Overview
PURPOSE OF THE EXPERIMENT
This experiment is meant to teach basic math skills required for lab throughout the semester.
INTRODUCTION- Part I – Significant Figures
We will learn next week that each piece of equipment has a specific number of digits we should report when using that device. When we read a number from left to right, the information given by each subsequent digit indicates a higher level of precision in the number. Equipment that measures out to more place values are more precise than those that measure less digits. Think about the number of attendees at football game. 36,572 people were at the game it would seem like a more exact answer than if I told you there were 36,000 people in attendance. Which of these answers if correct? It depends on how we measured the number of people. Did I estimate by looking at the size of the stadium, or did I count every individual person? When taking a measurement in the lab, we should always report all certain digits on a device plus one uncertain digit (we call this “reading between the lines”). The number of digits a device reports are known as significant figures and include all certain digits plus one uncertain digit. Significant figures tells us how exact a measurement is and are critical when reporting scientific data because they give the reader an idea of how well you could actually measure/report your data. We all know that 1.5135 in is a much better measurement (more precise) than 1 in, the fact that 1.5135 has more significant figures is a clue to this. In the example above, reporting that there are 36,572 people at the game implies that I measured each person individually while reporting 36,000 implies that I estimated. Knowing how to look at the amount of significant figures in a number can give a clue as to how it was measured. Zero digits tend to be tricky when discussing significant figures (what if I actually had counted all of the individual people at the game, but by coincidence 36,000 people were in attendance exactly? There are a few rules that govern whether or not a number counts as significant, or if it is just there to hold the place value of other numbers.
Rules for knowing which numbers are significant when looking at data directly:
1. All nonzero #’s are significant (zeros are too if they are in between!!!!)
a. 1.204 kg 4 significant figures
2. Trailing (ending) zeros are significant ONLY if there is a decimal somewhere in the #
a. 6.00 m 3 significant figures 600 m 1 significant figure
3. Leading (beginning) zeros are NEVER significant
a. 0.08 L and 002 L 1 significant figure
How do I round calculated numbers?
Rules about significant figures on measured quantities also affect calculated numbers that we obtain for measurements. For example, if I want to know the density of water I could measure the mass and volume of a sample and divide. The calculated density would still need to reflect the amount of certainty seen in the original measurements. There are rules for determining how to round calculated answers so that they correctly exhibit the amount of uncertainty in the original measurements. With that being said, a number should never be rounded during a calculation until you are ready to report the answer to the question. Never use mathematically rounded numbers in your problems.
Multiplying and dividing numbers (Where to round?)
The LEAST number of significant figures in any number of the problem determines the number of significant figures in the answer. Let's state that another way: a chain is no stronger than its weakest link. Your answer cannot have more sigfigs in your answer than the number with the least.
3.62 m X 4.259870000 m x 2.584 m = 39.84716477 m
3 sigfigs 10 sigfigs 4 sigfigs = answer should have no more than 3 sigfigs
Answer: 39.8 m
Adding and subtracting numbers with significant figures (Where to round?)
Add or subtract in the normal fashion. Round the answer to the LEAST number of places in the decimal portion of any number in the problem.
4.20 mL + 6.0 mL = 10.20 mL
2 decimals + 1 decimal = my answer can only have 1 decimal
Answer: 10.2 mL
Calculating sigfigs for multi step calculations can be tricky. Do your calculation all the way through and then round the number. To determine how to round, retract your steps of the calculation keeping track of how many figures each step should have. Then use them to round the final answer.
Part II – Units and Dimensional Analysis
Units are one of the most important aspects to any measurement taken in science. A unit must always be placed at the end of a number so that the meaning of the number is known. Some of the standard (SI units) for measurements are given below. Get familiar with the units, symbols, and what type of measurements they are associated with using the chart below.
Measurement
Unit
Symbol
Length
Meter
(m)
Mass
kilogram
(kg)
Temperature
Kelvin
K
Electric current
Ampere
A
Amount of something
Mole
mol
Time
Seconds
s
Luminous intensity
Candela
cd
The base units above can be used to makeup derived units with special names and symbols (I.e. volume m3 or velicity m/s). The units listed above can be added to metric prefixes to change the magnification of the unit. A metric prefix is a term that is placed before the base units name to signify a multiple or fraction of the expressed number by 10. For example, the distance from New Orleans to Baton Rouge is 130,000 meters. It may be more useful to express this number in kilometers than in meters. This distance between New Orleans and Baton Rouge in kilometers would be 130 km (a much easier number to digest). If I want to know the length of a piece of hair a smaller unit like centimeters (cm) may be more useful. The metric system is all based on the power of 10. If the prefixes are known, they work for all different types of units. Below are some of the common prefixes (many more symbols exist):
Prefix
Symbol
Unit Conversion (meter example)
giga- (1,000,000,000)
G_
1 Gm = 109 m
Kilo- (1000)
k_
1 km = 103 m
Deci- (10)
d_
1 dm = 10-1 m
Centi- (100)
c_
1 cm = 10-2 m
Milli- (1000)
m_
1 mm = 10-3 m
Any SI unit can be changed using the prefixes above. Some of the more common unit conversions that you should learn to switch between are listed below.
Length (ruler)
Mass (balance/scale)
Volume (glassware)
1 km = 103 m
1 kg = 1000 g
1 kL = 1000 L
1 cm = 10-2 m
1 mm = 10-3 m
1 mg = 10-3 g
1 mL = 10-3 L
1 in = 2.54 cm *
453.6 g = 1 lb *
1 L = 1.057 qt*
1 ft = 12 in
1 cm3 = 1 mL
*Show conversions from English to metric
We will use the process of dimensional analysis to convert between units and answer calculation based questions in this class. Dimensional analysis is a technique used in science to keep track of units when performing calculations. If is important that you learn to master this technique. If you know how to do dimensional analysis, you should rarely get a problem wrong (even if you don’t remember the equation). There are three rules to remember: 1) write out your units!! (units with per go on opposite sides of a fraction) 2) Units that match on the top and bottom of the fractions will cancel out 3) Work the numbers after the units when getting an answer (check units first, then plug in numbers). Numbers on the top get multiplied and numbers on the bottom are divided. Let’s practice doing unit conversions using dimensional analysis.
Unit Conversions
Example#1 How many inches are in 2.0 ft?
Step 1
We start by listing the given information at the left edge of the page and the unit of what we want to find at the right edge of the page. Thus our task will be to go from feet to inches.
GIVEN FIND
2.0 ft? = inches
Step 2
Determine your coversion : The conversion: 12 in = 1 ft or 12 in / ft We multiply by the fraction, "12 inches over 1 foot". This is based on one of the “conversion factors”. Feet are placed on the bottom of the fraction so that they will cancel with the feet given in the problem. Since the fraction we are multiplying by has a value of one, the answer so far (24 inches) is equal to the original 2.0 feet. Each time we have multiplied by a fraction equal to "1" so that the answer will be equal to the 2.0feet we started out with. Since 2.0 has 2 sigfigs our answer 24 should have 2 sigfigs as well.
2.0 ft
Example#2 How many in are in 2.0 km? (Sometimes we need to piece together multiple conversion factors to do the problem)
Step 1
We start by listing the given information at the left edge of the page and the unit of what we want to find at the right edge of the page. Thus, our task will be to go from km to cm. We need two factors to do this
GIVEN FIND
2.0 km? = in
Step 2
The conversions: 1 cm = 10-2 m and 103 m = 1km 1 in = 2.54 cm
We multiply by the fraction, "100 cm = 1m ". This is based on one of the “conversion factors”. Since this only gets us to the units of meters we must then use the second factor to get to cm.
(2 sigfigs)
Example#3 If mass of a sample of water is 5.02 grams and the volume is 5.26 mL, calculate the density in g/mL. Show all work including dimensional analysis and units. Answer with the correct number of significant figures.
To determine significant figures, 5.02 has (3) and 6.79 has (3). Our answer can only be reported to 3 significant figures.
Example#4 How long does it take (in minutes) to get from New Orleans to Baton Rouge if I drive 75 mph (75 mi/hr) ? The distance between the two cities is 81.0 miles. Show all work including dimensional analysis and units. Answer with the correct number of significant figures.
To determine significant figures, 81.0 has (3) and 75 has (2). Our answer can only be reported to 2 significant figures.
Name _______________________________________________________________
Numbers Lab
Report
1
Experiment
Part I: Significant Figures
1. Give the number of significant figures (0.5pt each):
a)101.00mL _______________________________
b)350kg_______________________________
c)0.00002kg _______________________________
d) 8080.09in _______________________________
e) 2.0092 x 1023atoms_______________________________
f) 5080cm ____________________________
2. Calculate and then round the answer to the correct place (0.5pt each):
a)(246) x (0.0015)=_______________________________
b)(974.59)÷ (14.020)=_______________________________
c)12.5849 + 100.26 +2.4 =_______________________________
d) 432.5 –0.0000690 =_______________________________
e) (1.0 x 106) x (3.504 x 10-4) =_______________________________
**numbers in scientific notation should always be entered in the calculate using the E/EE button, or parenthesis…to determine significant figures, only look at the coefficient which is the number before the x**
f) (3.0 x 4.52) ÷10.058____________________________________
Part II: Dimensional Analysis and Units
Unit Problems – To receive credit, you must show dimensional analysis and use only the conversions from the introduction of your packet. Answer with the correct number of sigfigs. (1pt each)
3. Convert 0.20200pounds (lbs) into milligrams(mg).
4. Convert 12.2 kilometers (km) intofeet (ft).
5. Convert 750.05 cubic centimeters (cm3) into liters (L).
6. Convert17.4quarts (qt) into kiloliters (kL).
7. The density of lead is 10.678 g/cm3. If I have sample with a volume of 19.83mL, what is its mass in grams (g)?
8. A high-speed train can move with an average speed of 220. Km/hr. If a trip from Baton Rouge to Boston take about 12.4 hours, how many miles does the train travel during this time?(1 mile = 1.609 km)T