Solid-Liquid Equilibrium

Solid-Liquid Equilibrium of the Naphthalene (1) / Diphenylamine (2) System Using Thermal Analysis to Generate the Solid-Liquid Equilibrium Phase Diagram Abagon, Ma. Victoria1, Buendia, Neil Daniel2, Caracas, Corine Jasper3 Department of Chemical Engineering, College of Engineering University of the Philippines - Diliman Quezon City, Philippines 1 [email protected] | [email protected] | [email protected]

Abstract—Using thermal analysis, cooling curves for eight different compositions of the naphthalene (1) / diphenylamine (2) system were generated. By plotting the temperature at thermal arrest versus concentration, the solid-liquid equilibrium phase diagram for the system was constructed and shown in Fig. 1. The first three points of the plot represent the diphenylamine-rich mixtures, and the last six points constitute the naphthalene-rich mixtures. Following thermal analysis, two separate second-degree curves were fit to the respective data points. The eutectic point is determined to be the intersection of the two curves, with composition 𝒙𝒆 = 𝟎. 𝟑𝟎𝟔𝟎 at 300.15 K. The eutectic composition deviated 14.99% from the literature value while the eutectic temperature differed by 1.71%. The deviations may have been incurred from material loss and sample contamination.

𝑥𝑖 𝛾𝑖𝑙 𝑓𝑖𝑙 = 𝑧𝑖 𝛾𝑖𝑠 𝑓𝑖𝑠 Defining the ratio, 𝑓𝑖𝑠 /𝑓𝑖𝑙 as 𝜓𝑖 : 𝑥𝑖 𝛾𝑖𝑙 = 𝑧𝑖 𝛾𝑖𝑠 𝜓𝑖

The ratio of fugacities at the temperature and pressure of the system may be written in expanded form: 𝑓𝑖𝑠 (𝑇, 𝑃) 𝑓𝑖𝑠 (𝑇, 𝑃) 𝑓𝑖𝑠 (𝑇𝑚𝑖 , 𝑃) = ⋅ 𝑓𝑖𝑙 (𝑇, 𝑃) 𝑓𝑖𝑠 (𝑇𝑚𝑖 , 𝑃) 𝑓𝑖𝑙 (𝑇𝑚𝑖 , 𝑃) 𝑓𝑖𝑙 (𝑇𝑚 , 𝑃) ⋅ 𝑙 𝑖 𝑓𝑖 (𝑇, 𝑃)

INTRODUCTION

Most substances found in the market are made up of multiple compounds and rarely of a pure chemical. In manufacturing these substances, it is important to know their physical and chemical properties especially in ensuring that the desired state of the final product is achieved. [1] Taking crystallization for example, it is important to know the required temperature to crystallize a solid of a certain composition. However, physical and chemical property data like melting points are usually only available for pure species and when they interact with other pure species in mixtures, they may behave differently. The study of solid-liquid equilibria (SLE) for a multiplecomponent system is useful in determining different thermodynamic properties exhibited by various mixtures. The simplest way to represent SLE would be a binary solid-liquid phase diagram which gives information about the phase, temperature and composition of the binary mixture. [2] Theoretically, SLE is represented by the uniformity of the temperature, pressure and fugacity for each chemical species throughout both phases. Mathematically it is shown as: 𝑓𝑖𝑙 = 𝑓𝑖𝑠

(1)

Expressing the previous equation in terms of the activity coefficient gives:

(3)

where 𝑥𝑖 is the composition in the liquid phase, 𝑧𝑖 is the composition in the solid phase, and 𝛾𝑖 is the activity coefficient for the liquid and solid phase respectively.

Keywords-eutectic temperature, eutectic composition, solidliquid equilibrium, thermal analysis, naphthalene, diphenylamine

I.

(2)

(4)

where 𝑇𝑚𝑖 is the melting point of pure species 𝑖. The second term is equal to unity since at the melting point the fugacity of the species is equal for both phases. The equation could then be simplified to: 𝜓𝑖 =

𝑓𝑖𝑠 (𝑇, 𝑃) 𝑓𝑖𝑠 (𝑇𝑚𝑖 , 𝑃)

⋅

𝑓𝑖𝑙 (𝑇𝑚𝑖 , 𝑃) 𝑓𝑖𝑙 (𝑇, 𝑃)

(5)

This equation may be evaluated leading to an expression where 𝜓𝑖 is a function of temperature: 𝜓𝑖 = exp

Δ𝐻𝑖𝑠𝑙 𝑇 − 𝑇𝑚𝑖 ( ) 𝑅𝑇𝑚𝑖 𝑇

(6)

Knowing 𝜓𝑖 , (3) may be used once the dependence of the activity coefficient, 𝛾𝑖 on temperature and composition is defined. Two limiting cases may be considered: 1.

An ideal solution behavior is assumed for both phases (both activity coefficients are equal to 1 for all temperatures and compositions).

2.

An ideal solution is assumed for the liquid phase (𝛾𝑖𝑙 = 1) along with a complete immisciblity of both species in the solid phase (𝑧𝑖 𝛾𝑖𝑠 = 1). [3]

Experimentally a solid-liquid phase diagram may be generated using thermal analysis. A mixture of two solids is heated until melting and a cooling curve is obtained for different mixture compositions. This curve is then used to locate significant temperature values. A change in slope, which is also a change in the cooling rate, represents supercooling. This temperature where the solid starts to separate from the liquid mixture is the break point. However, when the temperature of the mixture no longer changes, thermal arrest happens. This temperature and composition point is called the eutectic point. At this point, the presence of two solid phases and a liquid phase is indicated. [4][5] In this experiment, the solid-liquid phase diagram of diphenylamine and naphthalene is obtained using thermal analysis. The data obtained were analyzed and compared with the theoretical values. II.

MATERIALS AND METHODS

The solid samples used were prepared using an analytical balance and petri dish. The first mixture, which was composed of pure naphthalene, was transferred to a 6-inch test tube. To measure the temperature inside the test tube, a thermocouple was inserted through the rubber stopper used to seal the test tube. The test tube was placed in a boiling water bath to heat the sample until all of the solids have melted. The test tube was removed from the water bath and temperature was taken at five-second intervals until all the samples in the test tube solidified, or when temperature remained constant for three consecutive readings. The second mixture was prepared as specified in Table 1, and the process of heating the sample and obtaining temperature measurements were again administered. The succeeding mixtures were then prepared accordingly. However, mixtures 6, 7, and 8 were melted and recrystallized using a different 6-inch test tube. A cold water bath is used to recrystallize the diphenylamine-rich mixtures. III.

RESULTS AND DISCUSSION

Temperature data recorded from the experiment were used to obtain the melting point of pure naphthalene and diphenylamine. The following shows the analysis of the obtained melting point temperatures and the comparison to theoretical data as well as the calculations undertaken to generate the solid-liquid phase diagram of the system. A. Interpretation of Data Gathered The set of temperature readings per mixture was plotted against time, and the cooling curves produced could be viewed in the appendix. After plotting the data, the thermal break points and thermal arrest points were determined by calculating the slopes of the cooling curves. The decrease in the slope of the cooling curve

TABLE 1. NAPHTHALENE-DIPHENYLAMINE MIXTURE PREPARATION Mixture

Diphenylamine, g

Naphthalene, g

1

0

3.75

2

0.75

3.75

3

1.875

3.75

4

3.75

3.75

5

7.5

3.75

6

3.75

0

7

3.75

0.75

M6 + 0.75 naphthalene

8

3.75

1.2525

M7 + 1.2525 naphthalene

Mixture preparation

M1 + 0.75 g diphenylamine M2 + 1.875 g diphenylamine M3 + 3.75 g diphenylamine M4 + 7.5 g diphenylamine

corresponds to the thermal break point, wherein the appearance of the first solid could be observed. On the other hand, the point where the temperature value becomes constant is defined to be the thermal arrest point. This point, which is also the melting point of the mixture, was used to generate the solid-liquid phase diagram for the naphthalene (1) / diphenylamine (2) system. The curves produced are shown in Fig. 1. The first three points in the plot represent the diphenylamine-rich mixtures and a second-degree polynomial curve was fit into the three points. The same procedure was observed for the last six points in the plot, which constitute the naphthalene-rich mixtures. The intersection of the two curves is the experimentallydetermined eutectic composition 𝑥𝑒 and temperature 𝑇𝑒 of the naphthalene (1) – diphenylamine (2) system. Under these conditions, three phases could be observed. Hence, a horizontal line passing through 𝑥𝑒 could be drawn, since at this point solidphase pure naphthalene ( 𝑥 = 1 ), solid-phase pure diphenylamine ( 𝑥 = 0 ) and liquid-phase naphthalene – diphenylamine mixture (𝑥 = 𝑥𝑒 ) coexist. As shown in Fig. 1, the curves divided the phase diagram into four two-phase regions. Mixture compositions lying on the first region is expected to split into a pure naphthalene solid and a liquid with composition predicted by the curve fit. A similar behavior is exhibited by mixtures within region II, but the solid i1n equilibrium with the liquid is pure diphenylamine. Moreover, in region III lies a single liquid phase, while in region IV exists two distinguishable solid phases. B. Comparison to Theoretical Data Assuming that the system behaves as an ideal solution in the liquid phase and is completely immiscible in the solid phase, then 𝑥1 = 𝜓1 and 𝑥2 = 𝜓2 . Hence, using 𝜓 as defined in (6), the following equations also apply. 𝑥1 = exp

Δ𝐻1𝑠𝑙 𝑇 − 𝑇𝑚1 ( ) 𝑅𝑇𝑚1 𝑇

𝑥1 = 1 − exp

Δ𝐻2𝑠𝑙 𝑇 − 𝑇𝑚2 ( ) 𝑅𝑇𝑚2 𝑇

(7)

(8)

Temperature vs. x(1) 350 y = -69.377x2 + 148.1x + 263.22 R² = 0.9838

Temperature, Kelvins

340 330

y = -343.52x2 + 22.454x + 325.45 R² = 1

320 310 300 290 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x(1) Figure 1. SLE curve for the naphthalene (1) - diphenylamine (2) system at 101.325 kPa (constructed from experimental data)

The eutectic temperature and composition could be calculated by simultaneously solving (7) and (8) while using the theoretical values for diphenylamine and naphthalene shown in Table 2. Shown below is the resulting equation from combining (7) and (8). 1 − exp

Δ𝐻2 𝑅𝑇𝑚,2

(

𝑇−𝑇𝑚,2 𝑇

) = exp

Δ𝐻1 𝑅𝑇𝑚,1

(

𝑇−𝑇𝑚,1 𝑇

)

(9)

Using (9), the eutectic temperature and composition was determined, since the said equation is could only be satisfied by the conditions at the eutectic point. The deviation of the theoretical values obtained were compared to the experimental value determined from the intersection of the curves shown in Fig. 1. A summary of calculated values is displayed in Table 3. C. Sources of Deviation The major sources of errors in this experiment are material loss and sample contamination. The analytical balance used to weigh the samples may incur deviations due to poor maintenance of the instrument. The mass of the irrelevant particles present in the platform contributed to the recorded TABLE 2. PURE-SPECIES LITERATURE DATA Δ𝐻 𝑠𝑙 , kJ/kg 146.79 105.63

naphthalene diphenylamine

𝑇𝑚 , °C 80.2 52.9

Data obtained from National Institute of Standards and Technology

TABLE 3. SUMMARY OF OBTAINED VALUES 𝑥𝑒 𝑇𝑒 , °C

Experimental 0.306026561 27

Theoretical 0.360001299 32.21532052

deviation 14.99293 % 1.707896 %

mass, hence the actual amount of sample being processed is less than recorded. This leads to a lower crystallization temperature reading, as well as lower arrest and break temperature measurements. During the transfer of solids from the petri dish to the test tube, material loss might have been incurred. A watch glass should have been used instead of the plastic petri dish, as suggested by [6]. The lack of glass instruments led to using the petri dish to which the solids has higher affinity to. This caused difficulty in transferring the sample from the dish to the test tube which in turn led to material loss. IV.

CONCLUSIONS AND RECOMMENDATIONS

The data obtained by utilizing the thermal analysis method has been used to plot a phase diagram for the naphthalene – diphenylamine system, as shown in Fig. 1. From the phase diagram constructed, the behavior of the system at various combinations of mixture composition and temperature could be predicted. The eutectic point of the system at atmospheric pressure was also determined using the data gathered. The experimental eutectic composition and temperature are 𝑥𝑒 = 0.3060 and 27 C respectively. This is the intersection of the two curves constructed from the locus of diphenylamine-rich mixture points and the locus of naphthalene-rich mixture points. These values obtained from experimentation were compared to literature values, and deviations were recorded in Table 3. The errors could be attributed to accumulated material loss mainly from the transfer of samples from the balance, to the petri dish, and to the test tube. Also, sample contamination might have occurred which would have affected the temperature readings. It is recommended that for the future experiments utilizing the thermal analysis method, more mixtures of different compositions should be prepared so that there would be more

data points involved in constructing the SLE curve. Also if possible, at least two trials per mixture should be conducted so that the data that would be used for calculations and plot construction would be more accurate. Moreover, appropriate glassware and equipment should be used in performing the experiment to decrease the possibility of material loss.

REFERENCES [1] [2]

[3]

[4] [5] [6]

“Binary Solid-Liquid Phase Diagram” in CHEM 366, vol I, pp 1-8. J. Gallus, Q. Lin, S. Freiss, R. Hartmann, and E. Meister, “Binary SolidLiquid Phase Diagrams of Selected Organic Compounds,” Journal of Chemical Education, vol. 78, no. 7, July 2001. J.M. Smith, H.C. Van ness and M.M. Abbott, “Introduction to Chemical Engineering Thermodynamics,” McGraw Hill, New York, pp. 220-222, 2005. “Solid Liquid Equilibrium,” http://www.scranton.edu/faculty/baumann/courses/labs/solliq.pdf Department of Chemistry and Biochemistry, “Binary Solid-Liquid Phase Diagram,” in Chemistry 4581, pp 1-7. “ChE 124 Chemical Engineering Thermodynamics Laboratory Manual,” Quezon City, 2013, pp. 39-42.

APPENDIX Appendix A. Pure-Species Properties Diphenylamine

Naphthalene

Density (g/cm3)

1.2

1.16

Melting Point (deg C)

53

80.2

169.22244

128.17052

Molecular Weight (g/mol)

Appendix B. Relevant Equations used 𝑥1 =

𝑔𝑛𝑎𝑝ℎ𝑡ℎ𝑎𝑙𝑒𝑛𝑒 × 𝑀𝑊𝑛𝑎𝑝ℎ𝑡ℎ𝑎𝑙𝑒𝑛𝑒 𝑔𝑛𝑎𝑝ℎ𝑡ℎ𝑎𝑙𝑒𝑛𝑒 × 𝑀𝑊𝑛𝑎𝑝ℎ𝑡ℎ𝑎𝑙𝑒𝑛𝑒 + 𝑔𝑑𝑖𝑝ℎ𝑒𝑛𝑦𝑙𝑎𝑚𝑖𝑛𝑒 × 𝑀𝑊𝑑𝑖𝑝ℎ𝑒𝑛𝑦𝑙𝑎𝑚𝑖𝑛𝑒 % 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 =

𝑎𝑐𝑡𝑢𝑎𝑙 𝑣𝑎𝑙𝑢𝑒 − 𝑡ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 𝑣𝑎𝑙𝑢𝑒 × 100% 𝑡ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 𝑣𝑎𝑙𝑢𝑒 Appendix C. Cooling Curves

Mixture 1

Temperature (deg C)

120.0 100.0 80.0 60.0 40.0 20.0

0.0 0

50

100

150 Time (s)

200

250

300

Mixture 2

Temperature (deg C)

120.0 100.0 80.0 60.0 40.0 20.0 0.0 0

50

100

150

200

250

Time (s)

Mixture 3 Temperature (deg C)

120.0 100.0 80.0 60.0

40.0 20.0 0.0 0

100

200

300 400 Time (s)

500

600

700

Mixture 4 Temperature (deg C)

120.0

100.0 80.0 60.0 40.0 20.0 0.0 0

100

200

300 Time (s)

400

500

Mixture 5 Temperature (deg C)

120.0 100.0 80.0 60.0 40.0

20.0 0.0 0

50

100

150

200 Time (s)

250

300

350

400

250

300

350

400

250

300

350

400

Mixture 6 Temperature (deg C)

100.0 80.0 60.0 40.0 20.0 0.0 0

50

100

150

200 Time (s)

Mixture 7 Temperature (deg C)

100.0 80.0 60.0 40.0 20.0 0.0 0

50

100

150

200 Time (s)

Mixture 8 Temperature (deg C)

100.0 80.0 60.0 40.0 20.0 0.0 0

50

100

150

200 Time (s)

250

300

350

400