In chemistry, as in many other disciplines, graphs provide an important and efficient way of representing the numerical data gathered during an experiment. A graph is essentially a pictorial way of presenting sets of numerical data. Graphs allow an experimenter to easily observe and interpret trends or relationships in the gathered data. As we will see in this experiment, it is much easier to discern a relationship between two quantities when they are graphed than when the data are posted as just columns of numbers. In this particular experiment, you will perform two short laboratory exercises and learn how to properly construct a graph that will give a visual representation of the gathered data.
There are many different types of graphs which are seen every day in newspapers, magazines, and even on the Presidential Address to the Nation. Examples of such graphs are line graphs, pie charts, bar graphs, and many others. The most common type of graph used to illustrate physical science experiments is the line graph (or xy-graph or scatter graph) that shows the relationship between two different variables. Below are listed some simple conventions for the preparation of these line graphs.
To make a graph there are several rules that you need to follow.
The independent variable, i.e., the variable which you can control or which changes in a known way, is plotted on the horizontal or x-axis (known as the abscissa). For example, time proceeds linearly (at a constant rate) and is often the independent variable in physics experiments. The dependent variable, whose value depends upon the independent variable, is plotted on the vertical or y-axis (known as the ordinate).
1. A graph should be drawn on good-quality graph paper (NOT the paper in your lab notebook) with the smallest squares possible (10 mm/cm or 20 lines/inch is recommended). Graphs drawn on lab notebook paper will not be acceptable. Graphs should be drawn neatly in INK. Use a ruler for the drawing of any straight lines.
2. Each graph should have a descriptive title which helps to identify the nature of the experiment or place the graph in a context. For the example below your title might be, "Density of Liquid Mercury" or "Mass of Liquid Mercury As a Function of Mass". Avoid "Y vx. X" titles such as "Volume vs. Mass," since they merely repeat information which is already shown on the axes and donít tell the reader enough about the graph he or she is studying. When one than one graph is present, also include a figure number as part of the title.
3. The most difficult decision in plotting a graph is the selection of the scales. The scales of the graph should be chosen such that the data points that you are plotting will fill most of the page. By increasing the size of the graph, the data points can be located more precisely and all of the graph will contain useful information. It is not necessary that the axes begin at zero unless there is some need to show the intercept. Do not number every major division on the graph, as this will make it appear crowded and reduce legibility. Instead, number about 5-6 major divisions and place tick marks on minor divisions.
4. Both axes should be clearly labeled with the name of the quantity plotted and the UNITS.
5. The data points themselves should be quite small (for precision in location), but should be surrounded by a small circle to make it clearly visible.
6. If the data points appear to fall along what seems to be a straight line, use a ruler to draw a single straight line that DOESN'T go from point to point, but is a best AVERAGE fit of all the data points on the graph. In other words, if there is some experimental scatter, draw the line such that equal numbers of points appear on each side of the line. This is called a best-fit line.
7. If instead the data points do not represent a linear relationship but seem to follow a curve, then you should try to draw the best smooth curve through the points on your graph. Once again, if there is some experimental scatter, you should have equal numbers of points on each side of the line.
8. If the graph is
linear, we find the slope of the graph by picking two points on the
line and finding the difference between those two selected y-values
divided by the difference in their x-values. Another way of saying the
same thing is that the slope is the change in y divided by the change in
x (or rise/run). We can write this mathematically in the following form.
If you pick two points on your line (x1,y1) and (x2,y2)
then the slope is equal to;
slope = m = D y / D x = (y2 - y1) / (x2 - x1)
Be sure that the two points which you choose lie on the line (they should not be experimental points, but simply some convenient point, e.g., one that lies at the intersection of two lines on the graph paper). Also, choose two widely-spaced points.
To make this a bit clearer look at a simple example below. Let us say that you carefully measure out a series of volumes of liquid mercury and then weigh each sample which you have measured out. The data have been recorded in the table below and then they have been plotted in a line graph as shown. (Did you notice that the word data is is plural? The singular of this word is datum.)
Table I. Measurements
on Liquid Mercury
|Volume Hg, mL||Mass Hg, g|
Click HERE to see the example graph.
In this experiment the volume of the substance is the independent variable, the variable which you can control, since you picked the different volumes to measure out. It is thus plotted along the abscissa or x-axis. The mass, in this experiment, is dependent on the particular volumes you picked and is plotted along the ordinate or y-axis. We can see from the graph that there seems to be a trend in the data, namely, that as the volume increases the mass also increases by the same proportional amount. Using those data points a "best-fit" line is then drawn which represents an average of all the data points on the graph. If the graph is linear, as it is in this case, we say that there is a direct, or linear, relationship between the volume and the mass of the substance in this experiment. Note that, since the slope of the line has dimensions of mass divided by volume, it gives the best average value of the density of mercury without having to calculate individual values for all of the data points.
Look carefully at the graph and notice;
a. The descriptive title.
b. Both x and y-axes are labeled with the quantity and correct units (units very important).
c. The data points fill up the entire graph; they are not all bunched together in one corner.
d. The data points do not fall exactly along a straight line but seem to represent a linear relationship. The line that has been drawn to represent the average value of those points, does not go dot-to-dot. Some points lie on the line, some above, some below.
e. Notice that if we want to calculate the slope of this line we pick two points on the line relatively far apart. Be sure that these points are NOT the original data points. Let us say that we pick the points, (3,42 ) and (11, 145). Then slope would be equal to;
(145.0 - 42.0) grams / (11.0 - 3.0) mL = 12.9 g/mL
The slope of the line in the graph above is equal to 12.9 g/mL. Once again do not forget to add the units to the value of the slope. Why would we pick points on the line but not an actual data point? As discussed in class and in Chapter 1, this number is the approximate density of liquid mercury at room temperature in grams per milliliter. Heavy stuff!!!
PRE-LAB QUESTIONS. Answer in your lab notebook or on ordinary paper, as your instructor designates, before coming to lab.
1. List four
important points to consider when plotting a graph.
2. What are the three major chemical components of air?
3. In this experiment you will be working with pennies that are made, at least partially, out of copper. Why are pennies made out of copper instead of another metal, e.g., iron? List at least three reasons why this is practical.
PART A. - Weighing of Air
In this part of the experiment you will be weighing a plastic 2-liter bottle in which is contained air under different pressures. As discussed in class, air is a mixture of different molecules, all of which have mass. Even though we can not go into the lab and weigh every molecule individually we will, in this lab, examine the relationship between the mass of these air molecules inside the bottle and the amount of air pressure within the bottle.
To measure the mass you will be using the digital balance located in the chemistry laboratory. Your instructor will briefly explain how to use this type of balance at the beginning of the lab. There are several rules that should be observed when measuring the weight of an object. First, be sure to use the same balance for all of your measurements in one experiment. Different balances are most likely a "bit" different in their calibration. If you randomly select any balance to make your measurement then you could possibly introduce an error into your experiment. This is an easy error to eliminate if you are paying attention. Second, it is important to be sure that your digital balance reads zero before every measurement. If it does not read zero then you need to press the TARE (or ZERO) button on the balance. This serves to calibrate the balance every time and also will help eliminate a possible error. Record your mass data to the full precision (number of decimal places) available on the particular balance which you use.
1. Obtain a plastic 2-liter bottle from your instructor. The cap has been fitted with a tire valve so that you can easily fill the bottle with air. Be sure the cap fits tightly onto the bottle.
2. You will be using a standard tire gauge to measure the pressure in the bottle. If you do not know how to read this gauge please ask the instructor and he or she will explain the process. Remember the gauge measures pressure in pounds per square inch or (psi). In all of your pressure readings, you should estimate one additional digit than the smallest marked interval, i.e., to the nearest 0.1 psi. If you hear a hissing noise as you make the measurement, it means that air is leaking out past the gauge and your reading will be inaccurate. Press the head of the gauge more tightly against the valve stem of the bottle.
3. Attach the hose from the hand pump or compressor tank to your bottle valve and increase the pressure in the bottle and then measure the pressure with the tire gauge. You should put around 40-50 psi of pressure in the bottle.
4. Weigh the 2-liter bottle and record the mass in your lab book. Wait 5 minutes and then weigh the bottle once again. If the mass is the same as it was the first time then you know the cap is sealed tightly and the experiment can be started. If the mass of the bottle has decreased by more than 0.05 grams, then tighten the cap and repeat steps 3 and 4.
5. If the mass of the bottle and the air inside is the same, you can continue the experiment.
6. Now carefully measure the air pressure in the bottle using the tire gauge and record the value in your lab book. Now immediately weigh the bottle and record the corresponding value.
7. Reduce the air pressure in the bottle by a small amount by pressing on the valve stem. Measure and record the pressure and the corresponding mass once again.
8. Repeat step 7 until you have at least 5 sets of measurements of pressure and the corresponding mass. Your last reading should be somewhere around 10 psi.
9. Now let all the air out of the bottle so that the pressure in the bottle is the same as the surrounding atmosphere. Weigh the bottle once again.
10. Do the experiment
once again, repeating steps 3 through 9.
GRAPHING YOUR DATA - Weighing air
In Part A of the experiment you adjusted the amount of air that was in the bottle by changing the pressure. Each time you changed the pressure the bottle was then weighed to find its corresponding mass. In your lab book you should have two columns of data, one column of pressure in units of psi and another of mass in grams. Which one of these variables did you control? You controlled or set the pressure that was to be measured by letting out differing amounts of air from the bottle. Since this was the variable you controlled, we call it the independent variable and it will be plotted along the horizontal axis or x-axis. The mass, on the other hand, is the dependent variable; it is dependent on the amount of air in the bottle and thus the pressure. Since it is the dependent variable it will be plotted along the y-axis of our graph.
a. Carefully look at the range of pressures you have recorded. What is the highest? What is the lowest? For the horizontal axis on your graph, plot this range of pressures such that your data points will cover over most of the page.
b. Now look at the range of mass measurements that you have collected. These values are your dependent data points and will be plotted on the y-axis. Carefully choose an appropriate scale for the x-axis.
c. Once you have your axes labeled, carefully plot the data points on your graph. Use a sharp pencil to plot the points as precisely as possible, lightly at first and then, once you have confirmed that they are in the correct location, you can use ink so that they are easier to see.
d. The most important step in making a plot of your data is analyzing the points once you have graphed them. Look at the points carefully. Does there seem to be a linear relationship? If so, carefully use a straight edge and draw the best average fit through the data points. Most likely the line will not go through all the points. If this is the case then there should be equal number of points above and below the line.
e. Now determine the slope of this line. Select two points on the line (not data points) and widely spaced (near either end of the line) and carefully read off the x and y values for each. Use these points to calculate the slope of your line. Be sure to include units.
PART B. - WEIGHING OF PENNIES
Pennies have been around for over a century, and for most of those years they have been made out of pure copper metal that gives them their characteristic color. Around 1980, it was deemed too expensive to make a penny out of 100% copper, so the U.S. Mint combined copper with the much cheaper metal zinc. This has the advantage of retaining the characteristic color of the penny, while making it much cheaper to manufacture. In part B of this experiment you will weigh a series of pennies from different years and see if you can pinpoint the year this change in composition occurred.
Located in the lab is a "pile" of pennies. You will have to share this pile, so play nicely and return the pennies as soon as you are done. Select from this pile a series of pennies with a wide distribution of dates. You will want at least 10 pennies from before 1980 and at least 10 pennies after that date. Remember, we do not know exactly when the switchover took place. Hopefully, you will be able to discover the answer to this from your data. In addition to the above, pick several pennies that were made during the same year.
Make a table in your lab notebook in which you can record the date the penny was made and the mass of each. If needed, refer to the introduction to learn how to use the digital balance.
GRAPHING YOUR DATA - Weighing pennies
In Part B of your experiment you will be plotting the year of the penny on one axis and the mass of each penny on the other. Which one do you think should be on the horizontal axis? In this experiment the independent variable is the penniesí year and should be placed along the x-axis. The dependent variable is the mass of each penny and is placed along the y-axis accordingly.
a. Carefully look at the range of years you have recorded. What is the highest? What is the lowest? For the horizontal axis on your graph, plot this range of years such that your data points will cover the entire page.
b. Now look at the range of mass measurements that you have collected. These values are your dependent data points and will be plotted on the y-axis. Carefully choose a convenient set of units to be used along both axis.
c. Once you have your axes labeled, carefully plot the data points on your graph. Use a pencil to plot the points on the page, lightly at first and then, once you have confirmed that they are in the correct location, you can go over them with ink.
d. The most important step in making a plot of your data is analyzing the points once you have graphed them. Look at the points carefully. Does there seem to be a linear relationship among some of the points? Do some of your data seem to represent a straight line? How many straight lines are expected to appear on this graph? Carefully use a straight edge to draw the best average fit through each set of data points. Most likely the line will not go through all the points. If this is the case then there should be equal number of points above and below the line.
e. Think about what your graph should look like. Can you pinpoint on your graph the year that the U.S. Mint started adding zinc to the copper in pennies? How do you make this decision? Is there any uncertainty in your conclusion? Why or why not?
FINAL QUESTIONS . Answer at the end of your discussion section.
1. Are you satisfied that your graph of air mass vs air pressure does indeed give a straight line relationship? Explain why.
2. What is the slope of your graph in Part A? Donít forget the units!!
3. Name three experimental errors that could have had an effect on your graph in Part A.
4. After looking at your graph, in what year did they change the composition of the penny?
5. What is the average mass of a pure copper penny? What about a copper/zinc penny?
6. What does your graph tell you about the weight of zinc compared to copper?
7. Take a look at the pennies you weighed that were made the same year. Why is their weight different? Can you explain this difference?
8. How could you use
the data gathered in this experiment to find the density of pure copper?