Acolored dye compound decomposes to give a colorless product. the original dye absorbs at 608 nm and has an extinction coefficient of 4.7 ×10⁴ m⁻¹cm⁻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.

First Order reaction

k=  0.0067

Explanation:

To determine the rate law we will have to study the dependence of the rate of the reaction with respect to time.

So we need to calculate the concentrations given the data since  A= ε c l

where ε = molar absorptivity, c is the concentration moles/L, and l is the length of the cuvette.

c = A/εl

Time, min    M, (mol/L) x 10⁻⁵

0                         2.67

30                       2.00

60                       1.60

90                       1.43

120                      1.16

If the reaction were first order, the rate will have to remain constant which is not the case here since the rate of dissaperance is chnging:

Δmol/L/ΔT = (2.00 - 2.67 ) mol /30 min = -0.023 mol/Lmin ( between t= 30 and 0 min)

Δmol/L/ΔT = ( 0.160 - 2) /30 min =  = -0.061 mol/Lmin

If the rate is first order we will have to use the integrated rate law:

ln (A) t  = -kt + ln (A)₀

where (A) is the concentration, t is the time and k is the rate constant

Notice the equation  is of the form y = mx + b where

m= rate constant and b  (A)₀

To verify if our reaction is first order, we will graph the graph the data and see if it is a straght line and and if it is compute the slope which is k.

We could also perform a linear regression with an apprpiat program and obtain the results.

Working in excel the following results were obtained:

y = -0.0067 x  -10.582

k = 0.0067/ min

R² = correlation coefficient = 0.979

First Order reaction