Fundamentals: Graphing

1 lab period; work in pairs. Complete the Preparation page before laboratory.

Goals

To learn

1. The purpose and advantages of graphical analysis of data
2. Some of the functional forms that occur commonly in chemistry
3. How to cast a functional relationship into linear form
4. The meanings of the terms dependent variable, independent variable, Cartesian coordinate system, origin, slope, intercept, x-axis, y-axis, abscissa, ordinate, interval size, subdivisions of the graph paper, numerical limits
5. How to properly construct a graph, whether by hand or by computer, including
   • Correct choice of the quantity to be plotted on each axis
   • Good choice of numerical limits for each axis, to bracket the experimental points and allow for easy interpolation of readings between interval marks on the axes
   • Correct plotting of experimental points, with each point marked so it can be seen
   • Correct method for obtaining the best straight line or smooth curve through the data points
   • Method for determining the numerical value and units for the slope and intercept of a linear plot

To learn the use of a spreadsheet for organizing, manipulating, and plotting data.

Scientists experiment. Normally, the goal of an experiment is to determine how the value of one variable, called the dependent variable, changes as the experimenter systematically alters the value of another variable, called the independent variable. For example, a chemist may be interested in how the volume, V, of a gas sample varies as its pressure, P, is changed; or in how the rate of a reaction changes with the concentration of one of the reactants. There are two common ways of presentin g the results of experiments like these: tabular form and graphical form. Although both methods are commonly used, and are often used together, the graphical method is generally superior because it gives the scientist a visual feeling for the relationship between the dependent and independent variables.

Graphs are usually plotted on a Cartesian axis system as in Figure 1. The vertical axis is called the y-axis (or ordinate) and the horizontal axis, the x-axis (or abscissa). Generally, the value of the dependent variable (or some function of it) is plo tted on the y-axis, and that of the independent variable (or some function of it) on the x-axis. We assume that you have some familiarity with the Cartesian coordinate system and know how to plot a point represented by the ordered pair (x,y) in such an axis system.

A plot of the dependent variable, y, versus the independent variable, x, can be either linear or curved. For example, a plot of temperature in ^oC versus the corresponding temperature in ^oF is linear (straight line). On the other hand, a plot of the solubility of a salt such as K₂Cr₂O₇ (potassium dichromate) versus temperature is curved. Although both linear and curved plots can be useful, the linear plot is generally more so, for several reasons. First, linearity or deviations from it are readily seen by eye. On the other hand one cannot tell visually if a curve that resembles a hyperbola is truly hyperbolic. For example, although the curve in Figure 2 looks hyperbolic, it is not obvious from inspection that it actually is. Second, linear plots may be readily extrapolated (extended) into regions which are experimentally not accessible. Extrapolation allows the prediction of the value of the dependent variable under conditions other than those governing the experiment. For these reasons, scientists prefer linear plots. This workshop will focus on these.

The equation governing a linear relationship between two variables, x and y, is given in equation 1, where y is the dependent variable, x is the independent variable, m is a constant called the slope, and b is a constant called the y-intercept.

These quantities are illustrated in Figure 3. The trick to constructing a linear plot is to cast the functional relationship between dependent and independent variables into the form of equation 1. For example, the relationship between the pressure, P, and volume, V, of a fixed amount of ideal gas at constant temperature is in equation 2.

where K = a constant. This equation does not have the form of equation 1. However, dividing both sides by P gives

Equation 3 has the form of equation 1 where y corresponds to V, x to l/P, m to K, and b to 0. A plot of y (V) versus x (1/P) should be a straight line with slope m (K) and intercept zero; i.e., the line should pass through the origin.

Graphs may be constructed by hand or by computer. A professional scientist or engineer is able to use both methods. The purpose of this workshop is to provide guidance and experience in both plotting methods. The general criteria for con struction of a graph are the same regardless of method. Properly used, they will enable you to produce a plot that has a professional appearance and is easy to read and interpret.

If we agree to use y to represent the dependent variable (the variable measured in the experiment) and x to represent the independent variable (the variable controlled by the experimenter), there are six functional relationships between y and x which occur often in chemistry. These are detailed below. In the following discussion, do not be concerned if you do not understand where the equations come from and what the quantities mean. For now, we are interested only in the forms of the equ ations and how these translate into plots.

Case 1. y = mx + b. The value of y varies linearly with the value of x.

The volume of a sample of gas at constant pressure is related in this way to the temperature in ^oC. Illustrative data are shown in Table 1.

Clearly V increases as T increases, so a plot of V versus T might be linear. Figure 4 shows such a plot, created using a computer plotting software package. If we put an eye down near the paper and sight along the points, it is clear that they fall alo ng a straight line. The computer has calculated and drawn the straight line which best fits the data. Note that as many points as possible lie near the line, with roughly an equal number above and below it. The line represents the functional relation betw een the dependent variable, V, and the independent variable, T. Slope and intercept, calculated by the computer using the method of linear regression analysis (or method of least squares), are shown in the diagram:

The relationship between volume and temperature for a gas is called Charles' Law. The units of the slope, calculated by dividing DV in liters by DT in ^oC are L/^oC. Intercept un its are liters. Several important features of Figure 4 are listed below.

1. The entire window (for a manual plot, the window is the sheet of graph paper) is used for the plot. Generally, the bigger the graph the better, subject to the constraints in 2.
2. The upper and lower axis limits of the variable being plotted (0.65 L and 0.35 L for the y-axis in Figure 4) are chosen with two constraints.
   1. The lower and upper limits must bracket the range of experimental values being plotted; i.e.,the lower limit must be £ the smallest experimental value of the variable, and the upper limit must be ³ the largest experimental value of the variable. Thus the lower limit of the y-axis in Figure 4 must be < 0.403 and the upper limit > 0.625. Bracketing should be tight.
   2. The correspondence between the number of units spanned between the lower and upper axis limits and the number of intervals (steps) along the axis must allow for convenient plotting and reading of points. This means that the number of variable units per basic interval must be 2, 2.5, 5, or 10. (These numbers are generic--depending on the order of magnitude of the numbers plotted, the decimal point could be anywhere--e.g., 0.2, 0.25, 0.5, and 1.0).
3. The lower limit of each axis need not be zero. Although zero is a convenient lower limit for the T axis in this case, choice of zero for the lower limit of V would cause all the points to be cramped into a small area of the graph, as in Figure 5. The requirement that the plot fill as much as possible of the window takes precedence.
4. Last, although some of the points miss the line (experimental error), we do not "connect the dots". Always put the best smooth curve or line through the points.

Constraint 2b is usually easier to satisfy when plotting is done by computer. For manual plotting, it is more difficult, because we are restricted by the physical layout of commercial graph paper, with basic intervals of either cm or inches. Each basic interval is further subdivided into 5 or 10 parts. The graph paper you bought for this experiment has the cm as the basic interval with each subdivided into 10 parts. This paper is described as "10 squares (or subdivisions) to the centimeter." The re are usually 24 cm along the long dimension of the paper and 18 cm along the short dimension. In plotting a graph, one must establish a correspondence between the number of units spanned by the variable being plotted and the number of basic intervals available along the axis. The values of t range from 0 to 150 ^oC. The lower x limit must be £0 and the upper limit ³ 150. However, there are 18 basic intervals along the x-axis. To set up the x-axis so that the left end of the line corresponds to 0 and the right end of the line to l50 would require 150 - 0)/18 = 8.33 ^oC per cm.

This is an inconvenient number, so constraint 2b is not met. To satisfy both constraints, set the upper limit of the axis at 180 ^oC. This exceeds 150 and gives a convenient number of variable units per basic interval, specifically, 10 ^o C per cm. Setting the x-axis up to span the range 0 to 180^oC is as close as we can come to using the whole length of the axis, while maintaining plotting convenience. Similarly, V spans the range between 0.403 and 0.625 L, and there are 2 4 basic intervals along the y-axis. The number of variable units spanned is 0.625 - 0.403 = 0.222. The ratio of this number to 24 is inconvenient. Therefore picking lower and upper limits such that a variable range of 0.240 L is spanned gives a convenient correspondence of 0.01 L per cm. The limits used for the y-axis in a manual plot would be set up this way. Finally, the straight line in the manual plot is usually drawn by eye. To calculate the slope in a hand-plot, choose two points which lie on the li ne, not two experimental points. The best straight line averages the data, so a slope calculated from two points exactly on the line is more accurate than a slope based on two experimental points. The two points chosen should lie at the upper and lower ex tremes of the plot, rather than near one another on the line. Again, this is to minimize error in the slope due to error in reading y values from the line. Finally, we choose two points whose x values are convenient--that is, which correspond with lines delineating the basic intervals of the graph paper. Normally this procedure will give values close to the regression values produced in a computer plot.

Whether plotting by hand or computer, setting convenient limits becomes easy with practice, and is one of the goals of this workshop.

Case 2. y = m/x + b. y is inversely proportional to x.

A relationship of this type is that between the volume, V, of a sample of gas and its pressure, P, at constant temperature. Illustrative data is in Table 2.

As P increases, V decreases, suggesting an inverse relation. A linear plot might be obtained by plotting V versus 1/P. Calculated values of 1/P are given in the 3rd column of Table 2, and a computer plot of V vs. 1/P is in Figure 6. Upper and lower variable limits of the axes have been chosen by the guidelines above. The plot is clearly linear. The slope, calculated by regression analysis, is 0.368 L-atm. The intercept is zero, within experimental error, so the equation relating the variables, called Boyle's Law is

Case 3. y = mxⁿ, with n an integer. y is proportional to x raised to an integral power.

If n = 1, this equation reduces to case 1; if n = -1, it reduces to case 2 with b = 0. This type of functional relationship occurs frequently in chemical kinetics, where the rate of reaction is often proportional to an integral power of the concentration of a reactant. There are two steps to the graphical analysis of data related by a function of this form.

1. A plot of ln y vs. ln x is constructed. The reason for this is seen by taking the natural logarithm of both sides of the equation, y = mxⁿ:
   
   ln y = ln (mxⁿ) = ln m + nln x (6)

   A plot of ln y versus ln x should be linear, with slope n.

   There are two systems of logarithms in common use: logarithm to the base 10 (abbreviated log₁₀); and natural logarithm (abbreviated ln). Natural logarithm is also called logarithm to the base e, where e is the number 2.71828... . These logarithms are defined as follows:

   log₁₀ x is a number y such that 10^y = x.
   ln x is a number z such that e^z = x.

   It can be shown that the relationship between the two logarithmic scales is simple:

   ln x = 2.303log₁₀ x

   Once n has been determined,

2. y is plotted against xⁿ. m may be determined from the slope.

Table 3 shows the variation of the rate of the reaction in equation 7 as a function of the concentration of NOCl in moles per liter.

2NOCl ® 2NO + Cl₂ (7)

Assuming this data obeys y = mxⁿ, we first plot ln y vs. ln x to obtain n. Values of ln (Rate) and ln [NOCl] are in Table 4.

A computer plot of ln (Rate) vs. ln [NOCl] is shown in Figure 7. The regression slope is 1.997, very close to 2, so n = 2. We now return to the original data and plot Rate vs. [NOCl]² to obtain m from the slope.

Values of [NOCl]² are in column 3 of Table 3, and a plot of Rate versus [NOCl]² is in Figure 8. The slope of this plot is 3.96 x 10^-8 L/mole-sec. The original data obey equation 8.

Before proceeding, we should note something about the graph in Figure 8. The numerical values of rate (Table 8) are very small, involving an exponential factor of 10^-8. It is clumsy to write such small numbers along the axis of a graph. To avoid this, scientists label the axis using only the argument of the number (e.g., the argument of the number, 4.80 x l0^-8 is 4.80). The exponential term is acknowledged in the axis label. This has been done in Figure 8, where the vertical axis is labeled, Rate (moles/L-sec) x l0⁸. The number, l0⁸, is the number by which all the experimental rates were multiplied to convert them to the argument, which is actually plotted. Thus

For plots in which the numbers on one or both axes have implied exponentials, allow for these in calculating slope and intercept manually. Please see your instructor with questions about this.

Case 4. 1/y = mx + b.

Again, this form is common in kinetics. For a second-order reaction, the concentration of the reactant varies with time according to equation 9, where t = time elapsed since the beginning of reaction, k = the second-order rate constant, and [reactant]_o is the concentration of reactant at t=0, the beginning of the reaction.

A specific example of a second-order reaction is in equation 10. Table 5 shows how the concentration of C₄H₆ (butadiene) varies with time.

The third column lists calculated values of 1/[C₄H₆], which are plotted against time in Figure 9. The plot is linear, with slope 0.0141 L/mole-s and intercept 59 L/mole. From the intercept, [C₄H₆]_o = 1/intercept = 0.0170 M.

Case 5. ln y = mx + b.

Kinetics again provides examples of this type of function. For first-order reactions, concentration of reactant varies with time according to equation 12.

where t = time elapsed since the beginning of reaction, k is the first-order rate constant for the reaction, and [reactant]_o is the concentration of reactant at t=0.

An example of a first-order reaction is eqn 13.

Concentration-time data are in Table 6.

Values of ln [N₂O₅] are in the third column of the table, and a plot of ln [N₂O₅] vs. t is in Figure 10. The plot is linear. The rate constant, k (slope), is 7.0 x 10^-4 sec^-1 (be sure you understand how units are determined). ln [N₂O₅]_o (intercept) is 1.609.

Case 6. This is a very common functional relation in chemistry.

Some examples include how the equilibrium constant, K_eq, for a reaction, the rate constant for a reaction, and the vapor pressure of a pure liquid vary with temperature. Consider a particular example.

For most chemical reactions, the rate constant varies with Kelvin temperature according to eqn 15.

Here k is the rate constant, T is the temperature, E_a is the activation energy for the reaction, A is the frequency factor for the reaction, and R is the gas constant. (Do not worry about where this equation comes from or what all of the quantities mean--focus on the functional form and the graphical analysis).

Table 7 shows the rate constant as a function of temperature for reaction 16.

To plot ln k versus 1/T (K), we must a) convert temperature to the Kelvin scale; b) reciprocate the Kelvin temperatures; c) find the natural logarithms of the rate constants. The results are in Table 8.

A plot of ln k vs. 1/T is in Figure 11. Slope and intercept are

Since slope = -E_a/R, and R = 8.314 J/mole-K, we calculate E_a as follows

Since intercept = ln A, we calculate

In all of the previous examples, zero was a convenient lower limit for the range of the independent variable. In this example, however, zero is not a convenient lower limit for the abscissa if we want to utilize the full dimensions of the window. Since the zero of the abscissa does not occur where the two axes meet, the y-intercept cannot be obtained manually by extrapolating the linear plot to intersect the y-axis. In cases like this, the value of the y-intercept must be calculated. The standard form for a straight line, equation 1, can be rearranged to equation 17.

First, the slope is calculated from the best-fit line for the data. Then a single point on the line is chosen, and b is calculated by 17.

The intercept in Figure 11 may be calculated this way.

Choosing the point for which ln k = -7.23 and 1/T = 1.7 x 10^-3, we obtain

This completes our survey of common linear functional forms. Although other forms occur, familiarity with those discussed here will serve you well.

Before concluding this section, we summarize the practical aspects of constructing and analyzing plots.

1. For hand plots, use regulation graph paper, preferably ruled with 10 subdivisions per centimeter.
2. Decide how to plot the experimental data. If possible, plan a linear plot. How you plot the data (e.g., as y vs. x, y vs. 1/x, ln y vs. 1/x, etc.) depends on the expected relationship between the variables.
3. Perform calculations consistent with your chosen plotting method. For example, if you decide to plot the data as ln y vs. 1/x, calculate ln y and 1/x for each data point.
4. For a hand plot, set up a sheet of graph paper:
   • Darken the coordinate axes using a straight edge. Use the left vertical line as the y-axis, and the bottom horizontal line as the x-axis.
   • Label the axes with the quantities to be plotted.
   • Decide on the numerical limits (lowest and highest values) to appear on the axis for each variable.
     1. The limits should bracket the experimental values
     2. The limits should be chosen so there is a convenient number of variable units per basic interval of the graph paper.
     3. It is not necessary to start with zero on each axis. In many cases this causes compression of the data points into too small a region of the graph.
     4. Within constraints 4c 1 and 2, use as much of the window (graph paper) as possible for the plot.
5. Plot the points according to the chosen format (step 3). In a hand plot, circle each point to indicate its location, and draw the best straight line or smooth curve through the points. Use a straight edge or french curve.
   • Never "connect the dots." Not all points fall exactly on the straight line or smooth curve, since each point has experimental error. Deviation of a point from the best straight line or curve gives a measure of the error in the point.
6. If the graph is linear, determine the slope and intercept. For a computer plot, use linear regression analysis. For a hand plot:]
   • Slope =Dy/Dx. Use two widely separated points on the line.
   • Intercept. If the lower limit of the abscissa is zero, extrapolate (extend) the straight line to intersect the y-axis. The y value at the intersection is the intercept. If the lower limit of the abscissa is not zero, the intercept must be calculated: b = y - mx. Choose any point (x,y) on the line. Use x, y, and m to calculate b.

Focus Questions

What is the mathematical relationship among the four variables in your data set?

Equipment and Materials

Bring the following items to lab:

• 2 pieces of graph paper, preferably K & E type, with 10 subdivisions per centimeter.
• a straight edge
• several sheets of notebook paper
• one electronic calculator
• 2 sharpened pencils

Experimental

You will be given a table of experimental data relating the values of four variables. Your goal during the laboratory period is to use graphical methods to determine the functional relationship among the four variables. When you think that you have successfully achieved this, show the graphs to your instructor and explain how you used them to determine the functional relationship. S/he will point out errors, which must be corrected before leaving the lab.

During the two days following your lab period, use a spreadsheet to reproduce the plots that you made manually in the laboratory. No later than 4PM on the second day following your lab meeting, turn in your manual and computer plots and any attendant calculations.

Disposal Methods

No disposal required.

Preparation
Fundamentals: Graphing

Preparation Questions.