A&A 416, 187-190 (2004)
DOI: 10.1051/0004-6361:20031746

The heat capacity of water ice in interstellar or interplanetary conditions

L. M. Shulman

Isaac Newton Institute Kiev Branch, Main Astronomical Observatory of the National Academy of Sciences of Ukraine (MAO NASU), Golosiiv, 03680, Kyiv-22, Ukraine

Received 12 November 2002 / Accepted 24 November 2003

Abstract
Three empirical equations are proposed to describe the existing experimental data on the heat capacity of water ice for $0~{\rm K} \leq T \leq 273~{\rm K}$. This may be important for the rate of non-stationary thermal processes at very low temperatures in cometary nuclei or cosmogonical scenarios.

Key words: ISM: molecules - comets: general - stars: formation

The heat capacity of water ice is often used in cometary physics and in cosmogonical calculations. Water vapour is an important component of protostellar clouds. Its condensation into icy grains is a common feature of any cosmogonical scenario. Theoreticians usually take the heat capacity from glaciology. I used in my calculations (Shulman 1987) the relationship

$$C_{\rm p}=7.8\times 10^{-3} T\;\;{\rm J~ g^{-1}}~ {\rm K}^{-1}$$ (1)

which is given in the handbook by Bogorodsky & Gavrilo (1980), while other authors, e.g. Klinger (1981) and Prialnik (1992) used Klinger's formula

$$C_{\rm p}=7.49\times 10^{-3} T+0.09\;\;{\rm J~ g^{-1}~ K^{-1}}$$ (2)

which is based on the experimental data of Giauque & Stout (1936).

Equation (2) is in obvious contradiction with the Nernst heat theorem, but both (1) and (2) are wrong at low temperatures. Really, no one of them gives the low temperature heat capacity of ice which is described by the Debye-Sommerfeld equation

$$C_{\rm p}=\frac{12\pi^4k}{5m\theta_D^3}\cdot T^3$$ (3)

where k is Boltzmann's constant and m is the mass of a water molecule.

Giauque & Stout (1936) measured the heat capacity in the interval $16.43~{\rm K}\le T\le 267.77~ {\rm K}$. Measurements in the interval $2.144~{\rm K}\le T \le 27.034~ {\rm K}$ were carried out by Flubacher et al. (1960). However both (1) and (2) are valid only for $T>100~{\rm K}$ as shown by Fig. 1.

	Figure 1: The discrepancy between the experimental data (solid curve) and the empirical Eq. (1) - dash dotted curve and Eq. (2) - dotted curve.
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So we need a better approximation for the heat capacity of ice at low temperatures.

Table 1: The experimental data by Flubacher et al. versus Eqs. (4), (5).

To get such an approximation, the original data measured by Giauque & Stout (1936) was combined with those of Flubacher et al. (1960) and with the theoretical values calculated for $T\le2.5~{\rm K}$ by Eq. (3). The Debye temperature $\theta_{\rm D}=152~{\rm K}$given by Weast (1975) is not compatible with the Flubacher et al. data. The value $\theta_{\rm D}=222.2~ {\rm K}$ was taken for the vicinity of absolute zero. Then the whole set was processed to obtain the best accuracy with the minimal number of parameters.

A simple polynomial approximation is not applicable. A very good approximation is given by the formula

 

$$C_{\rm p}=7.73\times 10^{-3}T\left(1-{\rm e}^{-1.263\times 10^{-3... ...times 10^{-3}T^6+ 2.0825\times 10^{-7} T^4{\rm e}^{-4.97\times 10^{-2}T}\bigg).$$ (4)

The following expression with the same number of constants is simpler but gives worse accuracy:

 

$$C_{\rm p}=7.73\times 10^{-3} T\left(1-{\rm e}^{-1.263\times 10^{-3}T^2}\right)$$

$$\quad\quad -7.59\times t10^{-3}+2.509\times 10^{-3}T-1.472\times 10^{-5}T^2$$

$$\quad\quad-1.617\times 10^{-9}T^3+8.406\times 10^{-11}T^4.$$ (5)

The polynomial approximation could be improved by increasing the number of constants, e.g.:

 

$$C_{\rm p}=7.73\times 10^{-3}\left(1-{\rm e}^{-1.263\times 10^{-3}T^2}\right)$$

$$\quad\quad-6.22\times 10^{-3}+2.181\times 10^{-3}T-3.73\times 10^{-6}T^2$$

$$\quad\quad-1.286\times 10^{-7}T^3+6.7\times 10^{-10}T^4-9.34\times 10^{-13}T^5.$$ (6)

The first term, which ensures the asymptotic behaviour of the heat capacity, should not be omitted. The expressions give the heat capacity in the same units as Eq. (1). Tables 1 and 2 demonstrate the quality of the approximation.

Table 2: The experimental data by Giauque and Stout versus Eqs. (4)-(6).

The difference experiment-approximation is shown in the Fig. 2. These deviations are very small (Fig. 3).

	Figure 2: The difference experiment-approximation for the relationships (4) - solid line, (5) - dash dotted, and (6) - dotted line.
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	Figure 3: The experimental data (circles) versus the approximation (4) - solid line.
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	Figure 4: The same as in Fig. 1 for enthalpy. The solid line shows the enthalpy calculated from experimental data (the approximation (4) gives the same curve) while the dash dotted and dotted lines are obtained using (1) and (2).
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To calculate the enthalpy of ice

$$H=\int_0^TC_{\rm p}(T){\rm d}T$$ (7)

with $30\%$-accuracy in the low temperature region the simple relationships (1) and (2) may be applied as shown by Fig. 4.

Taking into consideration this correction to the low temperature heat capacity of water ice one can see that the same energy heats ice to a higher temperature. So any process whose rate is described by an activation law $\tau\propto \exp(E_{\rm a}/kT)$ proceeds in shorter time. The crystallization of amorphous ice is an example of such a process.

Equations (4)-(6) describe the heat capacity of crystalline ice but there is no reason for a big difference between these quantities for amorphous and crystalline ice. The heat capacity is formed by energy distribution on the degrees of freedom. Since water ice is a molecular crystal where molecules retain their individuality, the difference between the heat capacities should be of the same order as the difference between their heats of sublimation. The same assumption for the transfer properties of both states of ice would be wrong because the transfer properties are defined by the free path of the corresponding carrier: phonons for the heat transfer, H₃O⁺ and OH^- ions for electric conductivity, etc. Any free path is much shorter in amorphous ice which can be considered as having an extremely high concentration of dislocations.

Some authors mistakenly write that there are species for which the Nernst theorem is not fulfilled. The heat capacity of some species (for example, lanthanides and their compounds) has one or more sharp peaks at the extremely low temperature where the contribution of the inner degrees of freedom (the energy of inner electron envelops, nuclear spins etc) becomes essential. Nevertheless the heat capacity should go to zero at $T\rightarrow 0$. In the vicinity of these peaks the Debye-Sommerfeld formula (3) is incorrect but it does not disprove the third principle of thermodynamics.

Acknowledgements

The author is grateful to Prof H. Rickman and my colleagues from the Solar System group of the Astronomical Observatory of the Uppsala University for discussion of this problem, and to D. Prialnik who attracted my attention to the paper by Flubacher et al. The work was supported by a grant from the Royal Swedish Academy of Sciences and, partially, by the State Foundation for Basic Research of the Ministry of Education and Science of Ukraine.

References

 
• Bogorodsky, V. V., & Gavrilo, V. P. 1980, Ice, Physical properties, Modern methods for glaciology, Leningrad: Hydrometizdat, 384 (in Russian) In the text  
• Flubacher, P., Leadbetter, A. J., & Morrison, J. A. 1960, J. Chem. Phys., 33, 1751 In the text  
• Giauque, W. F., & Stout, J. W. 1936, J. Amer. Chem. Soc., 58, 1144 In the text  
• Klinger, J. 1981, Icarus, 47, 320 In the text NASA ADS  
• Prialnik, D. 1992, ApJ, 388, 196 In the text NASA ADS  
• Shulman, L. M. 1987, Comet nuclei (Moscow: Nauka), 232 (in Russian)  
• Weast, R. C. 1975, Handbook of Chemistry and Physics, 55th ed. (Cleveland, Ohio: CRC Press), D146 In the text