An exponential expression for gas heat capacity, enthalpy, and entropy

Experimental Thermal and Fluid Science 78 (2016) 249–253

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An exponential expression for gas heat capacity, enthalpy, and entropy Charles Bruel ⇑, François-Xavier Chiron 1, Jason R. Tavares, Gregory S. Patience ⇑ Department of Chemical Engineering, École Polytechnique de Montréal, 2900 boul. Édouard-Montpetit, Montréal, QC H3T 1J4, Canada

a r t i c l e

i n f o

Article history: Received 22 January 2016 Received in revised form 8 June 2016 Accepted 10 June 2016 Available online 11 June 2016 Keywords: Heat capacity Enthalpy Entropy High temperature Adiabatic flames Plasma

a b s t r a c t Gas heat capacity, CP, is a fundamental extensive thermodynamic property depending on molecular transitional, vibrational and rotational energy. Empirical four-parameter polynomials approximate the sigmoidal CP trends for temperatures up to 1500 K and adding parameters extends the range. However, the fitted parameters have no physical significance and diverge beyond their range at high temperature. Here we propose an exponential expression for CP whose fitted parameters relate to the shape of the CP versus T curve and to molecular properties: CP = CP0 + CP1[1 + ln(T)(1 + Ti/T)] exp(Ti/T). It accounts for more than 99% of the variance with a deviation of 1% from 298 K to 6000 K for linear C1–C7 hydrocarbons and N2, H2O, O2, C2H4, H2, CO, and CO2. We also provide an integrated form for enthalpy and an approximation to calculate entropy variations. This model replaces empirical polynomials with an expression whose constants are meaningful. Ó 2016 Elsevier Inc. All rights reserved.

1. Introduction Heat capacity (CP) is a critical property in thermodynamics spanning diverse fields related to states and phase transitions [1,2], heat transfer [3], spin measurements [4,5] or materials characterization [6,7]. The Dulong-Petit law [8] accounts for heat capacity of metals. Neglecting vibrational and rotational energies, heat capacity for ideal monoatomic gases is related to translational kinetic energy only [9] (3 degrees of freedom) and the molar

Abbreviations: DG, Gibbs free energy variation (J mol1); DH, enthalpy variation (J mol1); DS, entropy variation (J mol1 K1); e, empirical coefficient defined in Eq. (8); a, b, empirical coefficients defined in Eq. (3) (J mol1 K1); A, B, C, n, empirical coefficients defined in Eq. (1); A0, B1, C1, n1, B2, C2, n2, empirical coefficients defined in Eq. (2); AAD, Absolute Average Deviation (%); CP, heat capacity (J mol1 K1); CP0, empirical coefficient defined in Eq. (5) (J mol1 K1); CP1, empirical coefficient defined in Eq. (5) (J mol1 K1); D, deviance, sum of the squared differences between predicted and experimental values; Devi, deviation (%) between the ith experimental value and the ith predicted value; Ei, exponential integral; f, fi, yi, n, parameters defined in Eq. (7); nA, number of atoms in a molecule; R, gas constant; R2, correlation coefficient; Resi, residue of the fitting for the ith experimental value; S, S⁄, entropy (J mol1 K1); S0, empirical coefficient defined in Eq. (11) (J mol1 K1); S0⁄, empirical coefficient defined in Eq. (13) (J mol1 K1); T, temperature (K); T, dimensionless temperature expressed in Kelvin, T ¼ T=ð1 KÞ; Ti, empirical coefficient linked with Tinflection and defined in Eqs. (3)–(5) (K); Tinflection, temperature of inflection of the CP curve’s model (K). ⇑ Corresponding authors. E-mail addresses: [email protected] (C. Bruel), gregory-s.patience@ polymtl.ca (G.S. Patience). 1 Present address: Haldor Topsoe A/S, Haldor Topsøes Alle 1, DK-2800 Kgs. Lyngby, Lyngby, Denmark. http://dx.doi.org/10.1016/j.expthermflusci.2016.06.008 0894-1777/Ó 2016 Elsevier Inc. All rights reserved.

isobaric heat capacities are thus equal to 5/2 R. Theories for polyatomic gases poorly accounts for the variation of isobaric heat capacity with temperature. Textbooks [10,11] and specialized literature [12] report polynomial expressions that depend uniquely on the temperature (T). However, these expressions are valid over limited temperature ranges and deviate considerably for plasma applications [13] and adiabatic flames that can exceed 5000 K [14] (Fig. 1). NASA proposed a two range 5-term polynomial model [12] (200–1000 K and 1000–6000 K) to address the high temperature limitations (a total of 10 fitted parameters). Our model characterizes CP at very high temperatures whereas standard polynomials deviate beyond the specified temperature ranges (Reid et al. [10] and Yaws et al. [11]). The NASA 5-parameter model [12] deviates much less from the experimental CP than the other polynomial models. Models that approximate thermodynamic properties must meet several criteria [15,16]: all predicted values should be physically valid, and accurate over a wide range (1% average absolute deviation, AAD), the function should be integratable (to calculate changes in enthalpy, DH and entropy, DS), parsimonious (minimum number of parameters), and suitable for extrapolation. Among the non-polynomial models [17–19], Yuan and Mok [20] (1968) proposed a 4-parameter exponential expression:


 C C P ðTÞ ¼ A þ B exp  n T

ð1Þ

It fits polyatomic experimental data better than 5-parameter polynomials but relies on the exponential integral (Ei) to account

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C P ðTÞ ¼



 dH Ti Ti ¼ a þ b 1 þ lnðTÞ 1 þ exp  dT T T

ð4Þ

where the upper and lower bounds are lim C P ðTÞ ¼ a ¼ C 0P and T!0K

lim C P ðTÞ ¼ lim b lnðTÞ ¼ lim C 1 P lnðTÞ ¼ þ1.

T!1K

Fig. 1. Experimental methane heat capacity compared to model predictions from 298 K to 6000 K.

for DH and DS as a function of the temperature. The model accounts for over 99.5% of the experimental CP, even up to 6000 K (SI Table 1) but it tends toward a finite asymptote while the experimental values continue to increase. Furthermore, there is no coherent relationship between the 4 constants and molecular properties. Thinh et al. [21] expanded the exponential model to a 7-parameter group additive correlation:




C P ðTÞ ¼ A0 þ B1 exp 

C1 T n1




 B2 exp 

C2 T n2

T!1K

T!1K

Physically,

the

parameters CP0 (= a) and CP1 (= b), both in J mol1 K1, are thus respectively the limit of CP at low temperature and the growing rate of CP at high temperature, while Ti, in K, is a temperature related with the inflection point of the sigmoid and thus with the transition between low and high CP values. The inflection of a curve is mathematically defined as the point where the second derivative is null while its third derivative is different than zero. It happens for the CP model in Tinflection = Ti (0.353 + 0.234/Ti0.384) where Tinflection and Ti are both in K (R2 above 0.999). This expression comes from the resolution of the non-linear equation (d2CP/dT2)= 0 (Fig. 2). Replacing a and b with these limiting factors gives:



 Ti Ti C P ðTÞ ¼ C 0P þ C 1 exp  P 1 þ lnðTÞ 1 þ T T

ð5Þ


T 2 Ti DHT 1 !T 2 ¼ C 0P T þ C 1 T lnðTÞ exp  P T T1

ð6Þ

3. Methods

 ð2Þ

The number of parameters increases with the complexity of the molecule and can reach dozens of terms. Group additive correlations are often the most accurate [10,22] but also the most complex to handle [21,23–25]. Thinh et al. further developed additive models to calculate enthalpy, entropy and Gibbs free energy [26,27]. These correlations are accurate but cumbersome to implement because of the large number of parameters (not parsimonious). They are necessary for applications for which experimental heat capacity is nonexistent, sparse (as for polycyclic aromatic hydrocarbons) or for computed assisted engineering and modeling. (The thermodynamic package FactsageTM relies on up to 16 parameters to approximate gas CP between 298.15 K and 6000 K). Textbooks and handbooks [10,11] thus still report 4 to 5-term polynomials applicable over a narrower temperature-range that have no physical meaning.

Models were fitted to experimental data based on a Marquardt–Levenberg algorithm [31] to minimize the sum of the squared differences (deviance D) between the experimental data and the predicted values:

D¼

n X 2 ðf i  yi Þ

ð7Þ

i¼1

where n and yi are the number and the values of experimental data and f i values predicted by the model.

Resi ¼ f i  yi

ð8Þ

The deviation Devi between a prediction and an experimental value, in %, is calculated from the residues Resi:

Devi ¼ 100Resi =yi ¼ 100ðf i  yi Þ=yi

ð9Þ

The average absolute deviation, AAD, is the average of the absolute values of Devi, it is a measure of the average deviation a user can expect from the fitted expression:

2. Mathematical formulation of the CP model To reduce the empiricism of the polynomial models and account for the increase in CP above 3000 K, we modified the Yuan and Mok model with an expression that, contrary to Thinh et al. correlations, maintains its simplicity and can be integrated to calculate DH. We fit Eq. (1) to the CP of linear hydrocarbons (C1–C7), O2, H2O, H2, CO2, CO, N2, and ethylene (gas). All data we retrieved from the literature [28–30] and are either purely experimental [28,30] or interpolated [29]. With an exponential value of n = 1, we accounted for more than 99.6% of the variance in the data, which is as good as the fitting reported by Yuan and Mok [20]. Consequently, we integrated these expressions numerically to generate enthalpy as a function of temperature and fit the enthalpy with

H ðTÞ ¼ a T þ b T lnðTÞ expðT i =TÞ

ð3Þ

This expression accounts for more than 99% of the variance in the integrated CP where a, b and Ti are fitted parameters. Differentiating H⁄(T) gives:

Fig. 2. Methane model parameters deduced from experimental data: CP0  8/2 R, CP1 accounts for the growing rate of CP at high temperature, and Ti relates to the inflection point of the curve Tinflection.

C. Bruel et al. / Experimental Thermal and Fluid Science 78 (2016) 249–253

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Fig. 3. Model predictions compared to the experimental CP: a and b, the model accounts for 99% of the variance in the for linear hydrocarbons (C1–C7) and common gases (O2, H2O, CO2, N2, CO, H2).

AAD ¼

 n  n X X  jf i  yi j Dev i  ¼ 100  n  n jy j i¼1

i¼1

ð10Þ

i

4. Results and discussion

The coefficients CP0 and Ti of methane are closer to the other common gases, which is reasonable since its physico-chemical properties are distinct from the linear hydrocarbons. Furthermore, the best fit values of CP0 for diatomic gases – O2, N2, CO and H2 – are within 2% of the theoretical value for ideal diatomic gases [9] 7/2 R (29.1 J mol1 K1 or R/2 per degree of freedom in the molecule). In

The model (Eq. (5)) fits common gases (methane, dioxygen, nitrogen) from 298 K to 6000 K with a correlation coefficient R2 > 0.99 (Fig. 3a & b and SI Table 2). It fits C3–C7 linear hydrocarbons between 298 K and 1500 K and up to 3000 K for ethane equally well. Our model accounts for the experimental data equally as well as the 5-parameter polynomial model – AAD is mostly below 1% (SI Table 1, Fig. 4 and SI Figure 7 & 8). The Yuan and Mok [20] model accounts for the variance in the data slightly better than our exponential model but it has an additional fitted parameter. Moreover, its fitted parameters are not related to any molecular properties, whereas in Eq. (5) they can be linked with the number of atoms in the molecule, nA, and with its number of freedom degrees. Indeed, the coefficient CP1 increases linearly with nA: CP1 = 2.78 + 2.22 nA (R2 = 0.999) (Fig. 5a). For noble gases (monoatomic gases, nA = 1), Ti is infinite, thus CP  CP0 remains constant with T, which is the expected result for monoatomic gases. Similarly, CP0 and Ti vary linearly with nA for linear hydrocarbons (C2–C7): CP0 = 0.33 + 5.18 nA (R2 = 0.993) and Ti = 1132.2–7.3 nA (R2 = 0.965) (Fig. 5a & b).

Fig. 4. Average AAD for CH4, O2, H2O, CO, CO2, N2, H2 and C2H4 between 298 K and 6000 K for the exponential models (Eq. (5) and Yuan and Mok [20]), and polynomial models. Vertical bars represent the standard deviation of the AAD. Our model is significantly better than 4-parameter polynomials and is equivalent to the 5-parameter polynomial model.

Fig. 5. Model coefficients related to number of atoms, nA: In a, coefficients CP0 and CP1 are regressed based on the number of atoms in the molecule nA: CP1 = 2.78 + 2.22 nA (R2 = 0.999) for all gases. Restricting linear hydrocarbons CP0 to C2–C7 hydrocarbons gives: CP0 = 0.33 + 5.18 nA (R2 = 0.993). In b, the same restriction was applied for Ti: Ti = 1132.2–7.3 nA (R2 = 0.965).

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of the variance in the data with only three fitted parameters. The parameters reflect the characteristics of CP vs T curve—origin (CP0), inflection point (Ti) and growing rate (CP1). The fitted parameter CP0 is within a couple percent of the theoretical value of heat capacity tending toward 0 K. (For a diatomic gas CP = 7/2 R). The other two terms vary with the number of atoms in the molecule. It applies to all thermodynamic and thermochemical processes from heat transfer or adiabatic flame temperatures to Gibbs free energy calculations (DG = DH  TDS) to determine the direction and equilibrium of chemical reactions. Adding a fitted parameter increases the accuracy in entropy calculations (Eq. (12)). The model may apply to a larger variety of compounds and replace previous empirical polynomial expressions in handbooks and textbooks: the expression has fewer parameters, a wider temperature range, and equivalent accuracy.

Fig. 6. Entropy model predictions compared to the experimental data from 298 K to 6000 K. The model (Eq. (12)) deviates on average less than 1% for all the compounds.

fact, substituting the 7/2 R rather than fitting the value only drops the R2 from 99.4% to 99.3% for N2 in the range of 100–6000 K. For CO2, CP0 is also approximately equal to 7/2 R, whereas it is about 8/2 R for H2O, CH4 and C2H4. Thus, the experimental coefficients of our model are directly linked to some molecular characteristics and may be related to previous theories on ideal mono and diatomic gases. Associating physico-chemical properties to the parameters reduces the empiricism of previous models. Besides DH, DS is the other critical thermodynamic properties that depends on CP. Integrating CP/T (the Eq. (5)) leads to an Ei. Here, we propose a good approximation to it based on the same parameters we derived for CP (SI Table 2):



 
T 2 5T 2T Ti 1þ exp  DST 1 !T 2  C 0P lnðTÞ þ C 1 1þ P lnðTÞ 3 Ti 3 Ti T T1 ð11Þ


 

 5T 2T Ti þ S0 1 þ exp  SðTÞ  C 0P lnðTÞ þ C 1 lnðTÞ 1 þ P 3 Ti 3 Ti T ð12Þ CP0,

CP1

where and Ti are the same as the values for CP. This equation fits the experimental S [28,29] (SI Table 3, Fig. 6). The absolute deviation between the model and experimental data is less than 2% up to 6000 K for all compounds except ethylene (4%) (SI Figure 9). With the same coefficients that we derived for CP, the model accounts for 99% of the variance in the data with an average absolute deviations, AAD, of 1% over the whole temperature range. Introducing a fitted parameter e to replace the coefficients, 2/3 and 5/3, reduces that average deviation by factor of two (SI Table 3 and SI Figure 9):

S ðTÞ  C 0P lnðTÞ þ C 1 P lnðTÞ
 Ti þ S0 exp  T



 T T 1þe 1 þ ð1 þ eÞ Ti Ti ð13Þ

5. Conclusions The heat capacity of light gases and linear hydrocarbons (C2 to C7) varies as a function of temperature following an Arrhenius type dependency (Eq. (5)). The expression we derived to approximate CP and the integrated forms to give DH (Eq. (6)) and DS (Eq. (11)) are valid from 298 K up to 6000 K. These expressions account for 99%

Associated content Supporting Information Available: SI Fig. 7 and 8 & 9 related to deviation analysis and models comparison. SI Table 1, 2 & 3 providing experimentally fitted parameters for Eqs. (5), (6), (11) and (13) and models average absolute deviation AAD and correlation coefficients R2. This material is available free of charge via the Internet at http://www.sciencedirect.com/. Author contributions G.S.P conceived the project, C.B. and F.-X.C. did the literature review and collected the data. C.B. developed the model with assistance from G.S.P. The manuscript was written through contributions from C.B., F.-X.C., J.R.T and G.S.P. All authors have given approval to the final version of the manuscript. Funding This work was supported by the Fond de Recherche du Québec – Nature et Technologie (FRQNT). Acknowledgment We acknowledge the contribution of G. Lemieux and N. Amane as well as the FRQNT for its financial support. Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.expthermflusci. 2016.06.008. References [1] E.B. Moore, V. Molinero, Structural transformation in supercooled water controls the crystallization rate of ice, Nature 479 (2011) 506–509. [2] C.A. Angell, Insights into phases of liquid water from study of its unusual glass-forming properties, Science 319 (2008) 582–587. [3] M.W. Graham, S.-F. Shi, D.C. Ralph, J. Park, P.L. McEuen, Photocurrent measurement of supercollision cooling in graphene, Nat. Phys. 9 (2013) 103–108. [4] S. Yamashita, T. Yamamoto, Y. Nakazawa, M. Tamura, R. Kato, Gapless spin liquid of an organic triangular compound evidenced by thermodynamic measurements, Nat. Commun. 2 (2010) 275. [5] T. Liang, S.M. Koohpayeh, J.W. Krizan, T.M. McQueen, R.J. Cava, N.P. Ong, Heat capacity peak at the quantum critical point of the transverse Ising magnet CoNb2O6, Nat. Commun. 6 (2015) 7611. [6] A.C. Jacko, J.O. Fjærestad, B.J.A. Powell, A unified explanation of the KadowakiWoods ratio in strongly correlated metals, Nat. Phys. 5 (2009) 422–425. [7] M.H. Khademi, A.Z. Hezave, A. Jahanmiri, J. Fathikaljahi, An expression for ratio of critical temperature to critical pressure with the heat capacity for low to medium molecular weight compounds, J. Chem. Eng. Data 54 (2009) 690–700.

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