2.8: Graphing

Difficulty Level: At Grade Created by: CK-12

Student Behavioral Objectives

The student will:

• correctly graph data utilizing dependent variable, independent variable, scale and units of a graph, and best fit curve.
• recognize patterns in data from a graph.
• solve for the slope of given line graphs.

Timing, Standards, Activities

Timing and California Standards
Lesson Number of 60 min periods CA Standards
Graphing 1.0 None

Activities for Lesson 8

Laboratory Activities

• None

Demonstrations

• None

Worksheets

• Graphing Worksheet

• None

Answers for Graphing (L8) Review Questions

• Sample answers to these questions are available upon request. Please send an email to teachers-requests@ck12.org to request sample answers.

Multimedia Resources for Chapter 2

This website will help to build observation skills.

Introduction to the Metric System text.

Metric Conversion instruction and examples.

Metric Equivalents or conversion factors.

This video is an explanation of how to convert among the Celsius, Kelvin, and Fahrenheit temperature scales and includes a sample problem.

This video is an explanation of particle temperature, average temperature, heat flow, pressure, and volume.

Comparing and Ordering Numbers in Scientific Notation

This website has lessons, worksheets, and quizzes on various high school chemistry topics. Lesson 1-3 is on measuring matter, lesson 1-7 is on temperature conversion, lesson 2-1 is on the International System of measurements, lesson 2-3 is on significant figures, lesson 2-4 is on dimensional analysis, lesson 2-5 is on scientific notation, lesson 2-2 is on accuracy and precision.

The learner.org website allows users to view streaming videos of the Annenberg series of chemistry videos. You are required to register before you can watch the videos, but there is no charge to register. The website has a video that apply to this lesson called “Measurement: The Foundation of Chemistry” that details the value of accuracy and precision.

Density Determination

Pre-Lab Discussion

Density is defined as the mass per unit volume of a substance. The table below lists the densities of some well known substances.

Density of Some Common Substances
Substance Density Substance Density
Water \begin{align*}1.0 \ g/cm^3\end{align*} Aluminum \begin{align*}2.7 \ g/cm^3\end{align*}
Oxygen gas \begin{align*}0.0013 \ g/cm^3\end{align*} Iron \begin{align*}8.9 \ g/cm^3\end{align*}
Sugar \begin{align*}1.6 \ g/cm^3\end{align*} Lead \begin{align*}11.3 \ g/cm^3\end{align*}
Table salt \begin{align*}2.2 \ g/cm^3\end{align*} Gold \begin{align*}19.3 \ g/cm^3\end{align*}

Density measurements allow scientists to compare the masses of equal volumes of substances. If you had a piece of lead as large as your fist and a piece of gold as large as your thumb, you would not know which substance was innately heavier because the size of the pieces are different. Determining the density of the substances would allow you to compare the masses of the same volume of each substance. The process for finding the density of a substance involves measuring the volume and the mass of a sample of the substance and then calculating density using the following formula.

\begin{align*}Density = \frac{mass \ in \ grams}{volume \ in \ mL}\end{align*}

Example: Calculate the density of a piece of lead whose mass is \begin{align*}226 \ grams\end{align*} and whose volume is \begin{align*}20.0 \ mL\end{align*}. Also calculate the density of a sample of gold whose mass is \begin{align*}57.9 \ grams\end{align*} and whose volume is \begin{align*}3.00 \ mL\end{align*}.

Solution:

\begin{align*}\text{Density of Lead} & = \frac{mass}{volume} = \frac {226 \ g}{20.0 \ mL} = 11.3 \ \text{g/mL}\\ \text{Density of Gold} & = \frac{mass}{volume} = \frac {57.9 \ g}{3.00 \ mL} = 19.3\ \text{g/mL}\end{align*}

Methods of Measuring Mass and Volume

The mass of substances is measured with a balance. In the case of a solid object that will not react with the balance pan, the object may be placed directly on the balance pan. In the case of liquids or reactive solids, the substance must be placed in a container and the container placed on the balance pan. In order to determine the mass of the substance, the mass of the container is determined before hand (empty) and then the container's mass is subtracted from the total mass to determine the mass of the substance in the container. There are several common procedures for determining the volume of a substance. The volume of a liquid is determined by pouring the liquid into a graduated cylinder and reading the bottom of the meniscus. For a regularly shaped solid, the volume can be calculated from various linear measurements.

For an irregularly shaped object, a graduated cylinder is partially filled with water and the volume measured. The object is then submerged in the water and the new volume measured. The difference between the two volumes is the volume of the submerged object.

Equipment: Specific gravity blocks, graduated cylinders (\begin{align*}10 \ mL\end{align*} and \begin{align*}100 \ mL\end{align*}), thread, millimeter ruler, balance, distilled water, glycerol. (If you have an overflow can, it also works well for submersion.)

Procedure:

1. Obtain a regularly shaped object from your teacher. Measure and record its mass in grams and its dimensions in centimeters.
2. Add approximately \begin{align*}50 \ mL\end{align*} of tap water to a \begin{align*}100-mL\end{align*} graduate and record its exact volume. Tie a thread to the block and carefully immerse it in the cylinder of water. Record the new volume in the cylinder.
3. Measure and record the mass of a clean, dry, \begin{align*}10-mL\end{align*} graduated cylinder.
4. Add exactly \begin{align*}10.0 \ mL\end{align*} of distilled water to the cylinder. Measure and record the combined mass of the cylinder and water.
5. Repeat steps 3 and 4 with glycerol instead of water.

Data Table

Object

code name or letters ________________

width ________________

height ________________

length ________________

volume of water before block ________________

volume of water after block ________________

Distilled Water

volume ________________

combined mass ________________

Glycerol

volume ________________

combined mass ________________

Calculations

1. Find the volume of the solid object using the dimensions and appropriate formula.
2. Find the volume of the solid object using water displacement.
3. Find the density of the block using the volume from calculation 1.
4. Find the density of the block using the volume from calculation 2.
5. Find the density of the distilled water.
6. Find the density of the glycerol.
7. If your teacher gives you the accepted values for the densities in this lab, calculate the percent error for your values.

\begin{align*}\% \ \text{error} = \frac{(experimental \ value) \ - \ (actual \ value)}{(actual \ value)} \times 100\end{align*}

Teacher’s Pages for Thermometer Calibration

Notes:

Thermometers should be stored vertically when not in use. You can stand them upright in a large beaker or in tall test tube racks. When thermometers are stored horizontally, they sometimes suffer a separation of the liquid near the top. If you have thermometers with separated liquid, you can sometimes shake them down or bounce the bulb gently on a folded towel to rejoin the liquid.

1. Why is a mixture of ice and water, rather than ice alone, used in calibrating a thermometer?

You cannot be sure of the temperature of solid ice. It might be \begin{align*}-10^\circ C\end{align*}. When ice and water are both present and in equilibrium, you can be sure the temperature of the mixture is \begin{align*}0^\circ C\end{align*}.

2. Why does the boiling point of a liquid vary with the barometric pressure?

Water boils when its vapor pressure is equal to the surrounding pressure. If the surrounding pressure is above or below normal atmospheric pressure, then the boiling point of a liquid will be above or below its normal boiling point.

3. What is the approximate boiling point of pure water at 380 Torr?

Between \begin{align*}80^\circ C\end{align*} and \begin{align*}82^\circ C\end{align*}.

4. What is the approximate boiling point of pure water at 800 Torr?

Near \begin{align*}101^\circ C\end{align*}

5. Food products such as cake mixes often list special directions for cooking the products in high altitude areas. Why are special directions needed? Would a food product needing such directions require a longer or shorter time period to cook under such conditions?

At high altitudes, the atmospheric pressure is less than normal atmospheric pressure and therefore, the boiling point of water is below \begin{align*}100^\circ C\end{align*}. Since boiling water is less than \begin{align*}100^\circ C\end{align*}, cooking in boiling water will take longer.

Vapor Pressure of Water at Various Temperatures
Temperature in \begin{align*}^\circ C\end{align*} Vapor Pressure in mm of Hg Temperature in \begin{align*}^\circ C\end{align*} Vapor Pressure in mm of Hg
\begin{align*}-10\end{align*} \begin{align*} 2.1\end{align*} \begin{align*} 52\end{align*} \begin{align*} 102.1\end{align*}
\begin{align*} -5\end{align*} \begin{align*} 3.2\end{align*} \begin{align*} 54\end{align*} \begin{align*} 112.5\end{align*}
\begin{align*} 0\end{align*} \begin{align*} 4.6\end{align*} \begin{align*} 56\end{align*} \begin{align*} 126.8\end{align*}
\begin{align*} 2\end{align*} \begin{align*} 5.3\end{align*} \begin{align*} 58\end{align*} \begin{align*} 136.1\end{align*}
\begin{align*} 4\end{align*} \begin{align*} 6.1\end{align*} \begin{align*} 60\end{align*} \begin{align*} 149.4\end{align*}
\begin{align*} 6\end{align*} \begin{align*} 7.0\end{align*} \begin{align*} 62\end{align*} \begin{align*} 163.8\end{align*}
\begin{align*} 8\end{align*} \begin{align*} 8.0\end{align*} \begin{align*} 64\end{align*} \begin{align*} 179.3\end{align*}
\begin{align*} 10\end{align*} \begin{align*} 9.2\end{align*} \begin{align*} 66\end{align*} \begin{align*} 196.1\end{align*}
\begin{align*} 12\end{align*} \begin{align*} 10.5\end{align*} \begin{align*} 68\end{align*} \begin{align*} 214.2\end{align*}
\begin{align*} 14\end{align*} \begin{align*} 12.0\end{align*} \begin{align*} 70\end{align*} \begin{align*} 233.7\end{align*}
\begin{align*} 16\end{align*} \begin{align*} 13.6\end{align*} \begin{align*} 72\end{align*} \begin{align*} 254.6\end{align*}
\begin{align*} 18\end{align*} \begin{align*} 15.5\end{align*} \begin{align*} 74\end{align*} \begin{align*} 277.2\end{align*}
\begin{align*} 20\end{align*} \begin{align*} 17.5\end{align*} \begin{align*} 76\end{align*} \begin{align*} 301.4\end{align*}
\begin{align*} 22\end{align*} \begin{align*} 19.8\end{align*} \begin{align*} 78\end{align*} \begin{align*} 327.3\end{align*}
\begin{align*} 24\end{align*} \begin{align*} 22.4\end{align*} \begin{align*} 80\end{align*} \begin{align*} 355.1\end{align*}
\begin{align*} 26\end{align*} \begin{align*} 25.2\end{align*} \begin{align*} 82\end{align*} \begin{align*} 384.9\end{align*}
\begin{align*} 28\end{align*} \begin{align*} 28.3\end{align*} \begin{align*} 84\end{align*} \begin{align*} 416.8\end{align*}
\begin{align*} 30\end{align*} \begin{align*} 31.8\end{align*} \begin{align*} 86\end{align*} \begin{align*} 450.9\end{align*}
\begin{align*} 32\end{align*} \begin{align*} 35.7\end{align*} \begin{align*} 88\end{align*} \begin{align*} 487.1\end{align*}
\begin{align*} 34\end{align*} \begin{align*} 39.9\end{align*} \begin{align*} 90\end{align*} \begin{align*} 525.8\end{align*}
\begin{align*} 36\end{align*} \begin{align*} 44.6\end{align*} \begin{align*} 92\end{align*} \begin{align*} 567.0\end{align*}
\begin{align*} 38\end{align*} \begin{align*} 49.7\end{align*} \begin{align*} 94\end{align*} \begin{align*} 610.9\end{align*}
\begin{align*} 40\end{align*} \begin{align*} 55.3\end{align*} \begin{align*} 96\end{align*} \begin{align*} 657.6\end{align*}
\begin{align*} 42\end{align*} \begin{align*} 61.5\end{align*} \begin{align*} 98\end{align*} \begin{align*} 707.3\end{align*}
\begin{align*} 44\end{align*} \begin{align*} 68.3\end{align*} \begin{align*} 100\end{align*} \begin{align*} 760.0\end{align*}
\begin{align*} 46\end{align*} \begin{align*} 75.7\end{align*} \begin{align*} 102\end{align*} \begin{align*} 815.9\end{align*}
\begin{align*} 48\end{align*} \begin{align*} 83.7\end{align*} \begin{align*} 104\end{align*} \begin{align*} 875.1\end{align*}
\begin{align*} 50\end{align*} \begin{align*} 92.5\end{align*} \begin{align*} 106\end{align*} \begin{align*} 937.9\end{align*}

Thermometer Calibration

Background Information

The most common device for measuring temperature is the thermometer. The typical thermometer used in general chemistry labs has a range from \begin{align*}-20^\circ C\end{align*} to \begin{align*}120^\circ C\end{align*}. Most laboratory thermometers are constructed of glass and therefore are very fragile. Older thermometers contain mercury as the temperature sensing liquid while newer thermometers contain a red colored fluid. The mercury thermometers are hazardous if they break because mercury vapors are poisonous over long periods of inhalation and the mercury vaporizes slowly and so when it is spilled, the lab is toxic for several months unless every drop of mercury is picked up. The red colored liquid thermometers are also hazardous if they break because the liquid is flammable and may be toxic. Great care should be exercised when handling thermometers of either kind.

The typical laboratory thermometer contains a bulb (reservoir) of temperature sensing fluid at the bottom; it is this portion of the thermometer which actually senses the temperature. The glass barrel of the thermometer above the liquid bulb contains a fine capillary opening in its center, into which the liquid rises as it expands in volume when heated. The capillary tube in the barrel is very uniform in its cross-section all along the length of the thermometer. This insures that the fluid will rise and fall uniformly when heated or cooled.

(NOTE: laboratory thermometers look like clinical thermometers for taking people's temperatures but they are not the same. The clinical thermometer has a constriction in the tube so that after the temperature goes up and the thermometer is removed from the heat source, the liquid will not go back down. Such clinical thermometers must be shaken to lower the temperature reading before each use. Lab thermometers have no such constriction and hence the temperature reading immediately starts down when the heat source is removed. For that reason, lab thermometers must be read while the bulb is still in contact with the material whose temperature is being taken.)

Because thermometers are so fragile, it is a good idea to check them, now and then, to make sure they are still working properly. To check a thermometer, a process of calibration is used. To do this, you will determine the reading given by your thermometer in two systems whose temperature is known with certainty. If the readings of your thermometer differ by more than one degree from the true temperatures, it should be removed from use.

A mixture of ice and water which has reached equilibrium has a temperature of exactly \begin{align*}0^\circ C\end{align*} and will be used as the first calibration point. The second calibration point will be boiling water whose exact temperature must be determined using the barometric pressure in the lab.

Pre-Lab Questions

1. Why is a mixture of ice and water, rather than ice alone, used in calibrating a thermometer?
2. Why does the boiling point of a liquid vary with the barometric pressure?
3. What is the boiling point of pure water at 380 Torr?
4. What is the boiling point of pure water at 800 Torr?
5. Food products such as cake mixes often list special directions for cooking the products in high altitude areas. Why are special directions needed? Would a food product needing such directions require a longer or shorter time period to cook under such conditions?

Purpose

In this experiment, you will check a thermometer for errors by determining the temperature of two stable equilibrium systems.

Apparatus and Materials

• Thermometer
• \begin{align*}400 \ mL\end{align*} beaker
• \begin{align*}250 \ mL\end{align*} beaker
• distilled water
• ice
• hot plate
• stirring rod
• boiling chips.

Safety Issues

Mercury thermometers are hazardous if they break because mercury vapors are poisonous over long periods of inhalation and the mercury vaporizes slowly and so when it is spilled, the lab is toxic for several months unless every drop of mercury is picked up. The red colored liquid thermometers are also hazardous if they break because the liquid is flammable and may be toxic. Great care should be exercised when handling thermometers of either kind.

Procedure

Fill a \begin{align*}400 \ mL\end{align*} beaker with ice and add tap water until the ice is covered with water. Stir the mixture is a stirring rod for one minute. Dip the thermometer into the ice water mixture so that the thermometer bulb is approximately centered in the mixture (not near the bottom or sides). Leave the thermometer in the mixture for two minutes and then read the thermometer to the nearest \begin{align*}0.2 \ degree\end{align*} while the thermometer is still in the ice water bath. Record the temperature.

Allow the thermometer to return to room temperature by resting it is a safe place on the laboratory table.

Half fill a \begin{align*}250 \ mL\end{align*} beaker with distilled water and place it on a hot plate. Add 2 or 3 boiling chips to the water. Heat the water to boiling. Dip the thermometer into the boiling water making sure the thermometer does not get near the bottom, sides, or top of the water. Hold it there for \begin{align*}2 \ minutes\end{align*} and record the temperature reading to the nearest \begin{align*}0.2 \ degree\end{align*}.

Ask your instructor for the current barometric pressure reading in the laboratory room, look up the actual boiling point of water at this pressure and record.

Data

Actual freezing point of water = ____________

Freezing point determined by your thermometer = ____________

Difference between correct and trial values = ____________

Barometric pressure in the room = ____________

Actual boiling point of water at this pressure = ____________

Boiling point determined by your thermometer = ____________

Difference between correct and trial values = ____________

Post-Lab Questions

1. Calculate the percent error of your measurement of the freezing point of water.

\begin{align*}\% \ \text{error} = \frac{actual \ value - trial \ value}{actual \ value} \times 100 =\end{align*}

2. Calculate the percent error of your measurement of the boiling point of water.

\begin{align*}\% \ \text{error} = \frac {actual \ value \ - \ trial \ value}{actual \ value} \times 100 =\end{align*}

Actual Boiling Point of Water versus Various Room Pressures
Room Pressure (mm of Hg) Boiling Point of Water \begin{align*}(^\circ C)\end{align*}
\begin{align*}750\end{align*} \begin{align*}99.6\end{align*}
\begin{align*}751\end{align*} \begin{align*}99.7\end{align*}
\begin{align*}752\end{align*} \begin{align*}99.7\end{align*}
\begin{align*}753\end{align*} \begin{align*}99.8\end{align*}
\begin{align*}754\end{align*} \begin{align*}99.8\end{align*}
\begin{align*}755\end{align*} \begin{align*}99.8\end{align*}
\begin{align*}756\end{align*} \begin{align*}99.9\end{align*}
\begin{align*}757\end{align*} \begin{align*}99.9\end{align*}
\begin{align*}758\end{align*} \begin{align*}99.9\end{align*}
\begin{align*}759\end{align*} \begin{align*}100.0\end{align*}
\begin{align*}760\end{align*} \begin{align*}100.0\end{align*}
\begin{align*}761\end{align*} \begin{align*}100.1\end{align*}
\begin{align*}762\end{align*} \begin{align*}100.1\end{align*}
\begin{align*}763\end{align*} \begin{align*}100.1\end{align*}
\begin{align*}764\end{align*} \begin{align*}100.2\end{align*}
\begin{align*}765\end{align*} \begin{align*}100.2\end{align*}
\begin{align*}766\end{align*} \begin{align*}100.2\end{align*}
\begin{align*}767\end{align*} \begin{align*}100.2\end{align*}
\begin{align*}768\end{align*} \begin{align*}100.3\end{align*}
\begin{align*}769\end{align*} \begin{align*}100.3\end{align*}
\begin{align*}770\end{align*} \begin{align*}100.3\end{align*}

Density of Diet Soda vs. Regular Soda

Brief description of demonstration

A can of diet soda and regular soda are place into a clear container of water. The diet soda floats while the regular soda sinks.

Materials

• 12 oz. can of diet soda
• 12 oz. can of regular soda, preferably the same brand
• Clear container with enough volume so the can has room to sink totally

Procedure

Fill the clear container to within \begin{align*}5 \ cm\end{align*} of the top with water. Place the diet soda into the container. It will float. Place the regular soda into the container. It will sink.

Hazards

None.

Disposal

Pour the water down the sink.

Discussion

This is a good demonstration of density and for the discussion of dependent and independent variables. The only significant difference between the cans is their contents. One may want to try different sodas of both diet and regular varieties to show this.

Absolute Zero Determination Demo

Brief description of demonstration

An Absolute Zero Apparatus is placed in various liquids at different temperatures. The temperatures of each solution are known. The pressure is read from the pressure gauge on the apparatus. A graph is made with Celsius temperature on the vertical axis and pressure on the horizontal axis. The plot is then extrapolated to zero pressure. The extrapolated line will cross the temperature axis at absolute zero.

Apparatus and Materials

• Absolute zero apparatus (available from science supply companies for around \$150)
• 3 - Pyrex or Kimex \begin{align*}2.0 \ liter\end{align*} beakers
• Hotplate
• Ice
• Dry ice (you can find dry ice suppliers on the internet – dry ice can be stored in a Styrofoam cooler but do not put the lid on tightly)
• Ethanol – \begin{align*}600 \ mL\end{align*}
• If you have a mercury thermometer that covers the range \begin{align*}-100^\circ C\end{align*} to \begin{align*}+100^\circ C\end{align*}, you can use it to measure the temperatures of the baths.

Procedure

1. Fill one of the Pyrex beakers half full of tap water and place it on the hotplate to boil.
2. Fill the another Pyrex beaker half full of tap water and crushed ice.
3. Fill the third beaker about one-fourth full will broken pieces of dry ice and then add ethanol slowly (lots of fog) until the beaker is about half full.
4. Place the bulb of the Absolute Zero Apparatus into the boiling water and leave it there until a constant pressure is reached. Record the temperature of the bath (taken to be \begin{align*}100^\circ C\end{align*}) and the pressure on the gauge.
5. Place the bulb of the Absolute Zero Apparatus into the ice water and leave it there until a constant pressure is reached. Record the temperature of the bath (taken to be \begin{align*}0^\circ C\end{align*}) and the pressure on the gauge.
6. Place the bulb of the device into the dry ice and alcohol slush and leave it there until a constant pressure is reached. Record the temperature (known to be \begin{align*}-81^\circ C\end{align*}) and the pressure on the gauge.
7. Plot a graph of the temperatures and pressures recorded. Make sure that the temperature axis on your graph extends below \begin{align*}-280^\circ C\end{align*}.
8. After the three points are plotted on the graph, lay a straight edge on the graph line and extend it to the zero pressure line. You will get a graph similar to the one shown below.

Hazards

Do not handle dry ice with bare hands. Pot holders or thermal protection gloves are necessary to handle the boiling water beaker and the dry ice – alcohol slush beaker.

Disposal

Once the dry ice has all melted, the solutions can be poured down the sink.

Discussion

Pressure is caused by the collisions of gas particles with each other and the walls of their container. When the temperature is lowered, the particles move more slowly, decreasing the frequency and strength of these collisions. In turn, the pressure falls.

Absolute temperature can be defined as the temperature at which molecules cease to move. Therefore absolute zero temperature corresponds to zero pressure.

Extending a graph beyond actual data points is called extrapolation ... a not-always acceptable procedure.

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