Determining the Equilibrium Constant of Bromothymol Blue

Written by Lisa

Abstract

Bromothymol blue is an acid–base indicator whose color shows whether a solution is acidic, near neutral, or basic. In acidic solutions, it appears yellow, in basic solutions it is blue, and in mid pH ranges, it is green. By determining the absorbance values of solutions at all of these extremes, the equilibrium constant (K_c) of this indicator can be calculated. After taking the spectra of each the yellow, the blue and the green solutions, a K_c value of (1.2 ± 0.19) × 10⁻⁷ M and a pK_c value of 7.0 ± 0.2 were calculated. When comparing the calculated value to a literature pK_c value of 7.1, there is a percent difference of 1.4%.

Objective

Bromothymol blue is a chemical indicator that appears yellow in very acidic conditions, blue in basic conditions, and green in mid pH ranges. This experiment will determine the equilibrium constant of this indicator by comparing the absorbance of a buffer solution containing bromothymol blue at its yellow, blue and green states. After the spectrum of each solution is taken, the following equation can be used to determine the equilibrium constant (K_c).

However, to account for the fact that multiple species (different colors) of the bromothymol blue that may exist in the solution at one time, the revised equation below subtracts the absorbance of the invading species from the one of interest.

This gives a more accurate depiction of the calculated equilibrium constant of bromothymol blue.

Results

From the spectra of each solution (Figure 1), absorbance values at 453nm and 616nm were collected (Figure 2). The concentration of hydronium ions, [H₃O⁺], is determined from the initial pH reading of the sodium phosphate buffer made (Equation 1).

Figure 1: A graph showing the spectra from 375nm to 750nm of each of the different bromothymol blue solutions.

Figure 2: A table of absorbance values for each of the 3 bromothymol blue solutions at 453nm and 616nm.

Equation 1: Determining the concentration of hydronium ions.

With these values, the equilibrium constant can be calculated via the equation that adjusts for alternative absorbance values (Equation 2). The calculated K_c value can then be converted to a pK_c value for better comparison to a literature source (Equation 3).

Equation 2: Determining the equilibrium constant

Equation 3: Determining the pK_c value for literature comparison.

Thus, the calculated equilibrium constant for bromothymol blue is 1.12 × 10⁻⁷ M, with a pK_c value of 6.95.

Error Analysis

The error in pH values was determined to be ± 0.03 units, and for all absorbance values, there is an error of ± 0.002. With these error values, a total standard deviation associated with K_c can be calculated.

[H₃O⁺] relative variance: (0.03)(2.303)² = 0.159₁₁₄, variance = 0.159₁₁₄[H₃O⁺]

= 0.159₁₁₄(1.31) = 2.09₇₅₃

[H₃O⁺] = (1.3 ± 0.2) × 10⁻⁷ M

Determination of the total error for subtraction values:

0.128 ± 0.002 – 0.003 ± 0.002 = 0.125 ± 0.003

0.111 ± 0.002 – 0.046 ± 0.002 = 0.065 ± 0.003

With all corrected error values:

To incorporate errors into this calculation:

Variance in pK_c:

Comparing to the Literature Value:

Discussion and Conclusion

According to the data, the calculated equilibrium constant of the bromothymol blue indicator was (1.2 ± 0.19) × 10⁻⁷ M, which translates to a pK_c of 7.0 ± 0.2. When compared to the literature pK_c value of 7.1, the percent difference is 1.4%. The calculated value may have disagreed slightly because the calibration curve of the pH meter used was only 90.0. This means that the hydronium concentration calculated may be slightly off, thus throwing off the equilibrium value. Error may have also occurred during the spectrum readings since a SpectroVis Plus reader that correlated with Logger Pro 3 was used instead of a more sophisticated UV-Vis spectrometer. To improve the results, a more sensitive spectrometer and a single cuvette could be used to gain all solution spectra.

How Absorbance Data Give the Indicator’s Equilibrium Constant

Bromothymol blue is diprotic. Across its color change, the yellow form is actually HIn⁻ and the blue form In²⁻, but for this single equilibrium they can be written simply as HIn (acid, yellow) and In⁻ (base, blue):

HIn + H₂O ⇌ H₃O⁺ + In⁻    K_a = [H₃O⁺][In⁻]/[HIn]

The constant called K_c in this report is this acid ionization constant, K_a. It has units of molarity, and the value found here is on the order of 10⁻⁷ M.

Step 1: Beer–Lambert Law

Absorbance is proportional to concentration: A = εbc, where ε is the molar absorptivity at a given wavelength, b is the path length, and c is the concentration. With the same total indicator concentration and the same cuvette, the absorbance at a fixed wavelength tracks how much of each form is present.

Step 2: Measure Each Form Near Its Absorbance Peak

In acid the indicator is essentially all HIn, which was measured at 453 nm in this experiment (literature peaks for the yellow form are usually reported near 430 nm). In base it is essentially all In⁻, which absorbs near 616 nm. These solutions set the reference values: A₆₁₆(blue) = 0.319 and A₄₅₃(yellow) = 0.141 (Figure 2). Measuring near each form’s absorbance peak gives the largest, most sensitive signal.

Step 3: The Ratio [In⁻]/[HIn] from Absorbance

In the green buffer, both forms are present. Subtracting the small contribution of the other form at each wavelength gives the fraction of each species:

• Fraction as In⁻ ≈ (0.128 − 0.003)/0.319 = 0.392
• Fraction as HIn ≈ (0.111 − 0.046)/0.141 = 0.461
• [In⁻]/[HIn] ≈ 0.392/0.461 = 0.850

This is the same calculation as Equation 2. Ideally the two fractions would add to 1. Here they add to about 0.85, a useful check that points to some experimental uncertainty, as discussed in the conclusion.

Step 4: Henderson–Hasselbalch and K_a

Taking the negative logarithm of the K_a expression gives the Henderson–Hasselbalch equation:

pH = pK_a + log([In⁻]/[HIn])

With the buffer pH of 6.88 and log(0.850) = −0.071:

• pK_a = 6.88 − (−0.071) ≈ 6.95
• K_a = [H₃O⁺] × 0.850 = (1.32 × 10⁻⁷ M)(0.850) ≈ 1.12 × 10⁻⁷ M

Both routes agree with the values reported above. The ratio is close to 1 because the buffer pH was near the indicator’s color transition, where both forms are present in measurable amounts. Measurements near the transition, where both forms are present, give the most reliable ratio.

Editor’s note on the original calculations: the K_c and [H₃O⁺] values are in units of 10⁻⁷ M, and those units have been added above. For error propagation, the standard relationship is σ[H₃O⁺]/[H₃O⁺] = 2.303 × σpH = 2.303 × 0.03 ≈ 0.069, which gives about ±0.09 × 10⁻⁷ M rather than ±0.2. Carrying that value through, and combining the relative errors in quadrature, gives K = (1.12 ± 0.10) × 10⁻⁷ M and pK = 6.95 ± 0.04. The original report’s ±0.19 × 10⁻⁷ M and ±0.2 are therefore overestimates. The reported values themselves are unchanged. The 1.4% difference uses the rounded pK_c of 7.0. With the unrounded 6.95, it is about 2.1%.

For a related lab report, see this spectrophotometric analysis of the same indicator.

Sources:

• LibreTexts: Beer’s Law
• OpenStax Chemistry 2e, 14.7: Acid-Base Titrations (indicators, HIn/In⁻)
• OpenStax Chemistry 2e, 14.6: Buffers (Henderson–Hasselbalch)

References

Klotz, E., Doyle, R., Gross, E., Mattson, B., J. Chem. Ed.,2011 88 (5), 637-639.