Chapter 5 Chapter 5 Measurements and Calculations
Scientific Notation • Section 5.1 • Objective: • To show how very large numbers or very small numbers can be expressed as the product of a number between 1 and 10 and a power of 10.
Very large/very small numbers 93,000,000 miles to the sun! Dictionary of Measurement
Very large numbers can be awkward to write. For example, the approximate distance from the earth to the sun is ninety three million miles. This is commonly written as the number "93" followed by six zeros signifying that the "93" is actually 93 million miles and not 93 thousand miles or 93 miles.
Scientific notation (also called exponential notation) provides a more compact method for writing very large (or very small) numbers. In scientific notation, the distance from the earth to the sun is 9.3 x 107 miles. .
Using Scientific Notation • When the decimal point is moved to the right, the exponent of 10 is negative. Ex. 0.000167 is 1.67 x 10-4 • If the decimal point is moved to the left, the exponent of 10 is positive. Ex. 238,000 is 2.38 x 105
Very small numbers can be as awkward to write as large numbers. A paper clip weighs a bit more than one thousandth of a pound (0.0011 LB). This would be expressed in scientific notation as 1.1 x 10-3 lb. The negative sign indicates that the decimal point is moved to the left.
Representing a number using scientific notation The number 2,398,730,000,000 can be written in scientific notation as (the starred representation is most common): 0.239873 x 1013 ****2.39873 x 1012**** 23.9873 x 1011 239.873 x 1010
The number 0.003,483 can be written in scientific notation as (the starred representation is most common): 0.3483x 10-2 ****3.483 x 10-3**** 34.83x 10-4 348.3x 10-5 3483.x 10-6
Multiplying numbers written in scientific notation • To multiply 4 x 104 and 6 x 105: • Multiply the decimal parts together: • 4 x 6 = 24 • Add the two exponents: • 4 + 5 = 9 • Construct the result: • 24 x 109
Adjust the result so only one digit is to the left of the decimal point (if necessary): 24 x 109 = 2.4 x 1010
MULTIPLYING General Rule(a x 10x)(b x 10y) = ab x 10x+y
Dividing numbers written in scientific notation • To divide 6 x 105 and 4 x 104: • 6/4 = 1.5 • Subtract the two exponents: • 6/4 = 1.5 • Construct the result: • 1.5 x 101 • Adjust the result so only one digit is to the left of the decimal point (if necessary)1 • 1.5 x 109 = 1.5 x 101 No adjustment necessary
DIVIDING General Rule (a x 10x)/(b x 10y) = a/b x 10x-y
Now, neither the decimal part or the exponential part combine together in any obvious manner (as they did with multiplication and division). When adding or subtracting numbers written in exponential notation, the numbers must first be rewritten so the exponents are identical. Then, the numbers can be added or subtracted normally
To add 3.4 x 10-3 to 2.1 x 10-2: 1. Adjust one of the numbers so that its exponent is equivalent to the other number. In this case, change 2.1 x 10-2 into a number which has 10-3 as it's exponential part. 2.1 x 10-2 = 21 x 10-3 2. Add the decimal parts together: 21 + 3.4 = 24.4 3. The exponential part of the result is the same as the exponential parts of the two numbers, in this case, 10-3: 24.4 x 10-3 4. Adjust the result so only one digit is to the left of the decimal point (if necessary): 2.44 x 10-2
LINKS Printable WorksheetComputer-Graded Quiz
Survival Skills forQuantitative Courses Algebra & Arithmetic • http://www.brynmawr.edu/nsf/tutorial/ss/ssalg.html • http://www.brynmawr.edu/nsf/tutorial/ps/pstips.html
Units • Section 5.2 • Objective: To compare the English, Metric and SI systems of measurement.
Units • Scientists became aware of the need for common units to describe quantities such as time, length, mass, and temperatures. • Two systems: English System (U.S.) and the International System (the rest of the world).
Measurements of Length, Volume and Mass • Section 5.3 • Objective: To demonstrate the use of the metric system to measure, length, volume and mass.
Measuring • Volume • Temperature • Mass • Length
Reading the Meniscus Always read volume from the bottom of the meniscus. The meniscus is the curved surface of a liquid in a narrow cylindrical container.
Volume • Amount of three-dimensional space occupied by a substance. • 1cm3 = 1 mL
Try to avoid parallax errors. Parallaxerrors arise when a meniscus or needle is viewed from an angle rather than from straight-on at eye level. Correct: Viewing the meniscusat eye level Incorrect: viewing the meniscusfrom an angle
Graduated Cylinders The glass cylinder has etched marks to indicate volumes, a pouring lip, and quite often, a plastic bumper to prevent breakage.
Measuring Volume • Determine the volume contained in a graduated cylinder by reading the bottom of the meniscus at eye level. • Read the volume using all certain digits and oneuncertain digit. • Certain digits are determined from the calibration marks on the cylinder. • The uncertain digit (the last digit of the reading) is estimated.
Use the graduations to find all certain digits There are two unlabeled graduations below the meniscus, and each graduation represents 1 mL, so the certain digits of the reading are… 52 mL.
Estimate the uncertain digit and take a reading The meniscus is about eight tenths of the way to the next graduation, so the final digit in the reading is . 0.8 mL The volume in the graduated cylinder is 52.8 mL.
10 mL Graduate What is the volume of liquid in the graduate? 6 6 _ . _ _ mL 2
25mL graduated cylinder What is the volume of liquid in the graduate? 1 1 5 _ _ . _ mL
100mL graduated cylinder What is the volume of liquid in the graduate? 5 2 7 _ _ . _ mL
Self Test Examine the meniscus below and determine the volume of liquid contained in the graduated cylinder. The cylinder contains: 7 6 0 _ _ . _mL
The Thermometer • Determine the temperature by reading the scale on the thermometer at eye level. • Read the temperature by using all certain digits and one uncertain digit. • Certain digits are determined from the calibration marks on the thermometer. • The uncertain digit (the last digit of the reading) is estimated. • On most thermometers encountered in a general chemistry lab, the tenths place is the uncertain digit.
Do not allow the tip to touch the walls or the bottom of the flask. If the thermometer bulb touches the flask, the temperature of the glass will be measured instead of the temperature of the solution. Readings may be incorrect, particularly if the flask is on a hotplate or in an ice bath.
Reading the Thermometer Determine the readings as shown below on Celsius thermometers: 8 7 4 3 5 0 _ _ . _ C _ _ . _ C
Measuring Mass - The Beam Balance Our balances have 4 beams – the uncertain digit is the thousandths place ( _ _ _ . _ _ X)
Balance Rules In order to protect the balances and ensure accurate results, a number of rules should be followed: • Always check that the balance is level and zeroed before using it. • Never weigh directly on the balance pan. Always use a piece of weighing paper to protect it. • Do not weigh hot or cold objects. • Clean up any spills around the balance immediately.
Mass and Significant Figures • Determine the mass by reading the riders on the beams at eye level. • Read the mass by using all certain digits and one uncertain digit. • The uncertain digit (the last digit of the reading) is estimated. • On our balances, the thousandths place is uncertain.
Determining Mass 1. Place object on pan 2. Move riders along beam, starting with the largest, until the pointer is at the zero mark
1 1 4 ? ? ? _ _ _ . _ _ _ Read Mass
1 1 4 4 9 7 _ _ _ . _ _ _ Read Mass More Closely
Read the length by using all certain digits and one uncertain digit. Read it as 2.85cm.
Uncertainty in Measurement and Significant Figures • Section 5.4-5.5 • Objectives: To observe how uncertainty can arise and to use significant figures to indicate a measurement’s uncertainty.