Organizing and Analyzing Experimental Data

Transcription

1 CHAPTER 1 Organizing and Analyzing Experimental Data The Role of Models in Science Some of the physical, chemical, and biological systems we would like to investigate are too large, too small, too expensive, or too dangerous to have in our laboratories. In these cases, scientists and engineers turn to models. Models are a representation of the real thing, but of a size and cost that we can handle and afford. By studying the model, we can make some predictions about the behavior of the real thing. Physical models are used often in engineering. For example, models of the space shuttle were flown many times in wind tunnels before the shuttle s design was finalized and the shuttle was constructed (Figure 1). Automobile designers construct clay cars, and architects construct cardboard houses to help builders visualize the houses before construction begins. Figure 1. Molecular models help students and chemists understand the interaction between molecules and are often used to design therapeutic drugs (Figure 2). Behind each of these models, however, is a set of measured numbers from which the model is developed and from which predictions of the model s behavior are made. An equation is in fact a mathematical model, a statement of a perceived relationship between variable X and variable Y. The great advantage of mathematical models is the ability they provide to predict the behavior of a system beyond or between what we have observed or measured. One of the easiest ways to develop a mathematical model involves graphing of experimental data. 1

2 Chemistry 143 Experimental Chemistry H N O H N N N H N C-1 N N H N H O C Å Figure 2. Safety Precautions Wear your safety glasses at all times during this lab exercise unless instructed otherwise by your laboratory assistant. Materials Equipment Lab goggles 125 ml Erlenmeyer flask, 10 ml graduated cylinder, and a cm ruler (from your locker) Tygon tubing, plastic Beryl pipette, permanent marker, and electronic balance (supplied in lab) Thermometer, ball bearings, #4 stopper/single hole, thermistor temperature sensor, and caliper (check out from stockroom) Experimental Procedure This exercise consists of two parts. In both parts we will have pre-collected data that we will need to hand-enter into the MicroLAB software and analyze it. You will personally perform mini-experiments, where you will collect some data, solve for a mathematical function that fits the data, and finish by testing your mathematical model with actual experimental data. In Part I we will learn how to use a graph and the linear regression function to find a mathematical relationship between two simple variables in this case, the relationship between the Fahrenheit and Celsius temperature scales. We will then use this same approach in collecting other data and make some predictions about how the pressure of atmospheric gas changes with respect to temperature. Not all cause-and-effect systems are linear, so we look at some other data and learn how to use other mathematical functions to predict their behavior. Part II of the exercise will explore examples of nonlinear relationships that we will learn to fit with a number of mathematical functions. We will do this by hand-entering data, creating graphs, and performing curve-fits to find which functions offer the best fit. You will also apply 2

3 Organizing and Analyzing Experimental Data: The Role of Models in Science Chapter 1 this knowledge in mini-experiments. This will challenge your ability to fit nonlinear cause-and-effect relationships with a mathematical model, which will help you to predict experimental data. You will evaluate your math model by comparing its prediction with data measured to see if you were correct. There is an (optional) exercise at the end of the chapter that is an alternative approach to solving for mathematical relationships between two variables. Be sure to check with your lab instructor to see if this exercise applies to you. Part I. Using Graphs to Identify Mathematical Relationships Two major temperature scales exist in use throughout the world, the Fahrenheit and Celsius scales, as illustrated in Figure 3. Fahrenheit based his scale on the freezing point of water at 32, and the boiling point at 212, because he wanted body temperature to be 96 and wanted a temperature scale that could be divided into twelfths. The Celsius scale is based on the freezing point of water at 0 Celsius and the boiling point of water at 100 Celsius. The Celsius temperature scale has been adopted by all the world except the United States. As a result, in the U.S., we must often convert back and forth between the two temperature scales. How can this be done? Is there a simple mathematical relationship between the temperatures expressed by these two scales? Fahrenheit Celsius Hayden-McNeil, LLC Figure 3. Different scales. To explore this relationship, the experiment illustrated in Figure 4 was performed. We will use this data to construct a graph. We can use the linear regression function of the spreadsheet to draw the best-fit line through our experimental data points and to express the mathematical relationship between the x-value (horizontal axis) and the y-value (vertical axis). 3

4 Chemistry 143 Experimental Chemistry F C Fahrenheit Celsius Fahrenheit Celsius Fahrenheit y = mx + b? 20 Celsius Data Model Prediction Hayden-McNeil, LLC Figure 4. Fahrenheit and Celsius relationship. Please enter this data from Figure 4 into the MicroLAB spreadsheet by using the Hand Enter mode, press Accept Data, then drag and drop Fahrenheit to the y-axis and Celsius to the x-axis to construct the graph. (Refer to your measurement manual, page 44.) The fact that the data points appear to form a straight line is an indication that a mathematical relationship exists between the two variables. When data graphs form a straight line, we can write an algebraic equation relating the two variables. One form of this relationship is described by the slope-intercept equation: y mx b In this equation, y is called the dependent variable and is plotted on the vertical axis; x is known as the independent variable and is plotted on the horizontal axis; m is the slope of the line (Dy/Dx, or rise over run); and b is the y-intercept, the value of y when x equals zero. 4

5 Organizing and Analyzing Experimental Data: The Role of Models in Science Chapter 1 Figure 5. Fahrenheit vs. Celsius relationship. Click on the MicroLAB data analysis button, Analysis, at the bottom of the graph view. Choose the Add a curve fit from the Graph Analysis Tools pop up window, press OK, choose Degrees Fahrenheit for the y-axis and press OK. Place a linear regression or First Order equation on this set of data. This spreadsheet is illustrated in Figure 5. The linear regression is also called a least squares curve fitting technique. The best straight line is drawn through the data, and then the distances between each point and the line are determined. Some points will be above the line, and this distance will be positive; and some will be below the line, and this distance will be negative. However, if these distance values are squared, all of the signs will become positive. We then adjust the slope and position of the line until the total of all of the squares of these deviations from the line are minimized, hence, the term method of least squares. The computer will do this for you. The Correlation Coefficient The correlation coefficient is a measure of the quality of fit of the data that is observed with respect to the curve fit you have chosen. A perfect fit produces a correlation coefficient of either plus or minus Thus, the closer the correlation coefficient is to 1.000, the better the relationship between the experimental data and the mathematical function that was chosen in the curve fit. Prediction from the Model The slope-intercept equation is a mathematical model of the relationship between the Fahrenheit and Celsius temperature scales. From this model we can predict the Fahrenheit temperature that corresponds to any Celsius temperature using the regression equation. 5

6 Chemistry 143 Experimental Chemistry F C 1 32 After you have performed a curve fit, the equation shows up in the Data Sources and Formulas view under the ANALYSIS title. Right click on the correct equation and choose Interpolation/Extrapolation in order to predict values from the curve fit equation. Try a few different types of curve fits. Predict values by going to the predict location and placing a value in the x Value box and pressing Predict. The program will solve for y and display it in Predicted y value. With this, you can solve for Fahrenheit values if you know Celsius values. However, if you wish to convert Fahrenheit to Celsius, you can check the box that says, Enable Reverse Prediction (x from y), then you can enter a y Value, press Predict, and a Predicted x Value will be calculated for you. A Mini-experiment for You to Try Background As a constant volume of gas is cooled or heated, the pressure is affected. In this experiment, air is trapped in a 125 ml Erlenmeyer flask. The top of the flask is sealed with a #4 stopper and tube system, where the tube is connected to the pressure sensor on the back of the MicroLAB interface. See Figure 6 for apparatus setup. You will record the various pressures and temperatures of this system, solve for a mathematical correlation between the two variables, and test your math model by predicting data and checking against experimental data. Experimental Procedure A) Configuring the Computer Software Please choose the thermistor temperature sensor from the Add Sensor button in the Data Variables and Sources view and calibrate it with Celsius thermometer. Next, choose the pressure sensor in the same manner and press Finish without calibrating it. You will be recording the signal in atmospheres. Finally, choose the Keyboard from the Add Sensor, press the Next button, and type Enter degrees Celsius in the field provided. After configuring the temperature sensor and pressure sensor, you will drag and drop the pressure, temperature, and Keyboard variables into the Spreadsheet and Digital Display. Configure your pressure sensor to read 4 places past the decimal and your temperature sensor to read 2 places past the decimal. You will then place the pressure variable onto the y-axis and the Keyboard variable onto the x-axis and press Start. B) Collecting the Experimental Data Immerse your Erlenmeyer flask in different temperature water baths, as shown in Figure 6, while monitoring the output from the pressure sensor and the temperature from the thermistor temperature sensor in the digital display. Be sure to stir the water bath and add ice or hot water to maintain a constant water temperature. Once you observe that the temperature of the water bath and the pressure readings have stabilized, enter the temperature values, in degrees Celsius, via keyboard entry and press OK. Be sure to record four points, including the ice-point of water, room temperature, at least 10 6

7 Organizing and Analyzing Experimental Data: The Role of Models in Science Chapter 1 degrees above room temperature, and at least 10 degrees below room temperature. When the final entry is placed in the keyboard view, press STOP and your data collection will be complete. Once this data is collected, save it to your floppy disc. You may also record your data in Table 1. Temperature sensor (front of MicroLAB) To pressure sensor (front of MicroLAB) Hayden-McNeil, LLC Figure 6. Constant volume/variable temperature and pressure apparatus. Temperature (Celsius) Ice Water Solution 0 C Room Temp. Below Room Temp. Above Room Temp. Pressure Sensor (Volts) Table 1. Temperature vs. pressure data. Data Analysis 1. Use this data to find a relationship between the pressure of atmospheric gas and temperature. 2. Use your mathematical model to predict the pressure of gas (in volts) at a temperature between 0 C and your Above Room Temperature value. Does your mathematical model predict the proper value? 7

8 Chemistry 143 Experimental Chemistry Part II. Not All Graphs Are Linear Many cause-and-effect relationships are not directly proportional or linear. Other possibilities exist, such as inverse proportionalities, power relationships, logarithmic relationships, etc. For example, the amount of gas that can be dissolved in water decreases as the temperature increases. That is why cold soda pop has a lot of fizz (dissolved carbon dioxide), but warm pop is flat and stale. The amount of wood required to make a square tabletop is proportional to the square of the length of a side. For this reason, the relationship between cost and size of furniture shows cost increasing much more rapidly than size. In cases in which the rate of a process is dependent on how close it is to its final value, the relationship is often logarithmic. When a warm soda is placed in a refrigerator to cool, the rate of heat loss will be greatest right after the soda is placed in the refrigerator and will slow as the temperature of the soda approaches the temperature of the refrigerator. Case #1. Lacquering Volume vs. Length of Table (Hand-entered Data) Suppose that you were in the painting business and wished to bid for a large contract to lacquer the tops of small, square oak tabletops for a specialty doll house company. As you might expect, the longer the side of the tabletop, the more lacquer it takes. Because the tabletops come in many different sizes, but always square, you need to have some way of predicting how much lacquer will be required for each tabletop so that you can make an intelligent bid. To work up your bid system, you lacquered five tabletops of different sizes and measured how much lacquer was required for each. The actual data is found in Table 2. Lacquer (ml) Length of Table (cm) Table 2. Data for ml of lacquer used and size of table. Data Analysis We should be able to handle this pretty easily by entering the data into a MicroLAB spreadsheet, making a graph, and running a linear regression to find a relationship between the length of the side of the doll s table with respect to lacquer, right? In this case, we choose the volume of lacquer as the y-axis (the dependent variable), and the length of the side of the tabletop as the x-axis (the independent variable). The independent variable is the one that you control the value of. The value of the dependent variable then is determined by the value you set for the independent variable. As you look at your graph and Figure 7, it is obvious that this is not a linear relationship. The end points are above the regression line and the mid points are below it and the correlation coefficient is certainly not The amount of lacquer required increases a lot faster then the length of the side of the tabletop. The shape of the graph, however, is a good indication of the type of mathematical relationship involved. Consider the 8

9 Organizing and Analyzing Experimental Data: The Role of Models in Science Chapter 1 physical relationships involved. Your company is painting square tabletops. This implies that we must be dealing with area, which is length times width, or in this case, length squared, i.e., (side) 2. Figure 7. Nonlinear trend of lacquer vs. length of table. The most simple way of analyzing this data, referring to Figure 8, is to simply apply a Quadratic curve fit. This function will plot a best fit X^2 or quadratic equation over the set of data points and give the equation below the chart. Figure 8. A quadratic curve fit. 9

10 Chemistry 143 Experimental Chemistry Take a look at your mathematical function, y X^ X Notice that the only nonzero coefficient is in front of X^2 function. This implies that the characteristics of the Quadratic function correspond best to the data that is plotted. Now try fitting this data with a Power function. What is the power in X? What is the coefficient in front of the term in X? Does this information confirm that this data has Quadratic characteristics? What amount of lacquer would be used for a doll table that had a side length of 55 cm? (Use Predict.) Case #2. Mass of a Ball Bearing vs. Its Diameter ( Mini-experiment ) You will check out a few ball bearings of different diameters and a caliper from the stockroom. Collect the mass of all but one of the ball bearings in grams and measure the diameter of all of the ball bearings in millimeters, placing the data in Table 3. Your lab instructor will demonstrate how to use the balance and the calipers properly before you begin the experiment. Mass of Ball Bearing (grams) Diameter of Ball Bearings (mm) Predicted Value Actual Value Table 3. Mass and diameters of ball bearings. Data Analysis 1. Hand-enter all of the data and plot it on the graph. Be sure to determine which is the independent variable and which is the dependent variable. (The independent variable should be plotted on the x-axis.) 2. Perform a curve fit that best describes the relationship between mass and diameter of your sphere. Discuss the correlation coefficient and why you believe the function you have chosen gives the best fit to the given data. 3. Use this mathematical model to predict the mass of the ball bearing that was kept aside by using the measured diameter in your math model and record this in Table 3 (right click on analysis and left click on prediction). Now actually mass this bearing. Does the model predict your actual measured value? 10

11 Organizing and Analyzing Experimental Data: The Role of Models in Science Chapter 1 4. What function best fits your experimental data? Do you have an explanation for the reason this mathematical model fits your experimental data? If so, explain it in your report. If not, discuss this with your lab partner and other lab participants. 5. Discuss these results with the lab instructor. Case #3. Kinetic Energy of an Automobile as a Function of Its Speed. (Handentered Data from Table) The kinetic energy (energy of motion) of an automobile increases much more rapidly than its speed. A plot of speed versus energy produces a curved, rapidly rising line. The data in the spreadsheet, Table 4, shows the kinetic energy for a moving automobile that weighs about 3000 pounds. This is a mass of about 1364 kilograms. The English units, MPH (miles per hour) and pounds are converted to m/sec (meters per second) and kg (kilograms) to compute the kinetic energy into SI units, which is expressed in kjoules (kilojoules). Speed (MPH) Speed (m/sec) Kinetic Energy (kjoules) Table 4. A table of data from a 1364 kilogram automobile. Data Analysis 1. Hand-enter all of the data from Table 4 and plot it on the graph. Be sure to choose the data that relates to one another in terms of SI units. Determine which is the independent variable and which is the dependent variable. (The independent variable should be plotted on the x-axis.) 2. Perform a curve fit. Discuss the correlation coefficient and why you believe the function you have chosen gives the best fit to the given data. 3. Use this mathematical model to predict the speed of this automobile when it has kjoules of Kinetic Energy. Please give the answer in m/sec and MPH. (You may need to make a second graph to convert m/sec into MPH.) 4. Do you have an explanation for the reason this mathematical model fits this data? If so, explain it in your report. If not, discuss this with your lab partner and other lab participants. 5. Discuss your results with your lab instructor. 11

12 Chemistry 143 Experimental Chemistry Case #4. Pressure of a Gas as a Function of Depth ( Mini-experiment ) Take a tygon tube and, starting from one end, make 1 cm marks using a permanent marker and ruler. Once you made about 10 marks, securely connect the opposite end of the tube to your pressure sensor on the back of the MicroLAB interface. Configure your computer to collect data from the uncalibrated pressure sensor to 4 places past the decimal. Make a program that will collect Keyboard entries of depth in centimeters. Fill your 10 ml graduated cylinder with water and take a reading with the tube open to the atmosphere. Now begin lowering your tygon tubing into the water at 2 cm intervals while typing in the corresponding depth in centimeters. Make at least five measurements. On your last pressure measurement, press STOP. Depth (cm) Pressure (volts) Predicted Value Actual Value Table 5. Pressure with respect to depth. Data Analysis 1. Plot your data from Table 5 on the graph and be sure to determine which is the independent variable and which is the dependent variable. (The independent variable should be plotted on the x-axis.) 2. Perform a curve fit. Discuss the correlation coefficient and why you believe the function you have chosen gives the best fit to the given data. 3. Use this mathematical model to predict the pressure in volts when the tube is submerged 5.5 cm and record this value in Table 5. Now measure 5.5 cm on your tygon tube and perform this experiment. Do your experimental results match the predictions of your mathematical model of this system? 4. Do you have an explanation for the reason this mathematical model fits this data? If so, explain it in your report. If not, discuss this with your lab partner and other lab participants. 5. Discuss your results with your lab instructor. BONUS QUESTION: If you performed this experiment with a liquid that had a density less than water, would you expect the slope of this line to be greater than, less than, or equal to the data that was collected by water? 12

13 Organizing and Analyzing Experimental Data: The Role of Models in Science Chapter 1 Case #5. The Pressure of a Gas as a Function of the Volume (Hand-entered Data from Table) At a constant temperature, pressure increases when the volume decreases and decreases when the volume expands. From the data in the accompanying Figure 9, devise a mathematical model for this system. Pressure (Torr) Volume (ml) Hayden-McNeil, LLC Figure 9. Pressure and volume data. 1. What trend line do you choose for a best fit? 2. Predict the gas volume when the pressure is increased to 2500 torr. Print your data and include the prediction value. 13

14 Chemistry 143 Experimental Chemistry Case #6. The Analysis of Light that Shines Through Smoked Glass (Handentered Data from Table) When an LED is pointed toward a light detector, the detector can be calibrated to read 100%when nothing is obstructing its view. Smoked glass can be placed in between the light detector one plate at a time, as seen in Figure 10. %T is collected after successive plates have been placed between the light source and the light detector. The data collected is found in Table 6. (If the lab is equipped, you may collect your own data.) Light-emitting diode Light detector (phototransistor) Hayden-McNeil, LLC Figure 10. Light shining through smoked glass. Glass Plates (Quantity) % Transmitted Light (%T) % Transmitted Light (%T) Table 6. Percentage of light that is transmitted. 1. What trend line did you choose for a best fit? 2. If one plate of smoked glass has a width of 1.2 cm, what would be the % of Transmitted light shining through the same type of glass that was 3.0 cm in width? 14

15 Organizing and Analyzing Experimental Data: The Role of Models in Science Chapter 1 Case #7. The Cost of Boats (Hand-entered Data from Table) The cost of a boat is related to its length, but also many other functions as well. However, the cost usually increases much more rapidly than the length of the boat. From the data in the accompanying figure, determine the relationship between cost for this series of boats and their length. Bayliner Classic Series Boats Length Cost Length Cost 16 ft $7, ft $12, ft $19, ft $27,896 Cost ($) y = mx + b 28 ft? Length (ft) Data Model Prediction Hayden-McNeil, LLC Figure 11. Bayliner Classic data and schematic. 1. What mathematical function best correlates the cost of boats to their length? 2. Predict the cost of a boat that is 28 feet long. 15

16 Chemistry 143 Experimental Chemistry Case #8. Bacterial Growth (Hand-entered Data from Table) The number of bacterial colonies growing on a new culture are found to increase exponentially with time. Table 7 below gives the set of data from one experiment. Determine the mathematical relationship between time and bacteria counts. Time (minutes) Bacteria Count Table 7. Bacterial growth over a period of time. 1. What trend line did you choose for a best fit? 2. Calculate the number of bacteria that probably will be present after 9 minutes. 16