Revision of Scientific Title Will Alter Meaning
Beer’s law quantifies the relationship between the concentration of a certain substance in solution, and the amount of light this substance absorbs at a given length traveled by the incident light. In its general form, this law equates the product of incident light path length, concentration of the substance, and the substance-dependent molar absorptivity constant to absorbance, which is a logarithmic measure of the change of intensity the incident light endures as it progresses through the solution (Sheffield Hallam University, n.d.):
A=lCε= -log(I/I_0 )
For solutions with an identical solute and path length, this equation implies a constant ratio of absorbance to concentration. Such a relationship is derived from the principle that a longer path for the incident light, or a more concentrated solution, will tend to contain more particles of light-absorbing solute, resulting in more strongly decreased light intensity. Because this absorption is associated with the excitation of electrons to higher energy levels, large concentrations of solutes may exert significant intermolecular forces that render the relationship between absorbance and concentration nonlinear, thus Beer’s law breaks down (Clark, 2007). Successful spectrophoto-metry therefore requires relatively dilute concentrations of the analyzed substances.
The van’t Hoff equation reveals the significance of the equilibrium constant, which spectrophotometric analysis can derive, to the thermodynamic properties of a reaction (Oxtoby, Gillis, and Butler, 2012):
Because the natural logarithm of the equilibrium constant of a reaction has a linear relationship to the inverse of the temperature of the reaction, one can deduce the standard enthalpy and entropy changes of the reaction from equilibrium constants at multiple temperatures without imprecise calorimetric calculations. In these experiments we apply Beer’s law and the van’t Hoff equation to derive these thermodynamic quantities for the dissolution of bromopentaamminecobalt(III) bromide.
In this experiment, my lab partners and I followed the procedure outlined in Section 13, pp. 155-161, of the General Chemistry Lab Manual by Dr. Meishan Zhao, with the minor deviation of addition of 1 mm water above the mark of the volumetric flask for the stock solution preparation.
Table 1. Spectrophotometric Data for Stock Solution and Dilutions
Solution Concentration (M) ri (Ci/C0) Absorbance (A) A/ri
0 2.74 x 10-3 1 0.168 0.168
1 9.15 x 10-4 0.333 0.066 0.198
2 1.37 x 10-3 0.500 0.079 0.158
3 1.83 x 10-3 0.667 0.117 0.176
Absorbance values were empirically determined. Concentrations were calculated from the mass of cobalt salt sample and the dilution factor, detailed below. The range of A/ri values (0.04, or about 24% of 0.168), which should be constant if Beer’s law perfectly holds, suggests deviations from Beer’s law.
Table 2. Spectrophotometric Data for Saturated Solutions
Solution Saturation Temperature (°C) Absorbance (A) Concentration (M) Ksp
A 0 0.176 2.88 x 10-3 9.51 x 10-8
B 6 0.186 3.04 x 10-3 1.12 x 10-7
C 12 0.254 4.15 x 10-3 2.86 x 10-7
D 18 0.529 8.64 x 10-3 2.58 x 10-6
Concentrations were derived from relating the absorbance value of each solution to the absorbance and concentration of the stock solution 0, using Beer’s law, as shown below. These concentration values provided Ksp.
Table 3. Thermodynamic and Equilibrium Data for Saturated Solutions
Solution Saturation Temperature (°C) Saturation Temperature (K) 1/T (K-1) Ksp ln(Ksp)
A 0 273.15 3.6610 x 10-3 9.51 x 10-8 -16.2
B 6 279.15 3.5823 x 10-3 1.12 x 10-7 -16.0
C 12 285.15 3.5069 x 10-3 2.86 x 10-7 -15.1
D 18 291.15 3.4347 x 10-3 2.58 x 10-6 -12.9
These data enable calculation of further thermodynamic values for the dissolution of cobalt salt.
In the stock solution (0), 0.1053 g sample of cobalt salt was dissolved in about 100 mL deionized water, allowing for the following calculation of the concentration of the stock solution from the molar mass of cobalt salt (383.63 gmol-1) (Table 1):
Solution 0: (0.1053 g)/(0.100 L) * (1 mol)/(383.63 g)=2.74 x 〖10〗
(-3) M ± 1 x 〖10〗
Diluted solutions were prepared by adding to 10 mL of stock solution the following: for solution #1, 20 mL deionized water; for solution #2, 10 mL deionized water; and for solution #3, 10 mL stock solution and 10 mL deionized water. Concentrations of these dilutions were calculated as follows (Table 1):
Solution 1: (〖2.74 x 10〗
(-3) mol)/L*(0.010 L)/(0.030 L)=9.15 x 〖10〗
(-4) M ± 1 x 〖10〗
Solution 2: (〖2.74 x 10〗
(-3) mol)/L*(0.010 L)/(0.020 L)=1.37 x 〖10〗
(-3) M ± 1 x 〖10〗
Solution 3: (〖2.74 x 10〗
(-3) mol)/L*(0.020 L)/(0.030 L)=1.83 x 〖10〗
(-3) M ± 1 x 〖10〗
Supposing Beer’s law holds for these stock solutions, the concentrations of saturated solutions of cobalt salt for which absorbance values are known can be calculated from the data in Table 1, shown in Table 2:
A_0/C_0 = A_i/C_i
Solution A: C_A= (A_A C_0)/A_0 =((0.176)(2.74 x 〖10〗
(-3) M))/0.168=2.88 x 〖10〗
(-3) M ± 1 x 〖10〗
Solution B: C_B= (A_B C_0)/A_0 =((0.186)(2.74 x 〖10〗
(-3) M))/0.168=3.04 x 〖10〗
(-3) M ± 1 x 〖10〗
Solution C: C_C= (A_C C_0)/A_0 =((0.254)(2.74 x 〖10〗
(-3) M))/0.168=4.15 x 〖10〗
(-3) M ± 1 x 〖10〗
Solution D: C_D= (A_D C_0)/A_0 =((0.529)(2.74 x 〖10〗
(-3) M))/0.168=8.64 x 〖10〗
(-3) M ± 1 x 〖10〗
The expression for the solubility product Ksp of dissolution of cobalt salt is as follows:
K_sp=[〖(Co(〖NH〗_3 )_5 Br)〗
(2+) ] 〖[〖Br〗
Given that the stoichiometry of this dissolution reaction follows the equation:
[(Co((〖NH〗_3 )_5 Br)] 〖Br〗_(2(s))↔〖(Co(〖NH〗_3 )_5 Br)〗
x=[〖(Co(〖NH〗_3 )_5 Br)〗
For every mole of cobalt salt dissolved, a mole of (Co(NH3)5Br)2+ is produced, meaning the Ksp of each saturated solution can be calculated from the concentrations given in Table 2:
Solution A: K_sp=4〖(2.88 x 〖10〗
3=9.51 x 〖10〗
(-8)± 1 x 〖10〗
Solution B: K_sp=4〖(3.04 x 〖10〗
3=1.12 x 〖10〗
(-7)± 1 x 〖10〗
Solution C: K_sp=4〖(4.15 x 〖10〗
3=2.86 x 〖10〗
(-7)± 1 x 〖10〗
Solution D: K_sp=4〖(8.64 x 〖10〗
3=2.58 x 〖10〗
(-6)± 1 x 〖10〗
By plotting ln(Ksp) against 1/T in units of K-1, approximations of ΔH° and ΔS° can be calculated. For a system at equilibrium:
Substituting the second equation into the first and solving for ln(Ksp) gives the van’t Hoff equation:
This is a linear equation, given that ΔH° and ΔS° are not significantly temperature-dependent, in which:
Therefore, when a line of best fit is found for the data in Table 3, ΔH° and ΔS° can be derived for the dissolution of cobalt salt (Fig. 1):
o= -mR= -(-14249 K)(8.315 x 〖10〗
(-1) )= 118 kJ〖mol〗
o= bR= (35.502)(8.315 x 〖10〗
(-1) )= 0.295 kJK
From comparison of the ratios of absorbance to concentration for various degrees of dilution of the cobalt salt solution, Beer’s law did not hold perfectly. The ratio A/ri was predicted to be equal to A0, or 0.168, yet this ratio was as high as 0.198 and as low as 0.158 across the diluted samples, with a significant range of deviation (approximately 24% of the expected value). Additionally, based on the coefficient of determination for the plot of ln(Ksp) versus 1/T, approximately 0.83, there was a relatively poor linear relationship between these values, rendering the calculations of the standard enthalpy and entropy change of this dissolution reaction most likely inaccurate.
The high entropy change of the dissolution of (Co(NH3)5Br)Br2 is most likely a result of the chelate effect, which refers to the relative favorability of the formation of complexes with bidentate ligands than monodentate ligands (Lancashire, 1995). The denticity of a ligand (that is, a species that forms a complex with a metal) refers to the number of electron pairs the ligand donates to the central metal (Purdue University, n.d.). Bromide, for instance, provides one electron pair in complexes and is thus monodentate (Purdue University, n.d.). Compared with monodentate ligands, bidentate species can form stable structures called chelate rings when associating in complex with a metal (Lancashire, 1995). Because the formation of the cobalt complex from association of the monodentate ligand, bromide, is entropically unfavorable, the reverse reaction – dissolution of the complex – is highly entropically favorable, corresponding to the high standard entropy change.
Though error due to cuvette contaminations are likely even when great care is taken, this experiment could become less vulnerable to the effects of such errors with more trials. An excess of stock solution was already prepared according to the procedure, meaning it would not be inefficient to use as many solutions at constant temperature as possible from the 100-mL stock. Using more than four different saturation temperatures would also tend to create a better adherence of Ksp and 1/T data to a linear relationship.
Beer’s law reduces to A_1/C_1 =A_2/C_2 when the path length and molar absorptivity constant (which depends only on the chemical identity of the solute) are the same for the different trials under consideration. Since our experiment uses the same solute, the cobalt salt, across all trials and identical cuvettes, the following relations hold:
4.) Contaminations on cuvettes would block the transmission of light through the samples, reducing the final intensity value and therefore resulting in a higher absorbance value determined by the spectrophotometer than the intrinsic absorbance of the cobalt complex solution itself. Because this experiment uses the absorbance value of the initial stock solution to derive the concentrations and solubility constants of the saturated solutions, cuvette contaminations could in turn skew the enthalpy and entropy values.
The potential of this experiment to reveal the thermodynamic properties of cobalt complex dissolution was limited by apparent deviations from Beer’s law in the spectrophotometric data. Though literature values for the enthalpy and entropy changes of this reaction are not readily available, the accuracy of our data is doubtful because of inconsistent absorbance-to-concentration ratios and a nonlinear relationship between ln(Ksp) and 1/T. As discussed above, cuvette contaminations very likely distorted the absorbance data considering the sensitivity of this method. If inconsistently dirty cuvettes provided higher absorbance values than the real absorbances of some samples used, to greater degrees than in other samples, absorbance-to-concentration ratios would be inconsistent as well. This would give the impression that Beer’s law does not apply despite the use of a constant solute, constant path length, and low concentrations. Such an error would not be exceptionally unlikely, because Table 1 shows only one major outlier from the expected A/ri value, solution 1. Supposing there was contamination of the cuvette used for the stock solution as well, this inflated absorbance value carried over into Table 2’s data and provided lower concentration values than reality.
The assumption that ΔH° and ΔS° are temperature-independent is fairly valid for the small temperature range used in this experiment (Oxtoby, Gillis, and Butler, 2012). The poor linear regression seen in Fig. 1 therefore most probably resulted from a combination of inaccurate concentrations, altering the Ksp values from the actual ones. Aside from varying levels of cuvette contamination, concentration inaccuracies could also have developed from the small excess of water added to the initial stock solution, noted above. The concentration of the stock solution was thus lower than recorded, and though this universally inflated the concentrations derived for the saturated solutions, the effect on ln(Ksp) would not be uniform (and therefore negligible) because the stock solution concentration was a multiplicative factor. As the many mathematical relations detailed above show, the process of spectrophotometry can in theory illuminate fundamental thermodynamic properties of a reaction such as a complex dissolution. The efficacy of this method, however, only extends as far as the equipment is handled with sensitivity, and the tendency of small sample sizes to yield great variation must also be considered.
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