M8/102 App 02 Prep Interp of Graph.pdf

Appendix 2 PREPARATION & INTERPRETATION OF

GRAPHS

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All of you should have had some experience in plotting graphs. Some may have done it only in math courses where numbers are exact and significant figures were not considered. This appendix will serve as a review, as well as point out features of graphs used in science of which you may not be aware. 1. Selection of Appropriate Axis:

When we specify “A versus B,” by convention, “A” goes on the y-axis and “B” goes on the x-axis. (The y-axis is the vertical axis and the x-axis is the horizontal axis.)

When you are asked to plot a graph, read the assignment carefully to determine which data go on the x-axis and which go on the y-axis.

2. Selection of Range for the Scale:

It is important to note that all scales do not have to start at zero. If all of your data are in the range of 300 to 500, there is no reason to start with zero. The exception is if you need to find the y-intercept (the y-value when x equals zero) on the graph, then obviously you will need to have x equals zero showing on the graph. Choose a scale with a range that goes from a number smaller than any of the numbers in your data point, up to a number larger than any of the numbers in your data point. In other words, you want a scale that would not eliminate any of your data points. Also, keep in mind that the scale you select should be easily read. In other words, it should be labeled at regular, convenient intervals. Figure A is an example of a poorly selected scale. Figure B is an example of a better scale.

Figure A |_____|_____|_____|_____|_____|_____|_____|______ 25 28 31 34 ?

In Figure A, although the numbers are evenly spaced it would be difficult to read where the arrow is pointed. Figure B |_____|_____|_____|_____|_____|_____|_____|______ 24 26 28 30

y-axis

x-axis

A versus B

Appendix 2: PREPARATION & INTERPRETATION OF GRAPHS

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In Figure B, we can easily read the location of the arrow as being at 26.6 or 26.7. Common increments to use on a scale are as follows: 0, 1, 2, 3, 4… or 0, 2, 4, 6, 8… or 0, 5, 10, 15, 20… or 0, 10, 20, 30, 40…

Do NOT use increments such as 3, 6, 9, 12…or 4, 8, 12, 16… because we do not usually count in three’s and four’s.

Practice Exercise 1 If your x-data are as follows: 45.3 g, 58.4 g, 71.0 g, and 82.6 g, make a rough sketch of the x-axis, showing what you should have appear on the axis as minimum number, maximum number, other scale numbers and the minor tick marks between the numbers (as in Fig. B): Hint: You would not start with 0 g.

Practice Exercise 2 If your x-data are as follows: 235 g, 328 g, 471 g, and 582 g, make a rough sketch of the x-axis, showing what you should have appear on the axis as minimum number, maximum number, other scale numbers and the minor tick marks between the numbers (as in Fig. B): Hint: You would not start with 0 g.

3. Selection of an Appropriate Scale:

Select a scale for each axis such that the points are spread out over the whole graph (not clumped up together leaving large blank spaces in the graph). Figure C shows an undesirable graph where points are clumped together. Figure D shows a graph with a more appropriate scale for the x-axis. However, do not do this at the expense of ending with a scale that is too difficult to read. (For example, you do not want to have 6 squares for 5 mL. It is better to have 5 squares for 5 mL).

Figure C Figure D 4. Making a Plot of the Data points:

Always plot your points with a very sharp pencil. Mark each point with an X with the intersection of the X at precisely where your point should be. Do not mark the point with a big fat dot. You lose a lot of precision and accuracy as one cannot tell which part

x x x x

Appendix 2: PREPARATION & INTERPRETATION OF GRAPHS

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of the dot is precisely the point. If you wish you may circle the X to make it more visible.

5. Connecting Data Points:

In this class we will be dealing with only linear graphs. After you have plotted all the points, draw the best straight line (one line) through the points using a ruler and a very sharp pencil. If not all the points lie on the line, select a line such that it goes through most of the points, and with as many points above as there are points below the line. Extend this line all the way to the y-axis. Note that not all lines go through the origin. If one point seems way off, it may be that you misread the scale. It may also be due to an experimental error in your measurement. Consult with your instructor.

6. Label the Graph:

Label each axis with a title, including the units used. Label the graph with a title at the top of the graph. An example is shown below in Figure E:

7. Determination of the Slope:

The slope of the line is calculated from the coordinates of two points taken from the line. It is commonly referred to as “rise over run.” If we use the two points (x1, y1) and (x2, y2), the slope would be calculated as follows: (By convention, we list the coordinates of a point by the x-value first, and then the y-value.)

21 x- xy – y

slope =Δ

Δ=

In science, the x- and y-values usually have units. For the graph shown in Fig. E, the x-values have units of L and y-values have units of atm. Let us take the example of finding the slope from these two points: (3.15 L, 1.04 atm) and (6.82 L, 1.37 atm)

Pressure vs. Volume of a Gas

Fig. E Volume of a Gas (L)

Pres

sure

of a

Gas

(atm

)

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The slope would be calculated as follows:

atm/L 0.090 L 3.67 –

atm 0.33 –

L 6.82 – L 3.15atm 1.37 – atm 1.04

slope ===

Note that units are included in the setup and in the final answer. Note also how significant figures are treated in the calculations. Which points should you pick for the calculation of the slope? a) Select two points that lie exactly on the line and are easy to read. The common mistake is to just pick two data points for the calculation. This is all right if the data points happen to lie exactly on the line. If they do not then all you would be doing is finding the slope of the line that joins only those two points. It would be much easier to find two points that lie at the cross hairs of a vertical and a horizontal line of the graph paper (so that you don’t have to estimate between the lines). b) You should also pick two points as far apart on the line as possible. If you pick two points right next to each other, you would not have enough significant figures when you calculate Δx or Δy.

Determination of the Y-Intercept: Conceptually, the term, y-intercept, refers to where the line intersects with the y-axis. Technically, it is the y-value when x equals zero. Since it is a y-value, the y-intercept should have the same units and decimal places as the y-data. In the graph shown in Fig. E, the y-intercept would have the units of atm.

x x

Points selected are too close to each other.

Points selected are appropriately far from each other, giving more sig. fig. in the slope.

Fig. F Fig. G

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Practice Exercise 3 Answer the following questions based on the graph shown below. Watch your sig. figs. and units. Write down your answers before you check the correct answer. a) What is the slope? First write down the coordinates of two convenient points you are

going to use to calculate this slope. b) What is the y-intercept? c) What is the equation for the line?

Volume(mL)

Mass(g)

0.12 0.296

0.53 1.023

1.23 1.9281.69 2.546

2.04 3.0552.15 3.281

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Answers to Practice Exercises:

Answer to Practice Exercise 1 If your x-data are as follows: 45.3 g, 58.4 g, 71.0 g, and 82.6 g, make a rough sketch of the x-axis, showing what you should have appear on the axis as minimum number, maximum number, other scale numbers and the minor tick marks between the numbers (as in Fig. B): Hint: You would not start with 0 g.

40 45 50 55 60 65 70 75 80 85 90 Mass in g

Answer to Practice Exercise 2 If your x-data are as follows: 235 g, 328 g, 471 g, and 582 g, make a rough sketch of the x-axis, showing what you should have appear on the axis as minimum number, maximum number, other scale numbers and the minor tick marks between the numbers (as in Fig. B): Hint: You would not start with 0 g.

200 250 300 350 400 450 500 550 600 650 700

Mass in g

Answer to Practice Exercise 3 a) Possible points to use for the slope: (0.30 mL, 0.600 g) and (1.90 mL, 2.900 g)

Note that the x-values (volume) has the same decimal places and units as the data for volume, and the y-values (mass) have the same decimal places and units as the data for mass. The points, by convention, are listed with the x-value first, then the y-value.

(2.900 g – 0.600 g) 2.300 gslope = = = 1.44 g/mL

(1.90 mL – 0.30 mL) 1.60 mL

b) y-intercept = 0.19 g

The y-intercept is read off the graph where the line intersects with the y-axis, and recorded to the same decimal places as the y-data.

c) The equation for line is y = 1.44 g/mL (x) + 0.19 g To be more specific, the equation in terms of M and V where M = mass, V = Volume M = (1.44 g/mL) V + 0.19 g

M8/102 App 03 Scientific Method.pdf

Appendix 3 THE SCIENTIFIC METHOD

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The information gathered by chemists and other scientists is usually accurate and reliable. This is because scientists follow a clear set of guidelines in their pursuit of scientific knowledge. It has been shown that in general, scientists follow a number of procedural steps known as the scientific method in their pursuit of knowledge. The scientific method can be defined as a systematic way of gaining and organizing knowledge. It includes the following (not in any particular order):

• Making observations/addressing a question • Formulating a hypothesis • Conducting experiments to test the hypothesis • Predicting the outcomes of experiments to be conducted • Formulating a scientific principle, theory, etc.

Observations: What have you seen that makes you wonder? Observations are made with the senses. In chemistry, we make and record observations during experimentation. For example, we note physical properties of an object—color, odor, texture, solubility, density, etc.; and we note chemical properties of a substance—combustibility, inertness, thermal stability, etc. New discoveries are made when scientists observe inconsistencies with known or accepted facts, and then go out and investigate the cause of the inconsistencies. When something out of the ordinary is observed, scientists usually have a question…what…or how…? And to try to find some answers to these questions, they do a thorough search of the chemical literature to see if there is any information about what is observed. There is nothing to be gained by re-inventing the wheel! Hypothesis: What do you think is the reason for your observation? A hypothesis is an educated guess. It is a tentative or provisional explanation of what is observed. A hypothesis must be testable but it cannot be proven. A good hypothesis allows you to predict the results of experiments that you have not yet conducted. That is, you can say if this is done then that will happen. For example, if ice is placed in a glass of sugar-water solution with a density of 1.25 g/mL, it will melt slower than in a same-sized glass of water with a density of 1.00 g/mL. Experiments are generally conducted to gather data to support the hypothesis. If your experimental data supports your hypothesis, it is valid. If the experimental data does not support a hypothesis, the hypothesis must be either modified or abandoned. Formulating a hypothesis is essential in the pursuit of scientific knowledge because it allows scientists to focus on finding the solution to a specific problem. That is, it allows scientists to set up a simple cause and effect scenario which can be investigated through experimentation. Experiments: What will you do to test your hypothesis? Scientists must design experiments that can reliably give the expected outcomes. An experiment is the process

Appendix 5: SCIENTIFIC METHOD

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of collecting data. Each experiment should have just one variable and a control. In a control experiment, the variables are controlled so that so that the effects of varying one factor can be observed. A very important aspect of the scientific method is that if the hypothesis is sound, scientists should be able to design and predict the outcomes of experiments that they have not yet tried. This is where science differs from natural philosophy—science has the power to predict the outcome of an experiment while natural philosophy does not. Scientific theory: A scientific theory explains observed facts. A scientific theory is formulated after scientists test the hypothesis by performing many, many experiments and get consistent results. Examples of theories include the kinetic molecular theory, atomic theory, molecular orbital theory, etc. A student conducting an experiment (or set of experiments) in the laboratory would make a conclusion at the end of the experiment; not formulate a scientific theory. A scientific theory is formulated by trained scientists after they have conducted numerous experiments over a period of time. Note: A scientific theory should not be confused with a natural law. A law states the observed facts while a theory explains the facts. Examples of laws include the law of conservation of mass, the law of definite proportions, the law of conservation of energy, the law of gravitation, etc. Many scientific investigations do not end with theories or laws; still they provide useful information that enables us to learn about our world and us. The Case of the Dying Rhododendrons: The Scientific Method in Action Consider the following scenario: Priscilla Jones had a beautiful rhododendron garden (about 15 ft by 12 ft). One day, she noticed that some of the rhododendron leaves started to shrivel-up and fall-off prematurely. Priscilla was concerned because it was only about half way into spring and the weather had been fantastic–the other plants around her yard were basking in the sunlight and the rain, yet her beautiful rhododendrons were not doing well at all. On closer look, Priscilla noticed that the leaves of the affected plants had tiny holes in the veins and most had a small bug-like creature (that she had never seen before) clinging to their undersides. Let us help Priscilla apply the scientific method to solve this problem. What did she observe? Observation: The rhododendron leaves shriveled-up and fell off prematurely; the leaves had tiny holes in the veins; most of the affected leaves had a bug-like creature on their undersides; the weather was good; the other plants were healthy

Appendix 5: SCIENTIFIC METHOD

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Question: What is causing this to happen? Research: The bugs are probably causing the problem, but how? Now it's time to do some research. We would look up all that we can find about the bug-like creatures—see what they are called, what they feed upon, and what harm they cause…. Once we understand the problem, then we can formulate a testable hypothesis. Hypothesis: The rhododendron leaves are shriveling-up and falling-off (dying) because the bug is injecting them with a poisonous chemical. Question: How do we know that a chemical is causing the problem? What chemical is it? Now that we have a hypothesis, we can set about to test it. We can design experiments in which the “poisonous chemical” is extracted from the bugs and from the affected leaves and tested on unaffected plants. Experimental Design:

1. Collect a large number of bugs and soak them in a suitable solvent to extract out any potential chemical.

2. Collect a large number of the affected leaves and soak them in the same solvent to extract out the chemical injected by the bugs.

3. Collect the same amount of healthy leaves as affected leaves and soak them in solvent as a control. (If your hypothesis is correct, this extract should not contain the chemical found in the bugs and in the affected leaves.)

4. Compare the extracts from all three extractions to see if there is a chemical that is not in the healthy leaves but present in the bugs and the affected leaves.

5. Isolate the potential poison from the bug extract and inject it on healthy leaves. If the healthy leaves shrivel-up and fall-off after being injected with the bug extract, then the experimental result supports the hypothesis. Note: If this were a real situation, the chemical would be collected, purified, and identified. If it were never discovered before, at this point it would be named and characterized. Applying the Scientific Method In Module 5, you are required to use the scientific method to figure out the relationship between the color of a sample and the colors that are absorbed when the sample is placed in the path of white light. Make sure you do your research before attempting to formulate your hypothesis. Your hypothesis should be testable and you should test it to see if it is valid.

Appendix 5: SCIENTIFIC METHOD

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M8/Introduction- Spectroscopy.pdf

Experiment 8 SPECTROSCOPY: ABSORPTION AND

EMISSION OF LIGHT

Purpose: To investigate the relationship between the color of light absorbed by a substance and its color and to apply Beer’s law to determine the concentration of a colored ion in a solution. Introduction Visible light, called white light, has wavelengths between 400 and 700 nm. When light hits a substance, it may be refracted, transmitted, or reflected in part or the whole. When transmitted light is at an angle different from the angle at which it entered the medium, it is said to be refracted (see Figure 8.1). If the incoming light (the incident beam) looks different from the transmitted or reflected light, it is because part of the light has been absorbed by the substance. The color we perceive of a solution is based on the light being transmitted through it; whereas, the color we perceive of an opaque sample (such as a solid) is based on the light reflected off the surface.

Figure 8.1

When white light is passed through a prism of a diffraction grating, it is dispersed into a continuous spectrum of colors according to wavelength: red (610-720 nm), orange (600-610 nm), yellow (570-600 nm), green (490-570 nm), blue (450-490 nm), indigo (420-450 nm), and violet (400-420 nm). The ions of transition elements form compounds that are often colored due to the fact that they absorb certain wavelengths of visible light and transmit others. The color observed is not due to the color of the light absorbed but to the color of the light that is transmitted. Various color models are used to explain the observed color. According to one—the subtractive color model—the color that a surface displays depends on which parts of the visible spectrum are not absorbed and therefore remain visible. The table below shows the colors observed (transmitted light) when complementary colors of light are absorbed.

incident beam

Subs

tanc

Experiment 8: SPECTROSCOPY: ABSORPTION & EMISSION

Table: Color and Wavelength of Light Absorbed and Transmitted Color of light absorbed

Color of light transmitted

Color of light absorbed

Color of light transmitted

Violet (400-440 nm)

Yellow-green (560-580 nm

Green (500-560 nm)

Red (630-750 nm)

Blue (440-480 nm)

Yellow (580-600 nm)

Yellow-green (560-580 nm)

Violet (400-440 nm)

Green-blue (480-490 nm)

Red-orange (610-630 nm)

Yellow (580-600 nm)

Blue (440-480 nm)

Blue-green (490-500 nm)

Orange (600-610 nm)

Blue-green (490-500 nm)

In a color wheel, the colors that are opposite each other are called complementary colors. Complementary colors are pairs of colors which cancel each other out when they are combined. Which pairs of colors are complementary depends on which color model is being used. According to the subtractive color model, complementary colors are red and cyan (greenish-blue), green and magenta (purplish-red), and blue and yellow. For example as seen on the color diagram below, a magenta (or purplish) solution will absorb green light while transmitting blue and red lights.

When a transparent colored solution is placed in the path of a beam of white light, the amount of light absorbed depends on the wavelength of the light, the thickness of the solution, and the concentration of the absorbing substance in the solution. This relationship is described by Beer’s law. Beer's law states that at a given wavelength, the intensity of absorption of a solution is directly proportional to the concentration of the solution. The intensity of absorption is called absorbance (A).

A = εlc where A = absorbance ε = molar absorptivity (or molar extinction coefficient) l = the path length (diameter of the cuvette) c = concentration of the solution

Experiment 8: SPECTROSCOPY: ABSORPTION & EMISSION

Note that ε (pronounced epsilon) is a unique property of the solute and is constant for a given substance. l is also a constant as all the cuvettes (glass or plastic tubes holding the sample being examined) have the same depth. Thus we can combine the two constants (ε and l) and rewrite the equation as

A = kc where k is a constant (ε times l)

You should recognize that this is an equation of a straight line:

y = mx + b where y = absorbance (A) x = independent variable, (the concentration, c) m = slope (k = ε times l), and b = y-intercept = zero

Thus, if we plot Absorbance versus Concentration we should get a straight line, and the slope is equivalent to ε x l. The equation also tells us that absorbance is directly proportional to the concentration (i.e. the more concentrated the solution, the higher is the absorbance, proportionally.) Note that absorbance (A) has no units, because all the units cancel as shown below. A = L mol-1 cm-1 x mol L-1 x cm Using absorbance and concentration data from an experiment, one can make a calibration curve of Absorbance versus Concentration. This calibration curve (as shown in Figure 8.2 below) can be used to determine the concentration of the absorbing species in a solution whose concentration is unknown.

In this experiment, you will use a spectroscope to examine the absorption of light by aqueous solutions of ionic compounds containing transition metal ions. You will propose a hypothesis as to the relationship between the color of the solution and the color that is apparently being absorbed. You will then devise a plan to test your hypothesis.

Figure 8.2

M8/M8 Data and Results Page_updated_3_2022.pdf

Experiment 8 SPECTROSCOPY: ABSORPTION AND

EMISSION OF LIGHT

Data Collection and Results Pages Name: ____________________ (Answers must be in full sentences.) Date: ___________________

INTRODUCTION

1. Open the simulation. Explore the Beer’s Law screen for a few minutes. Try to figure out what all of the controls show and do.

2. How does Concentration affect how much light is absorbed and transmitted through the solution?

INVESTIGATING ABSORPTION AND CONCENTRATION

3. Predict what a graph of absorbance versus concentration would look like. Sketch your prediction.

Experiment 8: SPECTROSCOPY: ABSORPTION & EMISSION

4. Choose a solution from the simulation and measure the Absorbance for 5 different concentrations on the preset wavelength setting. Then draw a graph using your values in the table.

Solution Chosen __________________

Data from Simulation

Concentration (mM) Abs

5. How does your second graph compare to your prediction?

Experiment 8: SPECTROSCOPY: ABSORPTION & EMISSION

6. Based on Beer’s Law (A = elC, where A = absorbance, e = molar absorptivity, l = pathlength, and C = concentration), do you expect using different wavelengths of light would change the way your previous graph looks? Why or why not?

INVESTIGATING ABSORPTION AND WAVELENGTH – SCIENTIFIC METHOD

7. Compare three solutions of different colors with the same pathlength (width of container).

Solution Solution Color

Beam Color

Value (nm)

Abs Beam Color

Value (nm)

Abs

Preset Wavelength: Simulation default setting

Variable Wavelength: Set to same color as solution

Experiment 8: SPECTROSCOPY: ABSORPTION & EMISSION

8. a) Determine a hypothesis to explain the relationship between the color transmitted and the color absorbed as related to the color wheel. b) State your hypothesis. 9. a) Determine a plan to test your hypothesis, including predicted results, that can be carried out using the simulation. b) State your proposed plan with predicted results. 10. Post your stated hypothesis and proposed plan with results into the Scientific Method – Discussion Board. After confirmation from instructor, perform your test using the simulation. State your observations. Does your test show that your hypothesis is valid? Explain.

M8/M8 Post_Lab Assignment .docx

Name __________________________

Module 8

Post-Lab Assignment

Short Answer (20 points)

Answer the following questions based on material covered in this module.

(You must use complete sentences when answering each question.

1-point deduction per question not answered in complete sentences.)

1) (9 points)

a) What is the color of the beam at wavelength 520 nm?

b) What is the color of the drink mix solution at concentration 100 mM?

c) What is the color transmitted through the solution?

d) What is the color absorbed by the solution?

e) Why would it be incorrect to say that the color reflected by the drink solution is pink (light red)? What is a better way of describing what the pink/light red represents?

2) (5 points) What does Beer’s Law tell us about the relationship between the absorbance and concentration of a solution? Does your 2nd graph show that Beer’s Law is valid? Explain.

3) (6 points) Explain briefly why each of these graphs of Abs vs Conc would not fit Beer’s Law. Specifically what part of the graph is not in agreement with Beer’s Law?

concentration