Chapter 1R: Review

How do you quantify time? Depends on the application.

- 30 s (in the microwave)
- 50 min (of class)
- 10 h (until bedtime)
- 7 d (until next Friday)
- 4 m (until your birthday)
- 3 y (until graduation)

How do you quantify a length?

**Imperial System of Measurement**

- in (relatively small lengths)
- ft
- yd
- mile (relatively big lengths)

**Metric System**

- millimeter (relatively small lengths)
- centimeter
- meter
- kilometer (relatively big lengths)

Some *countries* still use the **imperial system**. *Science* has adopted the **metric system**, a system of measurement that uses units with ratios that are multiples of 10.

Source: Wikimedia

Science uses **SI Units** (International System of Units) to describe quantities. The **base SI units** are as follows:

Property | Unit | Abbreviation |
---|---|---|

Mass |
kilogram |
kg |

Length |
meter |
m |

Time |
second |
s |

Temperature |
kelvin |
K |

Amount of substance |
mole |
mol |

Electric current |
ampere |
A |

Luminous intensity |
candela |
cd |

Used to modify a basic unit

Prefix | Abbeviation | Meaning | Example |
---|---|---|---|

Tera- |
T |
10 |
1 terameter (Gm) = 1×10 |

Giga- |
G |
10 |
1 gigameter (Gm) = 1×10 |

Mega- |
M |
10 |
1 megameter (Mm) = 1×10 |

Kilo- |
k |
10 |
1 kilometer (km) = 1×10 |

Deci- |
d |
10 |
1 decimeter (dm) = 1×10 |

Centi- |
c |
10 |
1 centimeter (cm) = 1×10 |

Milli- |
m |
10 |
1 millimeter (mm) = 1×10 |

Micro- |
μ |
10 |
1 micrometer (μm) = 1×10 |

Nano- |
n |
10 |
1 nanometer (nm) = 1×10 |

Pico- |
p |
10 |
1 picometer (pm) = 1×10 |

Femto- |
f |
10 |
1 femtometer (fm) = 1×10 |

Which quantity would be most appropriate for describing the distance between Athens, GA and Akron, OH (*ca.* 665 mi)?

- 1.07×10
^{8}cm (107 million cm) - 1.07×10
^{6}m (1.07 million m) - 1,069 km (1.07 thousand km)

Source: Openstax

**°F → °C**

\[t/^{\circ}\mathrm{C} = \dfrac{\left ( t/^{\circ}\mathrm{F} - 32 \right )}{1.8}\]

**°C → K**

\[t/\mathrm{K} = t/^{\circ}\mathrm{C} + 273.15\]

**°C → °F**

\[t/^{\circ}\mathrm{F} = \left ( t/^{\circ}\mathrm{C} \times 1.8 \right ) + 32\]

**K → °C**

\[t/^{\circ}\mathrm{C} = t/\mathrm{K} - 273.15\]

What is 93.0 °F in °C and K?

33.9 °C

What is absolute zero (0 K) in °C and °F?

–273.15 °C

\[b^n = \underbrace{b \times b \times \cdots \times b}_{n~\mathrm{times}}\]

\[10^1 = 10 \] \[10^2 = 10 \times 10 = 100\] \[10^3 = 10 \times 10 \times 10 = 1000\]

\[b^{-n} = \underbrace{\dfrac{1}{b} \times \dfrac{1}{b} \times \cdots \times \dfrac{1}{b}}_{n~\mathrm{times}}\]

\[10^{-1} = \dfrac{1}{10} = 0.1\] \[10^{-2} = \dfrac{1}{10} \times \dfrac{1}{10} = 0.01\]

\[10^{-3} = \dfrac{1}{10} \times \dfrac{1}{10} \times \dfrac{1}{10} = 0.001\]

Used for small and big numbers

\[ 1,234 = 1.234\times 10^1 \times 10^1 \times 10^1 = 1.234\times 10^3 \]

\[ 0.01234 = 1.234 \times 10^{-1} \times 10^{-1} = 1.234\times 10^{-2} \]

\[ 0.1234 = \dfrac{1.234}{10^1\times 10^1} = 1.234\times 10^{-2} \]

Write the following numbers into scientific notation.

\[502\]

\[1,535,234\]

\[0.00008753\]

\[5.02\times 10^2\]

\[1.535234 \times 10^{6}\]

\[8.753\times 10^{-5}\]

\[3.4\times 10^{4} + 9.7\times 10^{5}\]

Transform each number into the same power of 10.

\[0.34\times 10^{5} + 9.7\times 10^{5}\]

Add or subtract the numbers.

\[10.04\]

Rewrite in scientific notation.

\[10.04 \times 10^{5} = 1.004 \times 10^6\]

Powers of 10 get added.

\[200 \times 100 = 20,000 = 2.00\times 10^{4}\]

\[\begin{align*} \left (2.00\times 10^{2} \right) \times \left (1.00 \times 10^{2}\right) &= \\[2ex] \left (2.00\times 1.00 \times 10^{2} \times 10^{2} \right) &= 2.00\times 10^4 \end{align*}\]

Powers of 10 get subtracted.

\[\begin{align*} \dfrac{200}{100} = 2 &= 2\times 10^{0} \\[4ex] \dfrac{2.00\times 10^{2}}{1.00 \times 10^{2}} &= 2\times 10^0 = 2 \end{align*}\]

Not all digits are *significant*. Here, the graduated cylinder is a “measuring stick” for volume. How accurate are we able to make a reading?

We know by inspection that the water is clearly greater than 21 mL but less than 22 mL. We estimate that the water is perhaps 21.6 mL (at the meniscus).

The “2” and the “1” are certain but the 0.6 is *uncertain*. All *certain/exact* digits as well as the *first uncertain/inexact digit* are considered significant. Our measurement has three significant figures.

Source: Openstax

The numbers you use in this class are treated as exact (infinite sig. figs.) or inexact numbers.

**Exact**

Obtained through counting

- 3 pennies
- 15 puppies
- 130 students

**Inexact**

Obtained through measurement

- length
- mass
- volume
- temperature

Imperial-to-metric conversions

**Hard**– inexact; conversion**Soft**– exact; substitution

See table for some official SI Units and Conversions

*For this class*, treat all imperial-to-metric conversions as being exact.

See Conversions notes.

**Precision** - indicates how well several measurements agree

**Accuracy** - the agreement of a measurement with the accepted value

Source: Openstax

Measurements have error.

**Experimental Error**

“Error in measurement” can be an average value of *n* measurements.

\[\mathrm{error~in~measurement} = \mathrm{measured~value} - \mathrm{accepted~value}\]

**Percent Error**

\[ \%~\mathrm{error} = \dfrac{\mathrm{error~in~measurement}}{\mathrm{accepted~value}}\times 100\% \]

A coin has a diameter of 28.054 mm. Two people make a series of measurements (reported in mm). What is the average diameter and percent error obtained in each case? Which person was more accurate on average?

Person 1 | Person 2 |
---|---|

28.246 |
27.9 |

28.244 |
28.0 |

28.246 |
27.8 |

28.248 |
28.1 |

**Person 1**

Average: 28.246 mm

\[\begin{align*} \mathrm{\%~error} &= \dfrac{28.246~\mathrm{mm} - \mathrm{28.054~\mathrm{mm}}}{28.054~\mathrm{mm}}\\[1.5ex] &~~~~\times 100\% \\[1.5ex] &= 0.684\% \end{align*}\]

**Person 2**

Average: 27.95 mm

\[\begin{align*} \mathrm{\%~error} &= \dfrac{27.\bar{9}5~\mathrm{mm} - \mathrm{28.054~\mathrm{mm}}}{28.054~\mathrm{mm}}\\[1.5ex] &~~~~\times 100\% \\[1.5ex] &= -0.4\% \end{align*}\]

Press `q`

for answer.

**Determinate**

identifiable; can be avoided

- using impure reagents
- underwashing glassware
- overwashing precipitates
- careless measuring

**Indeterminate**

random; arises from uncertainties in measurements

- random fluctuations in measuring devices (e.g. background noise from signals)

The std. dev. of a series of measurements is equal to the

- square root of the sum (\(\Sigma\)) of the squares of the deviations (\(\left ( x_i - \bar{x} \right )^2\)) for each measurement (\(x_i\)) from the average (\(\bar{x}\))
- divided by one less than the number,
*N*, of measurements

\[\mathrm{std.~dev.} = \sqrt{\dfrac{\sum{\left(x_i - \bar{x}\right)^2}}{N-1}}\]