A colored dye compound decomposes to give a colorless product. The original dye absorbs at 608 nm and has an extinction coefficient of 4.7 * 10^4 M^-1 cm^-1 at that wavelength. You perform the decomposition reaction in a 1-cm cuvette in a spectrometer and obtain the following data: Time (min) Absorbance at 608 nm 0 1.254 30 0.941 60 0.752 90 0.672 120 0.545. From these data, determine the rate law for the reaction 'dye → product' and determine the rate constant.

Verified step by step guidance

1. Step 1: Use the Beer-Lambert Law, A = εcl, to determine the concentration of the dye at each time point. Here, A is the absorbance, ε is the extinction coefficient (4.7 * 10^4 M^-1 cm^-1), c is the concentration, and l is the path length of the cuvette (1 cm). Rearrange the equation to solve for concentration: c = A / (εl).
2. Step 2: Calculate the concentration of the dye at each time point using the absorbance values provided. For example, at time 0 min, the concentration c = 1.254 / (4.7 * 10^4 * 1). Repeat this calculation for each time point.
3. Step 3: Plot the concentration of the dye versus time to visually inspect the data. This will help determine the order of the reaction. If the plot of concentration vs. time is linear, the reaction is zero-order. If the plot of ln(concentration) vs. time is linear, the reaction is first-order. If the plot of 1/concentration vs. time is linear, the reaction is second-order.
4. Step 4: Based on the plot that gives a straight line, determine the order of the reaction. Use the appropriate integrated rate law to calculate the rate constant (k). For a first-order reaction, use ln([A]_t/[A]_0) = -kt. For a second-order reaction, use 1/[A]_t - 1/[A]_0 = kt.
5. Step 5: Calculate the rate constant (k) using the slope of the linear plot from Step 3. The slope of the plot corresponds to -k for a first-order reaction or k for a second-order reaction. Ensure the units of k are consistent with the order of the reaction.