V. S. Iorish

Russian Academy of Sciences, High Energy Density Research Center of "IVTAN" Association, Glushko Thermocenter, Moscow

An empirical formula of heat capacity temperature dependence for solid organic substances in the range from room temperature up to melting point is offered. The formula is based on consideration of known heat capacity contributions and the analysis of experimental data. The standard deviation for the dependence is 0.5% and the maximum deviation is 1.2% for the substances used for the formula derivation. Good accuracy of the suggested formula is shown for substances, the data for which were not included into the primary data set.

The knowledge of heat capacity and its temperature dependence is very important for description of the thermodynamic properties of substances. In absence of experimental data group-additivity method is used for the estimation of heat capacity and other properties of the organic compounds. For gaseous substances such approach was used for many years and appropriate increments for heat capacity were repeatedly specified for a wide temperature range from 298.15 up to 1500 Ê. For organic substances in a solid state the opportunity of the additive approach has recently been shown in the work of E.S.Domalski and E.D.Hearing [1] for room temperature.

The questions of temperature dependence estimation of heat capacity for solid organic substances, as far as it is known to the author, were not so far considered in the literature. The approach, similar to the one used for gases, can not be applied here, as far as the melting temperatures of substances are various and it is impossible to find increments of an additive scheme for a set of given temperatures. At the same time importance of solution of this problem is not doubtful, as neglecting of temperature dependence of heat capacity can cause errors in estimation of all other thermodynamic properties at temperatures above 298.15K. It is especially important to take into account the temperature dependence of heat capacity for substances with high melting temperatures what are, for example, polycyclic aromatic hydrocarbons and polychlorinated dibenzo-p-dioxins.

In the present paper possible contributions into heat capacity of solid organic substances are briefly analyzed and simple formula for the prediction of temperature dependence from room temperature up to melting point (T_m) is offered. It is supposed, that the heat capacity value at room temperature is known from experimental data or estimated by additive or any other appropriate method.

According to the heat capacity theory for solid substances isobaric heat capacity of organic compounds can be presented as a sum of contributions [2]:

C_p = C_q + C_d, (1)

where C_q represents the vibrational (phonon) contribution, and C_d = C_p - C_v represents dilatation contribution.

The vibrational contribution is usually approximated by Debay function [ 2 ] n_fD(q / T) with empirically found Debay temperature q and number of degrees of freedom n_f.

The Debay temperature q is about 100 K and less for the most of organic substances [ 3-8 ]. It causes the first term in the formula (1) to approach a constant value at room temperature, therefore temperature dependence is connected with the second term, for which general theoretical expression is well known:

where a - the volume thermal expansion coefficient and b - isothermal compressibility. In absence of experimental data concerning these properties the considered contribution are calculated using several approximate expressions. The Nernst-Lindeman expression [ 2 ] is most frequently used for inorganic substances:

where T_m is the melting temperature.

The application of the expression to a number of organic substances to be appeared unsuccessful because of systematic deviations of experimental data from the values calculated by formula (3). It is possible, that the reason of such deviations is the additional anharmonic contribution, not taken into account in the initial formula (1). It is known, that this contribution also linearly depends on the temperature and has frequently negative value. For the best representation of experimental data equations (3) was a little changed and presented in the form:

where k - empirical constant, R - gas constant, and n - number of atoms in a molecule of organic substance.

Substituting (4) in (1) and taking into account practical constant value of the vibrational contribution at room and higher temperatures, we obtain:

If the heat capacity value at room temperature is known, expression (5) can be rewritten in the form:

Below we consider results of the empirical analysis of applicability (6) to the representation of heat capacity temperature dependence in the temperature range from T=298.15 K up to the melting point.

On Fig. 1 experimental data on the heat capacity of a number of organic substances (biphenyl ether[3], biphenyl [4], dibenzofuran [5], dibenzothiophene [6], 1,4-dichlorobenzen [7], thianthrene and phenoxathiin [8]) in a considered interval of temperatures in reduced coordinates are presented. One can see the good correlation( ² = 0.997) and the empirical factor = 1.49 0.04 is obtained from the graph. The standard deviation (s) of experimental data on the heat capacity is 0.5% and the maximum deviation (_max) is 1.2%. The tests of the offered formula on examples of carbamide[9] (s=1.0%, d_max=1.7%), 4,5,9,10 tetrahydropyrene (s=1.7%, d_max=2.8%) and 1,2,3,6,7,8 - hexahydropyrene [ 10 ] (s=1.7%, d_max=2.7%) are shown on Fig 2, Fig.3 and Fig.4. In all cases the good representation of experimental data is observed. It should be specially noted cases of phase transitions which take place for 4,5,9,10 tetrahydropyrene and 1,2,3,6,7,8 - hexahydropyrene. One can see from Fig. 3 and Fig. 4, the formula (6) represents successfully average behavior of the heat capacity in the ranges of phase transitions.

The experimental value of C_p(298.15) has been taken for all cases considered above. If the value is calculated the uncertainties of estimated heat capacities by formula (6) increase. If take into account the uncertainties of the room temperature heat capacity given in [1], the whole possible standard deviation can be estimated 3-5% and maximum deviation - 7-10%. It should be specially noted that the uncertainties estimation given in the present paper does not take into account of structural phase transitions where the uncertainties can be substantially more.

The offered empirical expression for temperature dependence of the heat capacity for solid organic substances has appeared rather successful for representation experimental data as for those substances, for which experimental data were used, as for those, which served for check of the formula. The use of data for greater number of substances, probably, will change obtained empirical factor, but whether will deliver under doubt general conclusion about applicability of offered expression. An interesting question, requiring further research, is applicability of obtained expression for temperatures below the considered range.

The present work was supported by Russian Foundation for Basic Researches (Grants N 96-02-16223 and N 96-07-89026). The author thanks Dr. G.A.Bergman for fruitful discussion during the work.

1. Domalski E.S., Hearing E.D. // J.Phys. Chem. Ref. Data. 1993. V.22. P.805.

2. Zhdanov G.S., Cristal Physics, Oliver and Boyd, Edinburgh, 1965.

3. Furukawa G.T., Ginnings D.C., McCoskey R.E., Nelson R.A. // J. Res. NBS, 1951, V.46, P.195.

4. Chirico R.D., Knipmeyer S.E., Nguyen A., Steele W.V. // J. Chem. Thermodynamics, 1989, V. 21, P.1307.

5. Chirico R.D., Gammon B.E., Knipmeyer S.E., Nguyen A., Strube M.M., Tsonoloulos C., Steele W.V. // J. Chem. Thermodynamics, 1990, V. 22, P.1075.

6. Chirico R.D., Knipmeyer S.E., Nguyen A., Steele W.V. // J. Chem. Thermodynamics, 1991, V. 23, P.431.

7. Dworkin A., Figuiere P., Ghelfenstein M., Szwarc V. // J. Chem. Thermodynamics, 1976, V. 8, P. 835.

8. Steele W.V., Chirico R.D., Knipmeyer S.E., and Nguyen A. // J. Chem. Thermodynamics, 1993, V. 25, P. 965.

9. Kozyro A.A., Dalidovich S.V., Krasulin A.P. // Zhurn. Prikl. Khimii, 1986. Vol.59, No 7, P. 1456.

10. Chirico R.D., Knipmeyer S.E., Nguyen A., Smith N.K., and Steele W.V. // J. Chem. Thermodynamics, 1993, V. 25, P.729.

Fig. 1.

Fig. 2.

Fig. 3.

Fig. 4.

Fig. 1. Experimental data correlation on heat capacity of solid organic substances in the temperature range from 298.15K up to the melting point.

Fig. 2. Comparison of experimental [9] and calculated data on heat capacity of carbamide.

Fig. 3. Comparison of experimental [10] and calculated data on heat capacity of 4,5,9,10 - tetrahydropyrene

Fig.4. Comparison of experimental [10] and calculated data on heat capacity of 1,2,3,6,7,8- hexahydropyrene