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Introduction

You have already been introduced to the idea of graphing data to find the relationship between two variables. In this activity you will practice the skills of graphing and data analysis.

You can see these data graphed in an excel file if you download and open this file.

For the following data sets you must:

1. Make a data table showing units; you may assume that all data sets have matched pairs of data
2. Decide which variable is dependent (y-axis) and which is independent (x-axis) and explain why
3. Make an easy-to-read graph that fills the space you make available for it (no blank spaces)
4. Determine whether the relationship between the variables is a direct proportion (a straight line) or an inverse proportion (an inverse curve)
5. If the graph is an inverse curve then transform it into a direct proportion by taking the inverse of one of the variables and graph it again
6. Use a ruler to draw a line of best fit: this line may not go through all or indeed any of the data points; it must only be as close as possible to all points
7. Using points on the line (not data points!) determine the slope of the line; the slope is the constant of proportionality
8. Choose two pairs of data points and use them to calculate the slope. Do they give the same slope as your best-fit line? As each other? What is the advantage of using the best-fit line over using a pair of data points?
9. Using your graph or your knowledge of algebra find the y-intercept of the line
10. Determine the correct units for both the slope and the y-intercept.
11. Write the equation of the line in y-intercept form using variables appropriate to the data
12. Answer the questions in each section

Density of Iron

You are already quite familiar with the concept of density. The following data were collected by making pieces of iron with each of the given volumes and measuring the mass of each piece.

Volume of Iron (V, cm³) ±0.5 cm³
2.5, 5.3, 9.8, 14.3, 25.7
Mass of Iron (m, g) ±2 g
21, 41, 76, 111, 204

Using the error limits given for volume and mass determine the error limits on the density of iron from any one pair of measurements. Compare that result to the fact that a detailed statistical analysis of these data reveal that the density of iron is 7.9 ± 0.1 g/cm³. What is the advantage of using graphs to find density vs. single measurements?

Gay-Lussac’s Law

A series of pressure and temperature measurements for a sample of gas are given below. The amount of gas (the number of moles) and the volume were held constant in the experiment. The experiment was performed by heating the gas in metal gas bottle from 0°C (273 K) to each of the temperatures given. Kelvin (K) is the absolte temperature scale which has its lowest point at absolute zero, the lowest possible temperature. Absolute zero is -273°C so to convert °C to K just add 273.

Temperature (T, K) ±1 K
273, 300, 327, 389, 415, 689
Pressure (P, atm) ±1 atm
3.10, 2.78, 3.66, 3.69, 4.56, 7.87

What would the pressure of the gas be under these conditions at a temperature of 575 K? At 100°C? Does your line go through all of your data points? Why or why not? Explain why the best-fit line does not need to go through all of the data points.

Newton’s Second Law, constant Mass

Force is a measurable quantity. It is measured in units called newtons (F, N) after Isaac Newton, one of the most influential figures in the history of science. The newton is defined as the force required to accelerate a 1 kg mass to 1 m/s². Mass is measured in kilograms (m, kg) and acceleration in meters per second per second (a, m/s²). So 1 N = 1 kg· m/s². The following data were collected by holding mass constant and looking at the amount of force required to produce a given acceleration.

Acceleration (a, m/s²)
0.8, 1.0, 1.3, 1.7, 2.4, 3.5
Force (F, N)
1.0, 2.0, 3.0, 5.0, 8.0, 13.0

What is the constant of proportionality? What are its units? What is the formula for this relationship? How much acceleration would there be for a force of 4.5 N?

Newton’s Second Law, constant Force

Now the force is held constant while the mass is varied. The measurements show the effect on the acceleration as mass is increased while force is held constant.

Mass (m, kg)
0.75, 2.7, 3.8, 6.2, 9.6, 14.5
Acceleration (a, m/s²)
13.63, 3.90, 2.93, 2.01, 1.44, 0.99

What kind of a proportion is this relationship? Try taking the inverse of the mass data and plotting it against the acceleration data. What happens? Perform the usual analysis on this new graph. Be very careful in you determination of the units of the slope.

Beer’s Law

Beer’s Law relates the concentration of a substance to the absorption of light at a particular wavelength. There is a constant of propotionality between concentration and absorbance called the molar absorptivity, or ε (the Greek letter epsilon). This constant can be used to measure the unknown concentration of a substance by measuring the absorbance and calculating the concentration. To find the molar absorptivity a series of solutions with known concentrations are made. Then their absorption of light at a particular wavelength is measured using an instrument called a spectrophotometer. As concentration changes, the absorbance changes. The following data can be used in an experiment to determine the caffeine concentration in a can of an energy drink.

Concentration of Caffeine (c, mol/L)
2.21 ×10^-05, 4.41 ×10^-05, 6.62 ×10^-05, 8.83 ×10^-05, 1.104 ×10^-04
Absorbance of Light at 273 nm (A, no units)
0.2075, 0.3985, 0.6083, 0.8098, 1.0052

What is the concentration of caffeine in a solution that gives an absorbance of A = 0.2796? If the caffeine had been diluted from the energy drink by a factor of 50, what is the molar concentration of caffeine in the drink? If the molar mass of caffeine is 194.19 g/mol then give the number of mg of caffeine in 250 mL of the drink.