Read this PURPOSE section first to understand WHY we are doing what we are doing today.
In most cases, line graphs are the most useful for scientists.
If data points lie roughly along a straight line, the x and y variables have a linear relationship or are directly proportional. This means as one variable increases, the other does too, in a constant proportion – as x doubles, y doubles; as x triples, y triples; etc. Directly proportional quantities, x and y, relate to one another through mathematical equations of the form y = mx + b, where m is a constant and b is zero. The equation for the directly proportional linear relationship is y = kx. Here m=k and b=0. For an indirect relationship, one variable increases while the other decreases at a constant rate. The mathematical expression for an indirect relationship is y = -kx.
If data points lie along a curve that drops from left to right, then the quantities have an inverse relationship or are inversely proportional. In an inverse relationship, one quantity increases as the other decreases. The mathematical relationship that expresses an inverse relationship is . The expression relating gas pressure and volume follows the expression PV=k. Note that an inverse relationship is nonlinear because the increase of one variable is not accompanied by a constant rate of decrease in the other variable. In summary:
k = the proportionality constant.
If two quantities are directly proportional k = y/x.
If two quantities are inversely proportional k = xy.
Purpose: to determine the relationship (linear or inverse) between mass and volume of aluminum; to determine the relationship (linear or inverse) between the pressure and volume of a gas
Procedure: Use Graphical Analysis program for entering and analyzing data. (All of the processes described below are for the Vermier Graphical Analysis program.)
First, install the program or decide what you will use.
- It is suggested you install the Vernier Graphical Analysis program. You can find the version for your device here. https://www.vernier.com/downloads/graphical-analysis/
- However, you could use any graphing program, even Excel, or just your hand and some graph paper.
TRY IT: A Best-Fit Straight Line
When Graphical Analysis opens, select Manual Entry. A blank data table will be on the right side of the screen. There are 3-dot menus next to the X and Y column headers. Click those and then click Column Options to change the name of each axis.
- Enter “Volume” for the x axis title and “cubic centimeters” for the units.
- Enter “Mass” for the y axis title and “grams” for the units. Enter the data from the data table below for the mass and volume of aluminum samples.
Enter the following data in the blank data table. The graph will instantly appear as you enter values.
|Volume (cm3)||Mass (g)|
After you have entered all the values, click on the Graph Options icon in the bottom left corner and click Apply Curve Fit. Select Linear and click Apply. This will put a linear info box on the graph showing the slope (m) and the y-intercept (b). This button also draws the best fit straight line for the data points (called a regression line).
Click the Graph Options icon again and scroll down to the bottom. Click Edit Graph Options to save the graph with a name. Title the graph “Volume vs. Mass.” Titles on science graphs are usually in the y vs. x format. Also on this screen, uncheck “Points.” This way the graph shows the best fit line for the data points and not a dot-to-dot line.
TRY IT: A Best-Fit Curved Line
Save the graph. Close it. And open a new graph and data table.
Enter the values for the pressure and volume of a gas as given below and create labels and units for the axes as well as a title for the graph.
|Pressure (atm)||Volume (L)|
Notice that the line does not look like a straight fit. The best fit for this line is Inverse. Click on Curve Fit, select Inverse, and click Apply. For this particular curve, the inverse relationship works best; this is .
Save a copy of this graph.
Answer the following questions, then check your answers by clicking and dragging your cursor (or double-clicking) over the space after the word “Answer.”
1. What is the slope for the first graph?
2. What type of relationship is indicated in Graph 1?
3. Write the equation for the line in the y=mx + b format, substituting the slope for m and zero for b.
Answer: y = 2.786x + 0
4. If the mass of the aluminum sample was 17 g, what volume would this correspond to? Read this from the graph. This is called interpolating data. Finding a data point outside the data range is called extrapolating data.
Answer: about 6.0-6.5 cubic centimeters
5. Using the equation from question 3, solve for the volume (x) when the mass (y) is 17 g.
Answer: x = 6.1 (Substitute 17 into the y = mx + b equation.)
Does this answer agree with what you read from the graph? If it does not, recheck your math or the graph.
Answer: yes (If they don’t, go back and check your values and your math.)
6. For Graph #2, what type of relationship is indicated?
7. Multiply each P and V point together to get a number. Average these values to get a constant “k”.
8. If the equation for Graph 2 is PV=k, rearrange this equation for V.
Answer: V = k/P