acta Crystallography[11]

2.0 (ZrO)2VP2O7 US Patent [12]

3.0 HfW2O8, Halfnium

Tungstate (orthorhombic) J. Appl. Phys. [13]

4.0 ZrMo2O8, Zirconium

molybdate, (cubic) Chem. Mater. [14]

5.0 Silicalite-1 &

Zirconium Silicalite-1 Mater. Res. Bulletin[15]

6.0 CuScO2 (delafossite structure) Chem. Mater. [16]

7.0 Polydiacetylene Crystal J of Polym. Sci. [17]

8.0 Graphite Fiber Composites Proc. of Royal Society[18]

7. Conclusions

Most materials expand upon heating. Some materials

have been reported as NTE, negative thermal expansion

coefficient materials. Very little thermodynamic analysis

has been done on these materials. In this study, Helmholz

free energy change during melting or solidication was

undertaken. Equation (10) was derived for expansivity at

equilibrium.

For physical changes, Equation (10) can be seen to be

always positive. This is because the lowering of pressure

as solid becomes liquid and the internal energy change is

positive resulting in a net positive sign in the RHS, right

hand side of Equation (10). No ideal gas law was as-

sumed. Only the first two laws of thermodynamics were

used and reversible changes were assumed in order to

obtain Equation (10). Thus reports of materials with ne-

gative thermal expansion coefficient are inconsistent

with Equation (10).

For ideal gases, Equation (23) would revert to the

volume expansivity,

would equal the reciprocal of ab-

solute temperature. This would mean that

can never be

negative as temperature is always positive as stated by

the third law of thermodynamics. So materials are with

negative values for

ab initio are in violation of the

combined statement of the 1st and 2nd laws of thermody-

namics.

Along similar lines to the improvement given by

Laplace to the theory of the speed of sound as developed

by Newton (as discussed in Section 2 Historical Note) an

isentropic volume expansivity is proposed by Equation

(24). This can be calculated using Equation (28) from

isobaric expansivity, isothermal compressibility and a

parameter

that is a measure of isentropic change of

pressure with temperature. Equation (34) can be used to

obtain the isentropic expansivity in terms of heat capaci-

ties at constant volume and constant pressure and iso-

thermal compressibility at a given pressure and tempera-

ture of the material.

Chemical changes have to be delineated from physical

changes when heating the material.

8. Acknowledgements

Acknowledgements are extended to my undergraduate

students in advanced standing in my CHEG 2043 Ther-

modynamics and CHEG 3053 Thermodynamics-II cour-

ses at Prairie View A & M University, Prairie View, TX

for valuable discussions on Clapeyron equation during

office hours.

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