Relationships Between Volume Thermal Expansion and Thermal Pressure Based on the Stacey Reciprocal K-Primed EOS

Indian Journal of Pure & Applied Physics Vol. 49, February 2011, pp. 99-103

Relationships between volume thermal expansion and thermal pressure based on the Stacey reciprocal K-primed EOS

S S Kushwah* & Y S Tomar Department of Physics, Rishi Galav College, Morena 476 001, MP, India *E-mail: [email protected] Received 25 February 2010; revised 22 December 2010; accepted 10 January 2011

It has been found that the Stacey reciprocal K-primed EOS is consistent with the experimental data for bulk modulus and thermal pressure as it yields correct values for volume expansion at high temperatures. The comparison of calculated values with the experimental data has been presented in case of NaCl, KCl, MgO, CaO, Al 2O3 and Mg 2SiO 4. It is also emphasized that the two equations mimicking the Stacey EOS recently used by Shrivastava [ Physica B , 404 (2009) 251] are in fact originally due to Kushwah et al . [ Physica B , 388 (2007) 20]. The results obtained for the thermal pressure using the Kushwah EOS are in good agreement for all the solids under study. Keywords : Equation of state; Bulk modulus; Thermal pressure; Thermal expansion

1 Introduction In writing Eqs (2 and 3), it has been assumed that Thermal pressure is a physical quantity of central the thermal pressure is a function of temperature importance 1 for investigating the thermoelastic only 1. At atmospheric pressure, i.e. at P(V,T) = 0, we properties of materials at high temperatures 2-7. The have: volume expansion of solids due to the rise in 8,9 temperature is directly related to thermal pressure . PVT( , ) = −∆ P …(4) In the present study, an analysis of volume expansion 0 th in terms of thermal pressure of solids using the Stacey reciprocal K-primed (SRKP) equation of state It should be mentioned that SRKP-EOS is valid for 10-12 both the isothermal bulk modulus as well as for the (EOS) has been performed. The SRKP-EOS can 11 be written as follows 10 : adiabatic bulk modulus . Eq. (1) on integration gives an expression for bulk modulus which has further been integrated to obtain 10 : 1 1 K '  P = +1 − ∞  …(1) KK' ' KK '  0 0 V  K' P   K '  P ln=0 ln1' −K +−0 1 …(5)   2 ∞    VK0  ∞' KKK  ∞ '  where K is the bulk modulus at pressure P, K0' and K∞' are the values of K′ = d K/d P at P = 0, the atmospheric pressure and P→∞ , respectively. Following the work Now, replacing P by –∆Pth, Eq. (5) can be written as: of Anderson 1, we can write : V  K'∆ PKP   '  ∆ ln  =0 ln 1 +K 'th −−0 1  th PVT(,)= PVT (,) + ∆ P …(2) 2 ∞  0 th VK0  ∞' KK  ∞ '  K …(6) where V is the volume, T the temperature and T0 is initial value of temperature taken here equal to 300 K. In the present study, Eq.(6) has been used The last term in Eq.(2) represents the difference in the successfully to predict volume expansion using the values of thermal pressure at two temperatures T and data on thermal pressure and bulk modulus at high T0, i.e. temperatures 1. This has been demonstrated here for

six solids viz. NaCl, KCl, MgO, CaO, Al 2O3 and ∆P = PT() − PT () th th th 0 …(3) Mg 2SiO 4. 100 INDIAN J PURE & APPL PHYS, VOL 49, FEBRUARY 2011

2 Method of Analysis K A=0 () K' − 2 K ' + 2 …(9) Eq. (6) has been used to calculate V/V0, which now 22 0 ∞ represents the volume expansion with the increase in temperature for solids at atmospheric pressure. Here and V0 is the volume at T = 300 K and V is the volume at K 0 2 elevated temperatures. Values of ∆Pth and K have A3=( KK 000 " +− K '3''6' KK∞ 00 + K been taken from the data for solids at different 6 …(10) 1 2 temperatures reported by Anderson . Values of input +3K∞ ' − 12 K ∞ ' + 6) parameters viz. K0, K0' and K∞' for different solid are presented in Table 1. Values of K∞' have been Thus, the coefficients A1, A2 and A3 in Eq. (7) are 14 determined using the relationship K∞' = 0.6 K0' identical with those taken by Shrivastava . K0 is the 12 supported by seismic data as well as solid state bulk modulus, K0' the first pressure derivative of bulk 13,14 data . modulus and K0'' is the second pressure derivative of It is found that the results for volume expansion bulk modulus at atmospheric pressure. From a V/V0 calculated from Eq.(6) based on the SRKP-EOS detailed analysis of seismic data for the lower mantle given in Table 2 present remarkably good agreement and core of the Earth as well as solid state data, it has with those based on the experimental density data been found empirically that the following relationship 11-14 reported by Anderson 1 in case of all the six solids for holds precisely well . the entire range of temperatures up to T ≥ 2 θD, where 3 θD is the Debye temperature. Values of ∆Pth and K at K'= K ' …(11) different temperatures are also included in Table 2. ∞ 5 0 The value of ∆Pth increases and K decreases with the increase in temperature. The variation is almost linear Values of K0K0'' are obtained from the Stacey for both, the thermal pressure and bulk modulus. relationship 10-12 as given below: Recently, Shrivastava 14 presented two forms of EOS mimicking the Stacey EOS. However, KK"=− KKK ' ( ' −′ ) =− 0.4 K ' 2 …(12) 14 00 00∞ 0 Shrivastava did not mention that these equations are originally due to Kushwah et al 15 . The generalized 15 Eq. (12) is based on the reciprocal K'-EOS due to Kushwah logarithmic EOS can be written as : Stacey 10 . Eqs(9) and (10) reduce to the following

forms on substituting the values of K∞' and K0K0'' K ∞' 2 Px(1− ) = A1 ln(1 ++ xA ) 2 [ln(1 + x )] from Eqs (11) and (12). …(7) 3 +A3[ln(1 + x )]   K 0' A2= K 0  − + 1  …(13) where x = (1−V/V0), V0 is the volume at atmospheric 10  pressure. The constants A1, A2 and A3 are determined by using the conditions at atmospheric pressure. and

K'2 K '  A = K …(8) 0 0 1 0 A3= K 0  − − + 1  …(14) 50 5  Table 1 — Values of input data 1 used in calculations; isothermal bulk modulus K 0 (GPa) and pressure derivative of isothermal bulk 15 modulus K 0' are at T = 300 K and atmospheric pressure. Values of Kushwah et al. used Eqs (7) to (14) for hcp iron K∞' are equal to 0.6 K 0' and found that the results presented good agreement with those based on the SRKP-EOS. Shrivastava 14 Solid K K ' K∞' 0 0 used the same formulation for some other metals and NaCl 24.0 5.38 3.23 confirmed that the Kushwah EOS, Eq. (7), is KCl 17.0 5.46 3.28 mimicking the Stacey EOS. Shrivastava 14 should have MgO 162 4.15 2.49 given the proper reference to the original work. CaO 111 4.85 2.91 In the present study,we make use of Eq.(7) to Al O 252 3.99 2.39 2 3 calculate the thermal pressure, replacing P by −∆Pth . Mg 2SiO 4 127 5.40 3.24 Thus, Eq. (7) takes the following form:

KUSHWAH & TOMAR: STACEY RECIPROCAL K-PRIMED EOS 101

Table 2 — Values of thermal pressure ∆Pth, and volume thermal expansion V/V 0 for different solids as a function of temperature

Solid Temperature Isothermal bulk Thermal pressure ∆Pth (GPa) Volume thermal expansion V/V 0 T (K) modulus K T (GPa) Kushwah EOS Experimental Stacey EOS Experimental Experimental Eq.(15) Ref. [1] Eq. (6) Ref. [1] Ref. [1]

NaCl 300 24.0 0.00 0.00 1.0000 1.0000 350 23.2 0.14 0.14 1.0060 1.0060 400 22.4 0.28 0.28 1.0122 1.0123 450 21.6 0.43 0.43 1.0191 1.0188 500 20.8 0.57 0.57 1.0258 1.0256 550 19.9 0.71 0.71 1.0330 1.0328 600 19.0 0.85 0.85 1.0406 1.0402 650 18.1 0.99 0.99 1.0486 1.0480 700 17.3 1.13 1.13 1.0568 1.0561 750 16.5 1.26 1.27 1.0653 1.0645

KCl 300 17.0 0.00 0.00 1.0000 1.0000 350 16.4 0.09 0.09 1.0054 1.0054 400 15.9 0.18 0.19 1.0116 1.0112 450 15.4 0.28 0.28 1.0175 1.0172 500 14.7 0.37 0.37 1.0238 1.0235 550 14.2 0.46 0.47 1.0308 1.0301 600 13.7 0.56 0.56 1.0374 1.0370 650 13.2 0.65 0.65 1.0442 1.0442 700 12.6 0.75 0.75 1.0523 1.0517 750 12.0 0.84 0.84 1.0601 1.0594 800 11.5 0.93 0.93 1.0679 1.0675 850 10.9 1.01 1.02 1.0764 1.0759

MgO 300 162 0.00 0.00 1.0000 1.0000 400 159 0.53 0.54 1.0034 1.0033 500 156 1.16 1.12 1.0071 1.0073 600 153 1.76 1.73 1.0111 1.0112 700 150 2.38 2.35 1.0153 1.0153 800 147 3.02 2.98 1.0196 1.0196 900 144 3.66 3.61 1.0241 1.0240 1000 141 4.28 4.24 1.0287 1.0284 1100 138 4.93 4.87 1.0334 1.0331 1200 135 5.58 5.50 1.0382 1.0379 1300 132 6.21 6.12 1.0431 1.0427 1400 129 6.83 6.74 1.0482 1.0476 1500 126 7.48 7.36 1.0533 1.0528 1600 123 8.12 7.97 1.0586 1.0581 1700 119 8.76 8.58 1.0644 1.0635 1800 117 9.37 9.20 1.0696 1.0688

CaO 300 111 0.00 0.00 1.0000 1.0000 400 109 0.36 0.36 1.0033 1.0033 500 106 0.72 0.74 1.0069 1.0066 600 104 1.14 1.13 1.0106 1.0106 700 102 1.54 1.53 1.0146 1.0145 800 100 1.96 1.94 1.0187 1.0186 900 98 2.35 2.34 1.0228 1.0226 1000 96 2.74 2.74 1.0270 1.0267 1100 94 3.15 3.13 1.0312 1.0311 1200 92 3.56 3.53 1.0357 1.0356

Contd — 102 INDIAN J PURE & APPL PHYS, VOL 49, FEBRUARY 2011

Table 2 — Values of thermal pressure ∆Pth, and volume thermal expansion V/V 0 for different solids as a function of temperature — Contd

Al 2O3 300 252 0.00 0.00 1.0000 1.0000 400 250 0.45 0.45 1.0018 1.0018 500 247 1.00 0.98 1.0039 1.0040 600 244 1.56 1.55 1.0063 1.0063 700 241 2.17 2.15 1.0088 1.0088 800 238 2.79 2.76 1.0114 1.0114 900 235 3.45 3.43 1.0143 1.0142 1000 231 4.06 4.01 1.0169 1.0168 1100 228 4.68 4.64 1.0197 1.0195 1200 224 5.32 5.27 1.0227 1.0223 1300 221 5.97 5.90 1.0257 1.0252 1400 217 6.56 6.53 1.0288 1.0279 1500 214 7.19 7.16 1.0318 1.0308 1600 211 7.81 7.79 1.0349 1.0337 1700 208 8.38 8.42 1.0381 1.0364 1800 205 9.01 9.05 1.0413 1.0394

Mg 2SiO 4 300 127 0.00 0.00 1.0000 1.0000 400 125 0.35 0.36 1.0029 1.0028 500 123 0.74 0.75 1.0060 1.0059 600 121 1.16 1.16 1.0094 1.0094 700 119 1.57 1.57 1.0128 1.0129 800 116 1.98 1.98 1.0164 1.0164 900 114 2.37 2.40 1.0201 1.0199 1000 112 2.80 2.83 1.0239 1.0238 1100 110 3.22 3.25 1.0277 1.0277 1200 107 3.67 3.69 1.0320 1.0320 1300 105 4.11 4.13 1.0361 1.0363 1400 102 4.54 4.56 1.0406 1.0407 1500 100 4.97 5.00 1.0449 1.0451 1600 98 5.40 5.43 1.0492 1.0498 1700 95 5.84 5.87 1.0541 1.0547

K ∞' 2 investigating the volume expansion of solids at high −∆−Pxth(1 ) = A 1 ln(1 ++ xA ) 2 [ln(1 + x )] temperatures. The same is true for the Kushwah EOS +A[ln(1 + x )] 3 3 which predicts the values of thermal pressure at high …(15) temperatures which is in agreement with the available Here V/V0 is now the volume expansion with the experimental data. Kushwah EOS is mimicking the increase in temperature for solids at atmospheric Stacey EOS because both of them are consistent with pressure ( P = 0). Values of thermal pressure ∆Pth Eqs (11) and (12). It should be mentioned that the calculated from Eq. (15) based on the Kushwah EOS Kushwah EOS can conveniently be used only for are also presented in Table 2. It is found that the determining values of thermal pressure if the volume results obtained for the thermal pressure using the expansion data are available. Kushwah EOS present good agreement with the experimental data in case of all the solids under study Acknowledgement for the whole range of measurements. One of the authors (YST) is grateful to University 3 Conclusions Grants Commission (UGC), Bhopal (MP) for To conclude, the results obtained in the present the financial assistance in the form of a research study support the validity of the Stacey EOS for project. KUSHWAH & TOMAR: STACEY RECIPROCAL K-PRIMED EOS 103

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