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# Simultaneous Determination of Several Thermodynamic Quantities Essay

I. INTRODUCTION
The solubility product constant, Ksp​, is the equilibrium constant for a solid substance dissolving in an aqueous solution. It represents the level at which a solute dissolves in solution. A more a substance dissolves, the higher the Ksp value it has.

In this experiment, a system of a sparingly soluble salt in water is studied. From the solubility information at various temperatures, the changes in standard enthalpy, standard entropy, and standard free energy were established. II. THEORETICAL BACKGROUND

The reaction that is studied in this experiment is the dissolution of borax in water. “Borax” is a naturally occurring compound; it is in fact the most important source of the element boron, and it has been used for many years as a water softening agent. Borax is a rather complicated ionic salt which has the chemical formula Na2B4O7•10H2O (Petrucci, 2007). When it dissolves, it dissociates as follows:

Na2B4O7 • 10H2O(s) 2Na(aq) + B4O5(OH)42(aq) + 8H2O(l) rnx (1)

The solubility product expression for this system is written below. Ksp = [Na]2 [B4O5(OH)42] (2) To determine a value for the solubility product, a method must be found to assay either the amount of sodium ion, or borate ion, in the sample mixture. The original equilibrium expression, and balanced solubility equilibrium reaction, leads to a convenient way to express either ion in terms of the other, so that it is possible to define Ksp in terms of either the concentration of sodium ion, or concentration of borate ion (Chang, 2010). A series of substitutions, based on the original balanced solubility equilibrium equation, gives the desired Ksp, expression defined in terms of the borate ion only: [Na] = 2 [B4O5(OH)42]

K = [ (2 [B4O5(OH)42] ) ]2 [B4O5(OH)42]
Ksp = 4 [B4O5(OH)42]3 (3) Finding the concentration of borate ion, in any sample at any given temperature, leads directly to a value for solubility product (Ksp) at that temperature. The concentration of tetraborate (B4O5(OH)42) can be determined through titration. Tetraborate is a weak base, so it can be titrated with a strong acid (Brown, 2012). The reaction for the titration is written as follows:

B4O5(OH)42– (aq) + 2 H3O+ (aq) + H2O (l)  4 H3BO3 (aq) (4)
The volume of the acid used can then be used to calculate the concentration of tetraborate using the formula written in the next page.

Where Mtetraborate = concentration of tetraborate
(M x V)acid = the product of the concentration and volume of the acid used Vtetraborate = volume of borax used
From the Ksp value, the Gibb’s energy can be calculated The relationship between the Ksp and Gibb’s energy is:
∆G° = -RT lnKsp (6)
where R is the gas constant (8.314 J/K.mol) and T is the absolute temperature (Brown, 2012). From the definition of Gibb’s energy the following can be written:
∆G° = ∆H° – T∆S° (7)
(8)
Clearly, ∆G° is a function of temperature. A plot of ∆G° vs T should yield a straight line with a slope of -∆S° and intercept of ∆H°. Likewise, a plot of ln K vs 1/T should also generate a straight line with a slope of -∆H°/R and intercept of ∆S°/R (Chaka & Madhugiri, n.d.). III. METHODOLOGY

A. Materials/Apparatuses/Chemicals
The materials/apparatuses/chemicals that were used in this experiment are as follows: analytical balance, volumetric flask, stirring rod, thermometer, beaker, hot plate, iron ring, iron stand, test tube, solid Na2B4O7•10H2O and 6M standardized HCl. B. Procedure

B.1. Preparation of Saturated Solution of Borax
About 30 g of borax reagent was weighed and it was dissolved with 150 mL of distilled water in a 250 mL beaker. The mixture was heated but the temperature was controlled so that it will not exceed 50°C. One the mixture has exceeded, however slightly, the 45°C mark, the beaker was removed to the bench top and it was replaced with a beaker containing 150 mL of distilled water. This water was also not allowed to exceed 50°C. Using a 5 mL pipet, exactly 5 mL of distilled water was added to each of ten test tubes. The level was marked with a masking tape and the water was poured out. The test tubes were marked to their assigned temperatures (45°C, 40°C, 35°C, 30°C and 27°C). B.2. Taking Samples of the Borax Solution

The borax/water mixture was stirred continuously until it has cooled to the assigned temperature. Then as quickly as possible, 5 mL of the solution was carefully poured out into the test tubes marked with the 45°C temperature. The temperature before and after the transfer was noted and the average was used if the readings differed by more than 1°C. B.3. Titration of the Borax Samples

When the water bath have reached at least 45°C, the samples was placed into it until any re-precipitated borax has dissolved. The dissolved borax solution was carefully poured into a 125 mL Erlenmeyer flask containing 50-mL of distilled water and 10 drops of bromocresol green indicator. The solution was then titrated with a standardized HCl until the yellow endpoint. (Note: The acid was standardized by the other group)

Procedure B.2 and B.3 were repeated for temperatures 40°C, 35°C, 30°C and 27°C.
A plot of ln K vs 1/T and T vs ∆G were then plotted. From the two plots, the ∆H and ∆S of the reaction were determined.

A saturated solution is the point at which the solution of a substance can dissolve no more of that substance and additional amounts of it will appear as a separate phase. In this experiment, saturated solution of borax was used to ensure that in each temperature, a maximum amount of borax can be dissolved. The dissolution of borax is an endothermic process. That is, as the temperature is increased, the dissolution of borax is favored. This is shown in Table 1. As the temperature increases, the concentration of tetraborate also increases. Moreover, as shown in Table 2, same trend is seen in Ksp values. As the temperature is raised, the formation of the products is favored thus Ksp is increased. While in cooling, the equilibrium shifts to the left which means the formation of the reactants is favored thus lowering the value of Ksp.

This is best presented by Figure 1. On the other hand, opposite trend is observed in the Gibb’s Energy values. The ∆G values is decreasing when the temperature is increasing. This just further proves that at higher temperatures, reaction (1) is getting spontaneous. In Figure 2, it is evidently shown that as the temperature increases, ∆G decreases. The thermodynamic quantities were calculated using the equations generated in each plot. The calculated values are shown in Table 3. It can be observed that the determined values from the two plots slightly differ from each other. This can be explained by knowing the fact that in the plot ln K vs 1/T, the change in entropy and enthalpy were calculated by multiplying the y-intercept and slope with the gas constant respectively.

The value of gas constant used was 8.314 J/K.mol. However, this value is already rounded off from its exact value. So during the calculations, an error called “round-off error” was committed. Round-off error is the difference between the calculated approximation of a number and its exact mathematical value. (Weisstein, 2014) The established enthalpy of reaction (1) is 83.45 kJ/mol as shown in Table 4. This proves that the reaction is endothermic, that is, heat is needed in order to dissolve the borax in the solution. Moreover, the calculated entropy of reaction (1) is 259.15 J/K.mol which means that the entropy of the system is positive further proving that the reaction is spontaneous. However, if these calculated values are to be compared with theoretical ones, it can be observed that the experimental values are lower than the theoretical values.

The computed percentage error for entropy is 31.80% while 24.14% for enthalpy. The first error that was committed during the experiment is the estimation of the volume of borax taken. Since the volume of borax is used in calculating the concentration of tetraborate (equation 5), then any changes in the volume (higher or lower) will lead to a different result in the concentration of tetraborate. The second error committed was in the titration process. The equilibrated borax samples was placed in an Erlenmeyer flask containing a distilled water. However, the distilled water used was not equilibrated to the same temperature where the borax samples was equilibrated.

As a result, for example, the borax sample is equilibrated at 45°C then during titration, its temperature will be lowered since the water added to it was in lower temperature. The last error is human error. These errors are the ones that are hard to control and difficult to detect. It can be justified by knowing that laboratory does not operate under ideal conditions. There is always a possibility of inaccuracies with measurement, perception of measurement, inaccuracies of equipment and the apparatus used throughout the experiment. VI. CONCLUSIONS AND RECOMMENDATIONS

The goal of this experiment was to calculate the thermodynamic quantities using the solubility product constants calculated at varying temperatures by plotting ln K vs 1/T and ∆G vs T. The calculated enthalpy and entropy in this experiment are 83.45 kJ/mol and 259.15 J/mol respectively. The established values differ from the theoretical ones due to error committed during the conduct of the experiment and due to the fact that the volume of borax used was only estimated.

One way to improve the result would be to perform more trials to improve the accuracy and help get rid of some of random error. Another way is directly pipet the borax sample and transfer it to the Erlenmeyer flask in order to lessen the error committed. Also, the water that will be used in the titration should be equilibrated to the same temperature where the sample was equilibrated.

VII. REFERENCES
Brown, T. et.al. (2012). Chemistry: The Central Science. 12th ed. United States of America: Pearson Education, Inc. Chaka, G. & Madhugiri S. (n.d.) Determination of Thermodynamic Quantities for a Chemical Reaction Chang, R. (2010). Chemistry. 10th ed. 1221 Avenue of the Americans, New York: McGraw- Hill Companies, Inc. Petrucci, Ralph H., et al. General Chemistry: Principles and Modern Applications. Upper Saddle River, NJ: Prentice Hall 2007. Weisstein, E. W. (2014). Roundoff Error. Retrieved July 22, 2014 from the world wide web: http://mathworld.wolfram.com/RoundoffError.html Experiment 17. Thermodynamics of Borax Solubility

Chemistry 212 Lab: Simultaneous Determination of Several Thermodynamic Quantities: K, ∆G°, ∆H°, and ∆S°

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