Physical Science Density and Measurements
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1 Physical Science Density and Measurements Name Date Density All matter has a mass that can be measured and a volume of space that it occupies. However, the relationship between mass and volume varies greatly from one kind of material to another. For example, a bucket of rocks has a much greater mass than an equal sized bucket of air. Therefore, a useful way to describe an object is to determine its mass per unit volume ratio, called density. Per refers to division, as in miles per hour (distance divided by time). Density is the measure of an object s mass divided by its volume (unit mass per unit volume). Scientists and mathematicians use the Greek letter rho (ρ) to represent density. Additionally, since the gram is a metric unit of mass, and the cubic centimeter (cm 3 ) is a basic unit of metric volume, density (ρ) is usually expressed in grams per cubic centimeter (g/cm 3 ). Metric System of Measurement People in different parts of the world have historically used different systems of measurement. For example, people in the United States have historically used the English system of measurement based on units such as inches, feet, pounds, gallons, and degrees Fahrenheit. However, for more than a century, most nations of the world have used the metric system of measurement based on units such as meters, newtons, liters, and degrees Celsius. In 1975, the U.S. Congress recognized the values of a global system of measurement and adopted the metric system as the official measurement system of the United States. This convention is obviously still not yet complete, so people in the US use both English and metric units. We will use only the metric or SI system; therefore, you must know how to convert between units. Each kind of metric unit can be divided or multiplied by 10 and its powers to form the smaller and larger units of the metric system. Therefore, the metric system is also known as a base-10 or decimal system. The International System of Units (SI) is the modern version of metric system symbols, numbers, base-10 numerals, powers of 10, and prefixes. Mass Earth materials do not just take up space (volume). They also have a mass of atoms that can be weighed or measured. You will use a gram balance to measure the mass of materials (by determining their weight under the acceleration of Earth s gravity). The gram (g) is a basic unit of mass in the metric system, but instruments used to measure grams vary from triple-beam balances to spring scales to digital balances. Consult with your instructor or other students to be sure that you understand how to read the gram balance provided in the lab. Linear Measurements You must be able to use a metric ruler or meter stick to make exact measurements of length this is called linear measurement. Most rulers in the US are graduated in both English and metric units. Note that the length of an object may not coincide with a specific centimeter or millimeter mark on the ruler, so you may have to estimate the fraction of a unit as exactly as you can. For example, if a measurement falls between two millimeter marks on the ruler, the most exact you can make the measurement is to the half millimeter. In other words you would report your measurement as X mm ± 0.5 mm. 1
2 Area and Volume An area is a two-dimensional space, such as the surface of a table. The long dimension is the length and short dimension is the width. If the area is a square or rectangle, then the size of the area is the product of its length multiplied by its width. The unit of an area with L= 4 cm and W= 2 cm is 8 cm 2. Three dimensional objects are said to occupy a volume of space. Box shaped objects have linear volume because they take up three dimensions of space- that is they have length, width, and height (or depth). For example, if a box has a length of 9 cm, a height of 4 cm, and a width of 4 cm, then its volume is 144 cm 3. Most natural materials, such as rocks, do not have linear dimensions, so their volumes cannot be calculated using linear measurements. However, the volume of odd-shaped materials can be determined by measuring the volume of water that they displace. This is often done in the lab using a graduated cylinder. Most graduated cylinders are graduated in metric units called milliliters (ml), which are thousandths of a liter. You should note that 1 ml of fluid volume is exactly equal to 1 cm 3 of linear volume (1 ml = 1 cm 3 ). When you pour water into a graduated cylinder, the surface of the liquid usually displays a curved meniscus in the cylinder and the volume is read at the bottom of the curve. If you drop an object into a graduated cylinder with a known volume of water, then it takes up space previously occupied by the water in the cylinder. This displaced water has nowhere to go but up; therefore, the volume of an object, such as a rock, is exactly the same as the water it displaces. Error Analysis In chemistry, we often wish to determine how close our experimental value is compared to the accepted or theoretical result. This is usually performed by finding the percent difference between the experimental value and the theoretical value. The percent difference (error) is given by: % Error = (experimental theoretical) x 100 theoretical Remember to review your data for any large difference between measurements or trials that could increase the error of your results. Repeat those measurements, if necessary. Experimental Error Personal: from personal bias or carelessness in reading an instrument (e.g., parallax), in recording observations, or in mathematical calculations. Systematic: associated with particular measurement techniques. - Improper calibration of measuring instrument. - Human reaction time. - Is the same error each time. This means that the error can be corrected if the experimenter is clever enough to discover the error. Random error: unknown and unpredictable variations. - Fluctuations in temperature or line voltage. - Mechanical vibrations of the experimental setup. - Unbiased estimates of measurement readings. - Is a different error each time. This means that the error cannot be corrected by the experimenter after the data has been collected. 2
3 Significant Digits Exact factors have no error (e.g., 10, π). All measured numbers have some error or uncertainty. Use all of the significant figures on a measuring device. For example, if a measuring device is accurate to 3 significant digits, use all of the digits in your answer. If the measured value is 2.30 kg, then the zero is a significant digit and so should be recorded in your laboratory report notes. Keep only a reasonable number of significant digits. Round these values correctly. As a general rule, the final answer should have no more significant digits than the data from which it was derived. Lab Exercises 1) Make the following unit conversions: a) 10.0 miles = kilometers b) 1.0 foot = meters c) 16 kilometers = meters d) 25 meters = centimeters e) 25.4 ml = cm 3 f) 1.3 liters = cm 3 g) 543 nanometers = meters h) 575 micrometers = millimeters 2) Use a meter stick to help you measure the length, width, and depth of your lab or classroom table top in centimeters, then calculate the area and volume of the table top (be as exact as you can and express your answer and units appropriately). 3) Using the instruments at your disposal, estimate the volume of the classroom in meters. 3
4 3) Exercises for the remaining portions of the lab are divided into stations. Each lab station must be completed by each group, but not necessarily in the order presented here. While you are allowed to work in small groups in lab, each student must complete each lab station (exercise) and turn in their own independent work. Station 1: Determining the Density of Water Using a small graduated cylinder and a gram balance, determine the mass of different volumes of water and input the data into Table 1. Then, calculate the density of each volume of water and the average density of water based on your measurements. Finally, answer the questions that follow. Table 1. Mass-Volume Relationship (Density) of Water Volume (ml) Mass (g) Density (g/cm 3 ) Average Density of Water Questions: a) Plot the data using Excel or any other appropriate spreadsheet software. Graph the mass versus volume data. Attach a copy of your graph to the lab handout. b) Insert a best-fit line (trendline) and explain why the line origin (0,0) should be included in your trendline. c) Determine the slope of the line. Include the units of your slope. Show all work, including the two points used from the graph. 4
5 d) What is the accepted or theoretical value for the density of fresh water? Use this value to calculate your percent error for the mean density value obtained above. Show your work below. Explain the difference (error). e) Use Excel to calculate the mean density value and the standard deviation for your collected data. Report these values below. f) What is the accepted value for the density of saltwater? Is this value more or less than the density of fresh water? Why? Explain your answer. Station 2: Determining the Density of Irregular-Shaped Objects (Rocks) Using a small graduated cylinder and a gram balance, determine the volume and mass of several different rock samples and input the data in Table 2. Then, calculate the density of each rock sample and the average density of the rock samples. Finally, answer the questions that follow. Table 2. Mass-Volume Relationship (Density) of Assorted Rock Samples Sample # Volume (ml) Mass (g) Density (g/cm 3 ) Average Density of Rock Samples 5
6 a) How does the density of these rock samples compare to that of the density of water? Can you think of any Earth materials (not necessarily rocks) that may have densities less than that of water? How do you know? b) How does your measured value of rock density compare to the average density of all rocks on Earth (~2.5 g/cm 3 )? Do you think our experiment is a valid representation of the density of all rocks on Earth? Explain. Station 3: Determining the Density of Common Substances Use a ruler to determine the volume dimensions of the following substances and then, determine the density of the following objects using their mass. Also, calculate the percent error and input those values below. The accepted or theoretical density values are provided for each object in parentheses. Table 3. Density of Selected Substances (Accepted Density Values Shown) Substance Mass (g) Volume (cm 3 ) Density (g/cm 3 ) % Error Aluminum Cube (2.7 g/cm 3 ) Brass Cube (8.0 g/cm 3 ) Nylon Cube (1.13 g/cm 3 ) Copper Cube (8.9 g/cm 3 ) Poplar Cube ( g/cm 3 ) 6
7 a) Brass is an alloy composed of two different pure metallic elements. Using the data you collected above, is it possible to identify one of these metal elements in brass? Explain your answer. b) Is the density of the poplar sample more or less than that of water? Given its density, should poplar sink or float in fresh water? Explain. Station 4: Determining the Density of Pennies Using a small graduated cylinder and a gram balance, determine the volume and mass of twenty (20) pennies minted prior to 1982 and twenty (20) pennies minted after 1982 and input the data in Tables 4 and 5. Repeat the procedure twice more. Then, calculate the density for each trial and the average density of the pennies. Finally, answer the questions that follow. Table 4. Density of Pennies Minted Prior to 1982 Trial Mass (g) Volume (ml) Density (g/cm 3 ) Average Density of Pennies Minted Prior to 1982 Table 5. Density of Pennies Minted After 1982 Trial Mass (g) Volume (ml) Density (g/cm 3 ) Average Density of Pennies Minted After
8 Questions: a) How does the density of each group of pennies compare to the density of copper given in Table 3? b) How does the density of the two groups of pennies compare to each other? Can you explain the difference? c) Use the results of this investigation to formulate a hypothesis about the composition of the two groups of pennies. How could you test your hypothesis? d) Why is it best to use the results of three trials rather than a single trial for determining density? e) What are some of the sources of experimental error or uncertainty that could be important to consider when analyzing your data in this experiment? 8
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