Journal of Materials Science

, Volume 50, Issue 2, pp 678–687 | Cite as

Approximated equations for molar volumes of pure solid fcc metals and their liquids from zero Kelvin to above their melting points at standard pressure

  • George Kaptay
Original Paper


Approximated equations have been constructed to describe the temperature (T) dependence of the molar volumes (V) of fcc solid metals (Ag, Al, Au, Cu, Ir, Ni, Pb, Pd, Pt, and Rh) and their liquids below and above their melting points at standard pressure of 1 bar from zero Kelvin. Below the melting point, the following new approximated equation is suggested for both the solid and liquid metals: V = a + b * T (to the power of n), where a, b, and n are semi-empirical parameters (at n larger than 1; this equation obeys the boundary condition that the thermal expansion coefficient becomes zero at T = 0 K). This approximated equation reproduces the measured molar volume of solids from zero Kelvin to melting point with an accuracy of 0.2 % or better. As a compromise, the derivative of this equation reproduces the measured thermal expansion coefficient of solids only with an accuracy 10 % or better and only above 100 K. Above the melting point, the following well-known equation is used for both liquid and solid phases: V = c + d * T, where c and d are semi-empirical parameters. This equation implies that the thermal expansion coefficient above the melting point has an approximately constant value. It is found that the volume change upon melting extrapolated to zero K is about 58 % of that at the melting point for all the 10 fcc metals. The tabulated 4 equations (below and above the melting point/for fcc and liquid states) are provided for each of the 10 fcc metals. These equations will be useful for estimating phase equilibria of nano-materials.


Melting Point Molar Volume Liquid Metal Thermal Expansion Coefficient Solid Metal 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



The authors acknowledge the financial support from the Hungarian Academy of Science, under the grant number K101781 and the “Science and Industry: the road from scientific achievements to social benefits” TÁMOP-4.2.3.-12/1/KONV-2012-0029 project supported by the European Union and co-financed by the European Social Fund. This work was partially carried out in the framework of the Center of Applied Materials Science and Nano-Technology at the University of Miskolc.


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of NanotechnologyUniversity of MiskolcMiskolcHungary
  2. 2.Department of Nanomaterials, BAY-LOGIBay Zoltan Nonprofit LtdMiskolcHungary

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