International Journal of Scientific & Engineering Research, Volume 6, Issue 3, March-2015 104

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Thermal expansivity of geophysical minerals at high temperatures

Monika Panwar, S.K. Sharma, Sanjay Panwar

Abstract— A new model for investigating the value of thermal expansivity is presented. It has been found that the values obtained in the present study are consistent with the experimental data. The excellent agreement between results obtained and experimental data show the validity of the present work.

Index Terms—Thermal pressure, Thermal expansivity, Bulk modulus, Equation of state, Volume expansion ratio, Geophysical minerals.

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1 INTRODUCTION

Thermal expansivity is a very important parametric quantity for interpreting the thermal and elastic properties of minerals at high temperatures because it has been emphasized [1] that most of the serious errors in the calculations of thermodynamic function arise due to uncertainty of thermal expansivity at high temperatures. Many researchers [2-5] have been made to estimate the temperature dependence of thermal expansivity to keep in mind its linear as well as non- linear dependence. Thermal expansivity is necessary parameter for solving many problems of material science and geophysics and many thermal and elastic properties can also be derived from it. Thermal expansivity may also be defined as

(1)

where , and are the volume, temperature and pressure respectively. Method of analysis is seen in section 2, result and discussion in section 3 and conclusion are presented in section 4.

2 METHOD OF ANALYSIS

Anderson [6] used the following relationship to estimate the value of thermal pressure

(2)

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• Monika Panwar, Department of Physics, MMEC, Maharishi Markandeshwar University, Ambala, Haryana-133207, India. PH+918059110033. E-mail: monikaverma05@gmail.com

• S.K. Sharma,, Department of Physics, Shivalik Institute of Engineering and Technology, Aliyaspur, Ambala, Haryana-133206, India. PH+918295055704. E-mail: sav31@rediffmail.com

• Sanjay Panwar, Department of Physics, MMEC, Maharishi

Markandeshwar University, Ambala, Haryana-133207, India. PH+918607715111. E-mail: dr.spanwar@gmail.com

(3)

where , , and are thermal pressure, thermal expansivity and isothermal bulk modulus and is room temperature or reference temperature. Thermal pressure is a physical quantity of central importance for investigating the thermo elastic properties of materials at high temperatures [6-8]. The volume expansion of solids due to the rise in temperature is directly related to thermal pressure [9, 10]. The equation of state (EoS) is effectively important in studying the properties of solids under high pressure and high temperature. The EoS yields pressure–volume– temperature (P–V–T) relationship for solids and helps in estimating a variety of properties under different conditions of pressure and temperature

(4)

where is the volume, the temperature and is initial value of temperature taken here equal to 300 K. In writing Eq. (2) and (3), it has been assumed that the thermal pressure is a function of temperature only [1]. At atmospheric pressure i.e. at P (V, T) = 0, we have

(5) It should be mention that Stacey reciprocal K-primed (SRKP)

equation of state (EoS) [11-13] is valid for both isothermal
bulk modulus as well as adiabatic bulk modulus. The Stacey
reciprocal K-primed (SRKP) equation of state (EoS) [11] can be written as follows

(6) Eq. (6) on integration gives an expression for bulk modulus

which has further been integrated to obtained [11]

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Now, replacing by , Eq. (5) can be written as:

where is the isothermal bulk modulus, is the first pressure derivative of isothermal bulk modulus at


atmospheric pressure i.e. and is an important
(7)
(8)
modulus () have been predicted from Eq. (10). Results obtained shown the consistency with the experimental data as shown in Table -2, 3, and 4 for MgO, Al2 O3 and Mg2 SiO4 .
As it is clear from Eq. (3) to investigate the values of thermal expansivity, the values of thermal pressure, and are necessary. By using the concept Eq. (9) takes the form

(11)

parameter which is first pressure derivate of isothermal bulk modulus at infinite pressure i.e. . Which can be determine by using [13-17]

where is the first pressure derivative of isothermal bulk modulus at atmospheric pressure i.e. and at room temperature.

Pallavi Sinha et al. [18] used the following data to calculate data
(9)
where

and other parameter are having their usual meaning and the following relationship have been used to be investigate the value of isothermal bulk modulus by Pallavi Sinha et al. [18]

where all the parameters are having their as usual meaning. The predicted values through Eq. (11) are compared with experimental data Anderson [6] in Table-2, 3 and 4 for MgO, Al2 O3 , Mg2 SiO4 . Comparison between results obtained and experimental data show the good agreement for all geophysical minerals. Using the Eq. (10) and (11), Eq. (3) becomes
(12) The results obtained through Eq. (12) are shown in Table-2, 3
and 4 with the experimental data for MgO, Al2 O3 , and
Mg2SiO4 respectively. Consistency the results obtained from
Eq. (12) and experimental data validate the present approach. For direct vision vs are plotted in Figures-1, 2 and 3 for MgO, Al2 O3 , Mg2 SiO4 respectively. It is clear from these figures that the results obtained through Eq. (11) are competent with experimental data [6]. vs are plotted in Figures- 4, 5 and 6 for MgO, Al2 O3 , and Mg2 SiO4 respectively. An excellent agreement between results obtained through Eq. (12) and experiment data shows the validity of present model.

3 RESULTS AND DISCUSSIONS

(10)

4 CONCLUSIONS

In the present study we introduce a new model to predict the values of thermal expansivity for geophysical minerals such as MgO, Al2 O3 , and Mg2SiO4. It is conclude that the results
We have computed the values of volume expansion by using
the Eq. (8) values of input parameter used in the calculation are shown in Table -2, for geophysical minerals MgO, Al2 O3 and Mg2 SiO4 . The values of have been taken from Anderson [6] to calculate volume expansion through Eq. (8) is shown in Table-1 and the results obtained through it. A close agreement has been found for all geophysical minerals used in the present study. The values of isothermal bulk
obtained in the present study support the validity of the present model.
Table 1 – Values of input data used in calculations: isothermal bulk modulus (GPa) and pressure derivative of isothermal bulk modulus are at T = 300 K and atmospheric pressure.

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Table 2 — Values of volume thermal expansion, thermal pressure (), and thermal expansivity () for MgO as a function of temperature

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Table 3 — Values of volume thermal expansion, thermal pressure (), and thermal expansivity () for Al2 O3 as a function of temperature

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Table 4 — Values of volume thermal expansion, thermal pressure (), and thermal expansivity () for Mg2 SiO4 as a function of temperature

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Figure 1: Comparison of theoretical and experimental value of temperature dependence of volume expansion ratio calculated by various Eqs. for MgO

Figure 3: Comparison of theoretical and experimental value of temperature dependence of volume expansion ratio calculated by various Eqs. for

Mg 2 SiO 4

Figure 2: Comparison of theoretical and experimental value of temperature dependence of volume expansion ratio calculated by various Eqs. for

Al2 O 3

Figure 4: The graph between thermal expansivity ( ) and temperature for

MgO between calculated values and experimental data

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Figure 5: Thermal expansivity ( ) for Al 2 O 3 calculated in present study and experimental data

[5] Fang Z H, Physica B 403 (2008) 1971.

[6] Anderson O L, Equations of State of Solids for Geophysics and Ceramic science, Oxford University Press, New York (1995).

[7] Wang Z, Lazer P, Saxena S K, Physica B 293 (2001) 408.

[8] Wang Z, Schott B, Lazer P, Saxena S K, J. Alloys Compd. 315 (2001) 51. [9] Kushwah S S, Shanker J, Physica B 253 (1998) 90.

[10] Shanker J, Kushwah S S, High Temp. High Press. 33 (2001) 207. [11] Stacey F D, Geophys J. Int. 143 (2000) 621.

[12] Stacey F D & Davis P M, Phys. Earth Planet. Int. 142 (2004) 137. [13] Stacey F D, Rep. Prog. Phys. 68 (2005) 341.

[14] Kushwah S S, Bhardwaj N K, J. Phys. Chem. Sol. 70 (2009) 700. [15] Srivastava H C, Physica B 404 (2009) 251.

[16] Kushwah S S, Shrivastava H C & Singh K S, Physica B 388 (2007) 20. [17] Kushwah S S, Tomar Y S, Ind. J. Pure and App. Phys. 49 (2011) 99-103. [18] Sinha Pallavi, Srivastava S.K. & Verma Nidhi, Physica B 406 (2011) 2488-

2491.

Figure 6: Thermal expansivity ( ) for Mg 2 SiO 4 calculated in present study and experimental data

REFERENCES

[1] Xia X & Xiao J, J. Phys. Chem. Sol. 54 (1993) 629.

[2] Singh K S & Chauhan R S, Physica B 315 (2002) 74. [3] Srivastava S K, Physica B 355 (2005) 32.

[4] Fang Z H, J. App. Phys. 102 (2007) 013523.

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