Reports of the Academy of Sciences of the USSR

1. Volume 162, No. 3

PHYSICS

I. E. Paukov, Corresponding Member of the Academy of Sciences of the USSR P. G. Strelkov,
V. V. Nogteva, V. I. Belyi

HEAT CAPACITY OF BLACK PHOSPHORUS

AT LOW TEMPERATURES

Black phosphorus was first obtained by Bridgman \((^{1})\) from the white modification of this element at a pressure of \(\sim 12\,000\) atm and a temperature of \(\sim 200^\circ\)C. Subsequently, other investigators \((^{2,3})\) established that there are two modifications of black phosphorus—crystalline and amorphous—and also determined the conditions for obtaining them. However, the physical properties of both modifications of black phosphorus have so far been studied incompletely.

The purpose of the present work was to investigate the true heat capacity of the crystalline modification of black phosphorus, and also to calculate the values of the absolute entropy and enthalpy under standard conditions. This modification was chosen as the object of study because it is apparently the most stable modification of phosphorus \((^{4})\). In addition, it is the only form of phosphorus for which the lattice parameters have been determined precisely \((^{5})\). The crystal consists of corrugated layers of phosphorus atoms, each layer consisting of two sublayers located at a distance of 2.28 Å from one another. Neighboring layers, situated at considerably greater distances, are bound to one another less strongly, which explains the layered character of crystalline black phosphorus, similar to the layered character of graphite.

The sample whose heat capacity was studied was obtained with the aid of a high-pressure bomb capable of operating up to pressures of 13,000–14,000 kg/cm² and temperatures of \(\sim 300^\circ\)C. To obtain the necessary pressure, the MOP-10000 apparatus was used.

Twice-distilled white phosphorus was used as the starting material for preparing the sample; chemical analysis of it showed the presence of impurities in an amount of less than 0.03%. The white phosphorus, in a sealed lead ampoule, was placed in the bomb and held at a pressure of 13,500 kg/cm² and a temperature of \(220^\circ\) for 15–20 min. Under these conditions it was practically entirely converted into black phosphorus. The samples thus obtained were subjected to chemical and spectral analyses, according to which the total phosphorus content in the sample was 99.12%. The principal impurities, carbon \(\sim 0.3\%\) and lead \(\sim 0.3\%\), were introduced during preparation of the sample in the high-pressure bomb. In addition, small amounts, about 0.01%, of Ni, Sn, Mg, Fe, and Co were present.

A phase x-ray analysis of the sample was also carried out, using an RKD chamber. The exposure was made with Cu \(K_\alpha\) radiation, with the Cu \(K_\beta\) radiation filtered out. It was found that the sample consists of a single phase of rhombic syngony, which is confirmed by the good agreement of the \(d/n\) values obtained in the analysis with the literature data \((^{5})\). The presence of other phases was not detected by x-ray analysis. The diffuse ring characteristic of amorphous black phosphorus was likewise not found on the x-ray pattern.

The apparatus and the method for measuring the true heat capacity at low temperatures were basically similar to those described earlier \((^{6})\). We note only that the temperature measurements were carried out with the aid of a platinum-

of the resistance thermometer, made of “Pobeda” platinum and calibrated at the All-Union Scientific Research Institute of Physicotechnical and Radiotechnical Measurements. The resistance of the thermometer at \(273.15^\circ\text{K}\) was \(R_0 = 91.712\) ohm, \(\alpha = 0.003925\).

Table 1

Adiabatic conditions during the calorimetric experiment were maintained automatically with the aid of an apparatus described in detail in (7).

The heat capacity of the empty calorimeter (more precisely, of the calorimeter filled with dry pure helium to a pressure of \(\sim 25\) mm Hg) was measured in the temperature interval \(12.9\)–\(297.5^\circ\text{K}\). Ninety-four calorimetric experiments were carried out, and from them a graphically smoothed curve \(c - T(^{\circ}\text{K})\) was constructed. The deviations of individual experimental points from the smoothed curve did not, as a rule, exceed \(0.2\%\) in the temperature interval \(45\)–\(297.5^\circ\text{K}\), but gradually increased below \(45^\circ\text{K}\) and became equal to \(\sim 1.5\%\) in the region \(13\)–\(20^\circ\text{K}\).

A total of 32.713 g of substance was placed in the calorimetric vessel, which amounted to 1.056 g-at. The linear dimensions of the individual crystals of the sample were several millimeters; in this form was almost the entire sample. A small part (\(\sim 5\%\) of the total amount) consisted of pieces of the sample less than 1 mm in size and of dust.

The heat capacity was measured in the temperature interval from \(13.0\) to \(293.8^\circ\text{K}\), and 118 calorimetric experiments were carried out. The measurement results, obtained from the graphically smoothed curve \(c_p - T\), are given in Table 1.* In Fig. 1, experimental values of \(c_p\) are given in part. The deviations of the experimental values of the measured heat capacities from the smoothed curve \(c_p - T(^{\circ}\text{K})\) averaged less than \(0.1\%\) in the interval \(40\)–\(293.8^\circ\text{K}\) and, gradually increasing, reached \(\sim 1.7\%\) at the lowest temperatures.

The calculation of the values of the absolute entropy at \(298.15^\circ\text{K}\) by numerical integration of the curve \(c_p - \ln T\) led to the value

\[ S_{298.15}^0 = 5.457 \pm 0.010 \text{ entropy units.} \]

The value of the enthalpy difference at \(298.15\) and \(0^\circ\text{K}\) was likewise calculated by numerical integration of the curve \(c_p - T\):

\[ H_{298.15}^0 - H_0^0 = 882.6 \pm 1.5 \text{ cal/g-at.} \]

Fig. 1. Temperature dependence of the heat capacity of black phosphorus in logarithmic coordinates

* In the calculations of \(c_p\), \(S_T^0\), and \(H_T^0 - H_0^0\), it was assumed that \(1\) cal \(= 4.1840\) J.

For the calculation of the quantities \(S^0_{298.15}\) and \(H^0_{298.15} - H^0_0\), the smoothed curve of the dependence of heat capacity on temperature was graphically extrapolated to \(0^\circ\) K. In our estimate, the error in the quantities \(S^0_{298.15}\) and \(H^0_{298.15} - H^0_0\) due to extrapolation should not exceed \(\sim 0.003\) entropy units and \(\sim 0.02\) cal/g-atom, respectively.

No anomalies were found in the behavior of the \(c_p - T\) curve. In Fig. 1 the temperature dependence of the heat capacity is presented in the coordinates \(\ln c_p - \ln T\). The straight line drawn through the experimental points in the temperature interval \(20\)--\(37^\circ\) K corresponds to the heat capacity in this region being proportional to temperature to the power 2.3. However, upon more careful examination of the curve, it is seen that the exponent changes gradually and, below \(20^\circ\) K, in the interval \(13\)--\(20^\circ\) K, becomes equal to 2.7. At lower temperatures the heat capacity will apparently be proportional to the third power of temperature, i.e., for black phosphorus Debye’s “cube” law will become valid. In any case, measurement of the heat capacity of black phosphorus at lower temperatures is highly desirable.

It should be noted that there have as yet been no literature data on measurements of the heat capacity of black phosphorus.

Institute of Thermophysics
Siberian Branch of the Academy of Sciences of the USSR

Received
17 II 1965

REFERENCES

1. P. W. Bridgman, J. Am. Chem. Soc., 34, No. 7, 1344 (1914).
2. R. B. Jocobs, J. Chem. Phys., 5, 945 (1937).
3. K. Patz, Zs. anorg. u. allgem. Chem., 285, 29 (1956).
4. Van Vezer, Phosphorus and Its Compounds, IL, 1962, p. 98.
5. R. Hulgren, N. S. Gingrich, B. E. Warren, J. Chem. Phys., 3, 351 (1935).
6. P. G. Strelkov, E. S. Idkevich et al., ZhFKh, 28, No. 3, 459 (1954).
7. Ya. A. Kraftmakher, P. G. Strelkov, Journal of Applied Mechanics and Technical Physics, No. 3, 194 (1960).