# 4.2: Beer's Law and Integrated Rate Law Lab


Learning Objectives

Goals:

• Calculate the concentration of a dilute aqueous solute using absorbance spectroscopy (Beer's Law).
• Determine order of reaction by measuring the concentration of a reactant as a function of time

By the end of this lab, students should be able to:

• Calibrate a spectrometer and create a linear Beer's Law plot
• Use the Beer's Law plot to determine the concentration of an unknown analyte.
• Determine the order of reaction by measuring the absorbance of a reactant at a specific wavelength as a function of time.

Prior Knowledge:

## Safety

• Emergency Preparedness
• Minimize Risk
• If you use a laptop
• Do not let it be near chemicals
• Due to COVID 19, only the owner of the laptop should touch the keyboard.
• Recognize Hazards
• Sodium hydroxide is caustic and may cause skin irritation and possibly burns.  If spilled on the skin immediately flush with water.
• Report any spill to your instructor
• Crystal violet is a biological stain.  Avoice contact with your skin or clothing.

## Equipment and materials needed

 VIS Spectrophotometer 6 cuvettes with covers LabQuest or Laptop w/LoggerPro 0.10 M NaOH 2.0x10-5M Crystal Violet 2 burettes 100 ml Beaker 4.0x10-5M Crystal Violet

## Overview

In this experiment you will essentially run three different tasks to determine the order of reaction for the bleaching of crystal violet in excess sodium hydroxide

Step 1: Obtain a Spectra of Crystal Violet.  This will allow you to choose a wavelength where you can run the experiments
Step 2: Make a Beer's law plot of your solution at a wavelength determined in step 1, so you can determine its concentration from absorbance measurements.
Step 3: Run a time resolved kinetics experiment at the wavelength from step 1, and determine if it is a zero, first or second order reaction.

## Background

This experiment will introduce you to the use of spectrometers, and techniques for using them.  The goal of this lab is to understand the kinetics for the bleaching of crystal violet by sodium hydroxide.  This is done by measuring how the blue color of the crystal violet solution becomes colorless over time

IMPORTANT

Before Proceeding Read Section 4.5.1-4 (Resources and Further Information) of this experiment.

1. Absorbance Spectroscopy Overview
2. Spectrometer Design
3. Absorbance
4. Beer's Law

In this lab you are responsible for the material in that section. Your prelab questions, postlab quiz and lab activity require you to be familiar with that material.

There are essentially three components to this lab.

1. Obtain an absorbance spectra.  This is a plot of the absorbance as a function of wavelength, and is sort of a fingerprint of a molecule, which is often used as a qualitative technique to identify if a molecule is present.
2. Obtain a Beer's Law plot.  This allows one to determine the concentration of a chemical species as a function of its absorbance.
3. Obtain a kinetics decay plot.  This is a plot of the absorbance as it changes over time, and can be used to determine the kinetics of a reaction. We are actually running this experiment before we cover this material in lecture and using this lab to introduce the concept of kinetics.

### Absorbance Spectra

An absorbance spectra is a plot of the light absorbed as it travels through a sample as a function of its wavelength. Figure $$\PageIndex{2}|) shows the spectral characteristics of the spectra of chlorophyll a and chlorophyll b. When visible light interacts with a molecule's electrons they absorb the energy of the photons and can be excited from low energy to a higher energy states if the energy gap between the orbitals is equal to the energy of the light absorbed. In gen chem 1 (section 6.2) we learned that this energy is related to the frequency and wavelength of the light by Planks constant \(E= (h\nu =h\frac{c}{\lambda}$$) and that the intensity (I) of light at a specific frequency was nh$$\nu$$.  We studied line spectra of atoms  (section 6.3) where we could identify the atom by its unique absorbance or emission spectrum.  Molecules have more complex orbitals and bonds that are in constant vibrational motion, so their energy gaps are constantly changing and they do not produce a discrete line spectrum like the gaseous atom, but a sort of blurring of the lines as seen in figure $$\PageIndex{2}$$), where each peak is associated with an electronic transition.  In the line spectra we only noted if light was absorbed (or emitted) at a wavelength, but in molecular spectroscopy we are also concerned with the number of photons absorbed by the molecule at that wavelength. So the Y-axis of the spectrum in figure  $$\PageIndex{2a}$$ is theaAbsorbance (A), which is logarithmically related to the reduction in intensity of light as it travels through the sample.

$A=log\frac{I_0}{I_t}$

where,

A=Absorbance
I0= light intensity entering sample
It = light intensity leaving (transmitted through) sample

This equation is derived in the Resources and Further Information (section 4.5.4) of this experiment.  It should be noted that what is measured by the spectrometer is the intensity of light, and there is a small range of absorbances where we can trust the value of the data from our spectrometer. The rule of thumb we will use is that we only trust Absorbance values between 0.05 and 1.0.  It is important to realize that any instrument has a range for which it is calibrated, and you need to know the accuracy (and precision) of any instrument you use in the lab.

### Beer's Law

Beer's law is followed if a plot of the Absorbance at a specific wavelength vs. the concentration of a molecule that absorbs light at that wavelength is linear. Figre $$\PageIndex{3}$$ shows a plot that follows Beer's Law, which is commonly expressed in the form of:
$A=\epsilon bc$

where

• $$\epsilon$$= the extinction coefficient
• b=path length
• c=concentration

Beer's Law is derived in section 4.5.4 of this lab. Deeper Dive 4.5.1 shows the calculus and you should not that mathematically this is the same kind of relationship that results in first order reaction equations that we are studying in the lecture.  That is, a logarithmic relationship arises when the rate at what something changes is proportional to the thing itself.  In first order kinetics the rate of change in concentration as a function of time is proportional to the concentration, $$\frac{\Delta [A]}{\Delta t}=-k[A]$$ and here it is the change in intensity of light as a travels is proportional to the intensity of light $$\frac{\Delta [I]}{\Delta x}=-k[I]$$.  As you advance in your studies you will frequently see these kinds of logarithmic relationships and you need to be comfortable with them (calculus is not a prerequisite for this class, but the equations we use come from the calculus, and you need to know that).

What is very important to realize in the lab that is that your instrument has a range where it can accurately make a measurement, and in this lab we only trust absorbance values between 0.05 and 1.0.

Exercise $$\PageIndex{1}$$

The extinction coefficient of a sample at 654 nm is 100000 cm-1M-1. What is the concentration of an unknown sample if it had an absorbance of 1.00 in a cuvette with a 1.0 cm path length?

$A=\epsilon bc$

$$\epsilon$$= 100000 cm-1M-1

b=1 cm

c=?

1.00=(100000 cm-1M-1)(1 cm)(c)

c=1.00/((100000 cm-1M-1)(1 cm))

c=0.000010 M

### Kinetics Decay Plots

In lecture we will first study the differential rate laws and then derive the integrated rate laws from them.  The integrated rate laws are easier to experimentally understand, and so we are going to run the lab on the integrated rate laws before doing the differential rate laws experiment. The rate of reaction describes how fast a product is produced or a reactant is consumed (section 14.1) and the rate law (section 14.3) is a power function.

$A \rightarrow Products \\ R=k[A]^m \\ \; \\ \text{describing the rate in terms of the reactant concentration} \\ \;\\ \frac{\Delta [A]}{\Delta t}=-k[A]^m$

As the above equation has a power, we would need to do a log/log plot, where the dependent variable (Y-axis) would be the log of the rate ($$\frac{\Delta [A]}{\Delta t}$$) and the independent variable (X-axis) would be the log of the concentration (log[A]), and we will run those experiments next week.

The integrated rate laws are covered in section 14.4 and simply describe how the concentration of a reactant changes over time. That is, we solve the following equation for [A] and time

$\frac{\Delta [A]}{\Delta t}=-k[A]^m$

This requires calculus and we are only going to look at three solutions, which are for the values of m = 0,1 or 2. Calculus is not a prerequisite for this course, but if you are interested the relevant integrals are in section 4.5.5

#### Zero Order Reaction

$Rate=\frac{\Delta[A]}{\Delta{t}}=-k[A]^0=-k$

When integrated, there is a direct linear relationship between the concentration and time

$A]_t = -kt + [A]_0$

#### First Order Reaction

$Rate=\frac{\Delta[A]}{\Delta{t}}=-k[A]^1=-k[A]$

When integrated this results in a linear relationship for the natural log of the concentration as a function of time

$ln[A]_t = -kt + ln[A]_0$

#### Second Order Reaction

$Rate=\frac{\Delta [A]}{\Delta{t}}=-k[A]^2$

When integrated this results in a linear relationship for the reciprocal log of the concentration as a function of time

$\frac{1}{[A]_t} = kt + \frac{1}{[A]_0}$

### The Kinetics Experiment

This lab will involve the bleaching of crystal violet dye by sodium hydroxide

$\underbrace{CV^+}_{blue} + OH^- \rightarrow \underbrace{CVOH}_{clear}$

Which has the rate law of

$R=k[CV^+][OH^-]$

In this experiment the [CV+]= 0.00002M and the [OH-]=0.10M, which means the [CV+] is 5000 times more concentrated than the hydroxide, and so the concentration of the hydroxide stays constant (by being in such excess it is effectively zero order and does not change over the course of the reaction).  So the rate law reduces to

$R=k[CV^+]$

In this lab you will mix 10 mL of the CV with mix with 10 mL of the sodium hydroxide and immediately transfer to the spectrometry that is set up to collect data every 5 seconds.  Your goal is to get data for the full range of absorbances from 1.0 to 0.05.  There is no problem if your initial absorbance is greater than 1.0, just do not use that data when you make your graphs.  But there are problems if your initial values are too low, as all plots will start to look linear as you approach 0.05 absorbance.  You will get the clearest results if your data spans the range the instrument is best suited to measure.

## Experimental Procedures

There are four parts to this experiment.  First you will use the technique of dilutions to make a total of 5 solutions (stock plus 4 diluted solutions).  Then you will take a spectrum of your stock and choose a wavelength for your Beer's Law plot.  You will then record the absorbance of all 5 solutions and make a Beer's law plot that has 6 data points (when you calibrated the instrument you set A=0 for the pure solvent, the point 0,0 is a value on your graph).

### Part I: Dilutions

1. Obtain stock solution of CV⁺ $$2.0x10^{-5}$$M  (Note that this is half the concentration we will use in Part IV)
2. Using a burette transfer 5 mL of $$2.0x10^{-5}$$M CV⁺ to a 25mL volumetric flask
3. Fill the flask a bit over half way to mark with deionized water and swirl to mix all of the solution. Continue filling to mark, using an eye dropper to add the last few drops.  Be sure all the solute is fully mixed. Pour solution into a clean 50mL beaker.
4. Fill cuvette 3/4ths full with solution and label the cap with number 1
5. Repeat steps 2-4 for volumes of 10mL, 15mL, and 20mL using a new cuvette and labeling each cuvette 2,3,4
6. Fill a cuvette with stock $$2.0x10^{-5}$$M CV⁺ and label cap 5

### Part II: Spectrum

1. Calibrate the spectrometer
• Warm spectrometer for 5 minutes
• Fill cuvette 3/4ths full of solvent (water), cap and label the cap 0 (zero)
• Place in cuvette in the cavity so the light path goes through the clear side
• From Experiment menu choose Spectrophotometer/calibrate
• Follow the instructions until the calibration is OK.
• Keep this "blank" solution in the cuvette until the experiment is over, as you may need to recalibrate the spectrometer
2. Generate a spectrum
• After calibrating spectrometer place stock solution (cuvette #5) into cuvette cavity
• Click <Collect> and once the spectrum is displayed click <Stop>
• Choose a wavelength for Beer's Law plot where A=1 for the stock solution, write this down in your data sheet
• Up on the top menu select File>Save. Enter name and save
• To Export: Plug in flash drive, File>Export. Enter name and save. Make sure to share file with all group members

### Part III: Beer's Law plot

1. Calibrate the spectrometer if needed (you can read the absorbance of the blank (solvent), if it is zero at the wavelength you are measuring you do not need to recalibrate.
2. Place each cuvette into the spectrometer and read the absorbance at the chosen wavelength (where A=1 for stock).
• Record values in data sheet
• You should have 6 values (5 for each of the solutions, and the blank, which should read 0)

### Part IV: Transient Kinetics

1. Calibrate the spectrometer if needed
2. From the meter tab click on "Mode:" and select "Time Based"
3. Set Interval to 5 seconds and Duration to 800s
4. Use a graduated cylinder to measure 5mL of $$4.0x10^{-5}$$M CV⁺ (Double check label! this conc is different from Beer's Law plot)
5. Use another graduated cylinder to measure 5mL of 0.100 M NaOH(aq)
6. Make sure Lab Quest is ready to take measurements.
7. Pour both solutions into a clean beaker and swirl to mix. Fill cuvette 3/4ths full with solution and place in spectrometer and press play. (You have about 15s from mixing)
8. If the initial absorbance is below 0.9 reset and try again.
9. When the absorbance is below 0.05 you can stop the labquest, save the data, then export the data to a flashdrive

## Data Analysis

### Cover page tab

The first page is always the cover page.

### Spectra tab

The second tab will contain your spectra.  Open the CSV (comma separated variable) text file of the spectrum that you saved to a flash drive on the labquest, and then copy and past the raw data into columns A & B of the second tab (keep the titles, so start on the second row).  Use "smooth line chart"

#### Importing Spectra csv

1. On your google sheet open the spectra tab and click on cell 1A
2. Click File>import
4. Click the drop down for import location
5. Select Append to current sheet
6. Then click import data

#### Making Spectra Chart

1. Highlight columns A and B by clicking and dragging on column labels (where it says A and B) or click A then shift+click B. This should highlight the entire column
1. (there is a lot of data and trying to scroll through the entire sheet will take a long time)
2. Click Insert>Chart
3. Open chart editor by clicking the three dots in the top right corner of the chart
4. Check that your chart type is a smooth line chart
6. Make sure your chart has a Title, Axes labels for x and y axis, and units.

### Dilution tab

In this tab you will calculate the concentration (Molarity) of the solution after each dilution. Use the drop down to label each column as the initial or final Molarity or volume.

### Beer's Law tab

Transfer the data from your data sheet to the orange cells and enter your values from the dilutions tab in the blue cells.  Do not rearrange these cells as they are connected to your instructor's workbook.  If needed you can download a new template, but then must resubmit its URL through the google form and alert you instructor (as there will be two spreadsheets with your name).

If needed you can also make another tab and name it "scratch" to do other work, However only answers in the boxes on the template pages will be graded.

Make a Beer's Law Plot using a scatter plot with line of best fit and appropriate equation. Use the equation to find the extinction coefficient, ϵ. 4.2.2 (Note the width of the cuvette is 1cm)

### Kinetics Tab

Import your decay plot data using the same steps as importing the spectra. Delete any rows where the absorbance is above 1 or below 0.05. (Delete rows not the individual cells otherwise your pdf will have several blank pages)

Make your plots for Zeroth, First, and Second Order (Summary of Rate Laws). Determine the order of reaction and report it on the coverpage

Use formulas to do algebraic calculations (Tips and short cuts)

4.2: Beer's Law and Integrated Rate Law Lab is shared under a not declared license and was authored, remixed, and/or curated by Robert Belford.