Mathematics and Measurements 1 (Worksheet)
- Page ID
- 3153
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Name: ______________________________
Section: _____________________________
Student ID#:__________________________
Work in groups on these problems. You should try to answer the questions without referring to your textbook. If you get stuck, try asking another group for help.
“Anything worth measuring is worth measuring well.”
Source unknown
To determine if a runner broke the world’s record for a marathon, you must carefully measure the time that passed between the start and finish of the race and compare it to the record time for that marathon. Since time can be measured and expressed as an amount, it is called a quantity. Ten seconds, two minutes, and five hours are examples of quantities of time. Other familiar quantities that are important in chemistry include mass (similar to the more familiar weight), length, volume, and density.
The International System of Units
In 1960, a group of scientists from many fields and many countries agreed upon a set of metric units that would serve as a standard for scientific communication. This standard set of units is known as the International System of Units and is abbreviated SI (the abbreviation is derived from the French spelling le Systeme International d’ Unites). Seven quantities are the foundation for SI, and each has a base unit in which that quantity is expressed. Table 1 lists the base units for length, mass, volume, temperature, time and chemical amount, along with their abbreviations and their relationships to common United States units.
Table 1: Seven fundamental units in the SI system. All other units can be decomposed to these units
|
Quantity |
U.S. |
SI Base Unit |
Chemistry |
|---|---|---|---|
| Mass (weight) | Pound (lb) | Kilogram (kg) | “Gram” (g, mg) |
|
Volume |
Gallon (gal) |
Liter (L) |
“Liter” (mL, L) |
|
Temperature |
Fahrenheit (oF) |
Kelvin (K) |
K & Celsius (oC) |
|
Length |
Mile (mi), feet(ft), Inches (in) |
Meter (m) |
“Meter” (cm, mm, nm) |
|
Time |
Second (s) |
Second (s) |
Second (s) |
|
Mole (mol) |
Table 2: SI Base Units Equivalents
|
Quantity |
Base Unit |
Abbreviation |
U.S. Equivalent |
|---|---|---|---|
|
Mass |
kilogram |
kg |
2.205 pounds |
|
Volume |
liter |
L |
0.946 quarts |
|
Length |
meter |
m |
39.37 inches |
The three SI base units for mass, volume and length in Table 1 were chosen because they correspond to magnitudes which are convenient for everyday measurement. They are well suited for measurements on a scale that we can directly relate to. However, chemists often work with tiny quantities such as those used to express the diameter of a hydrogen atom or huge quantities such as the number of particles in a kilogram of carbon. These numbers are beyond the range of our senses and cannot be conveniently expressed in standard notation in SI units. Thus, the system of scientific notation is used to express very small and very large quantities.
Scientific notation is a method of expressing numbers as a product of two factors. The first factor is a number that is greater than or equal to 1 but less than 10. The second factor is 10 raised to a power. The power of 10, or exponent, is positive for numbers greater than 10 and negative for numbers less than 1. Table 2 gives examples showing both the ordinary decimal form and the exponential form for some quantities. Notice how scientific notation eliminates the need to write a long list of zeros in very small and very large numbers.
Table 2. Examples of Quantities Expressed in Scientific Notation
|
Quantity |
Ordinary Decimal Form |
Scientific Notation |
|---|---|---|
|
Diameter of a hydrogen atom |
0.000 000 000 074 m | 7.4 x 10–11 m |
|
Mass of a hydrogen atom |
0.000 000 000 000 000 000 000 000 11 kg | 1.1 x 10–25 kg |
|
Number of molecules in 2.0 g of hydrogen |
600 000 000 000 000 000 000 000 | 6.0 x 1023 |
Metric Prefixes
To further simplify the expression of measured quantities, scientists use prefixes with metric units to represent powers of ten. The following Tables list metric prefixes with a range of 25 orders of magnitude and those frequently used in chemistry. Notice that each prefix has an abbreviation and an equivalent power of ten. You should know those prefixes in bold face italics and their associated powers of ten listed in Table 3. You may need to memorize them if you can't remember them.
Table 3. Commonly used prefixes and Equivalent Powers of Ten. Know the bold italics.