Chemistry Chapter 2: Analyzing Data Study Guide

International System of Units (SI) and Base Units

The International System of Units, known as SI, is a standardized system based on natural phenomena that can be measured consistently anywhere in the universe. There are seven fundamental base quantities and their corresponding units:

1. Length is measured in the meter ($m$), which is defined based on the speed of light.

2. Mass is measured in the kilogram ($kg$), which is defined based on Planck’s Constant.

3. Time is measured in the second ($s$), defined by the oscillations of Cesium atoms.

4. Electrical current is measured in the ampere ($A$).

5. Temperature is measured in the kelvin ($K$), which is based on the lowest possible theoretical temperature (absolute zero).

6. Amount of a substance is measured in the mole ($mol$).

7. Luminous intensity is measured in the candela ($cd$).

Metric Prefixes and Scale of Units

Metric prefixes are used when a base unit is either too large or too small to describe a measurement conveniently. These prefixes signify factors of ten and are applied to the standard SI units. Common conversational scales include ten years in a decade, ten dimes in a dollar, one hundred cents in a dollar, one hundred years in a century, and one thousand years in a millennium.

Large scale prefixes (Multiples of the base unit):

• Yotta ($Y$): $10^{24}$ (septillion)

• Zetta ($Z$): $10^{21}$ (sextillion)

• Exa ($E$): $10^{18}$ (quintillion)

• Peta ($P$): $10^{15}$ (quadrillion)

• Tera ($T$): $10^{12}$ (trillion)

• Giga ($G$): $10^{9}$ (billion)

• Mega ($M$): $10^{6}$ (million)

• Kilo ($k$): $10^{3}$ (thousand)

• Hecto ($h$): $10^{2}$ (hundred)

• Deka ($da$ or $D$): $10^{1}$ (ten)

Small scale prefixes (Fractions of the base unit):

• Deci ($d$): $10^{-1}$ ($1/10^{th}$, or tenth)

• Centi ($c$): $10^{-2}$ ($1/100^{th}$, or hundredth)

• Milli ($m$): $10^{-3}$ ($1/1000^{th}$, or thousandth)

• Micro ($\mu$): $10^{-6}$ (millionth)

• Nano ($n$): $10^{-9}$ (billionth)

• Pico ($p$): $10^{-12}$ (trillionth)

• Femto ($f$): $10^{-15}$ (quadrillionth)

• Atto ($a$): $10^{-18}$ (quintillionth)

• Zepto ($z$): $10^{-21}$ (sextillionth)

• Yocto ($y$): $10^{-24}$ (septillionth)

Example measurements for standard length units involve relationships such as $1\,m = 10\,dm = 100\,cm = 1000\,mm$. For larger measurements, $1\,km = 1000\,m$, $1\,Hm = 100\,m$, and $1\,Dm = 10\,m$ .

Temperature Conversions

Temperature is measured across three primary scales: Fahrenheit ($^\circ F$), Celsius ($^\circ C$), and Kelvin ($K$). The relationship between Kelvin and Celsius is defined by the following formulas:

$T(K) = T(^\circ C) + 273.15$ $T(^\circ C) = T(K) - 273.15$

Practice conversion tasks include changing $100^\circ F$ to $^\circ C$, $425^\circ F$ to $^\circ C$, $25^\circ C$ to $^\circ F$, and $-195^\circ C$ to $^\circ F$. Additionally, converting from Kelvin involves examples such as $295\,K$ to $^\circ C$ and $400^\circ C$ to $K$.

Derived Units and Density

A derived unit is a unit defined by a combination of base units. Examples include speed, measured in meters per second ($m/s$); volume, measured in cubic meters ($m^3$); and density, measured in kilograms per cubic meter ($kg/m^3$).

Volume refers to the amount of space an object occupies. Common units include the Liter ($L$), milliliter ($mL$), $m^3$, and $cm^3$. Other historical or non-SI units include the cup, quart, gallon, peck, bushel, barrel, and hogshead. Essential conversions include: $1\,mL = 1\,cm^3$ $1000\,L = 1\,m^3$

Density is the ratio of mass to volume and is a characteristic property that can identify a substance: $\text{Density} = \frac{\text{Mass}}{\text{Volume}}$

Specific densities include:

• Aluminum: $2.71\,g/cm^3$

• Iron: $7.87\,g/cm^3$

• Copper: $8.96\,g/cm^3$

• Brass (a mixture, not a pure substance): $8.4\,g/cm^3$ to $8.7\,g/cm^3$

Experimental density problems include finding the identity of a metal with a mass of $1000\,g$ and volume of $128\,cm^3$, determining if a cube with volume $5\,cm^3$ and mass $20\,g$ is pure aluminum, calculating the volume of a $10\,kg$ gold bar, and determining the mass of copper required to occupy $35\,cm^3$.

Scientific Notation and Dimensional Analysis

Scientific notation is used to express very large or very small numbers by multiplying a number between 1 and 10 by a power of 10. Examples of conversions into scientific notation include:

• $123 = 1.23 \times 10^2$

• $4\,567 = 4.567 \times 10^3$

• $2\,067\,000 = 2.067 \times 10^6$

• $3.49 \times 10^{34}$ (representing 34 followed by 33 zeros)

• $0.16 = 1.6 \times 10^{-1}$

• $0.000049 = 4.9 \times 10^{-5}$

• $6.022 \times 10^{-24}$

Dimensional Analysis is a systematic problem-solving approach using conversion factors to move from one unit to another. A conversion factor is a ratio of two equivalent units. The process follows three steps:

1. Write the current ratio in fraction form.

2. Select a conversion factor that cancels the unwanted unit and leaves the desired unit.

3. Simplify and multiply.

Example calculations include converting $72\,in$ to feet ($6\,ft$) and finding the inches in $8\,yards$ ($288\,in$). Further conversion practice includes:

• $530\,ms \rightarrow s$

• $25\,cm \rightarrow m$

• $29\,kg \rightarrow g$

• $47\,Hm \rightarrow m$

• $4.6\,ds \rightarrow ms$

• $8.4\,km \rightarrow cm$

• $8\,hours \rightarrow seconds$

• $12.7\,kg \rightarrow mg$

• $94\,cm \rightarrow Hm$

Uncertainty in Data: Accuracy and Precision

Accuracy refers to how close a measured value is to the accepted or true value. This is typically assessed by checking the average of several trials. Precision refers to how close a series of measurements are to each other, which is assessed through the standard deviation of the data runs.

Experimental errors can be categorized as:

• Random errors: Result in a spread of data around a mean; high random error leads to high uncertainty in single measurements even if accuracy of the average is high.

• Systematic errors: Occur when measurements are consistently off in one direction; these result in good precision but low accuracy.

Percent error is a formula used to evaluate the accuracy of experimental data: $\%\,error = \frac{|\#\text{experimental} - \#\text{actual}|}{\#\text{actual}} \times 100$

Example iron measurement: Lab value of $7.35\,g/cm^3$ vs. accepted value of $7.87\,g/cm^3$. Example mercury measurement: Lab value of $14.2\,g/cm^3$ vs. accepted value of $13.45\,g/cm^3$.

Significant Figures and Rounding

Measurement precision depends on the device used. A beaker might allow a measurement of $3 \pm 1\,mL$, whereas a graduated cylinder allows $25.2 \pm 0.1\,mL$, and a buret provides $25.28 \pm 0.01\,mL$. Significant figures represent all known digits plus one estimated digit.

Rules for Significant Figures:

• Non-zero digits are always significant.

• Zeros between non-zero digits are significant.

• Zeros in front of non-zero digits (leading zeros) are not significant.

• Zeros at the end of a number to the right of a decimal point are significant.

• Zeros at the end of a number to the left of a decimal point may or may not be significant depending on measurement precision.

• Exact values (e.g., a count of 24 students) have an infinite number of significant digits.

Calculation Rules:

• Multiplication and Division: The answer should have the same number of significant figures as the measurement with the fewest significant figures.

• Addition and Subtraction: The answer is determined by the fewest decimal places in the measurements.

Representing Data with Google Sheets

Google Sheets is a digital tool for handling and visualizing data. Formulas always begin with the equal sign ($=$). Basic operations include addition ($+$), multiplication ($*$), division ($/$), and the use of parentheses. The AVERAGE function calculates the mean of a range, denoted as =AVERAGE(A3:A7), where the colon indicates a range of cells.

Data Visualization Procedures:

• Charts: Highlighting data and using the "Insert >> Chart" menu allows for the creation of line charts, column charts, and pie charts.

• Formatting: Use the Chart Editor to customize titles, vertical axis labels (e.g., "Temperature (F)"), and colors. Use the "+ Plus" sign to add new sheets within a document.

• Trendlines: In a scatter plot of mass vs. volume, a trendline equation represents the density of the substance ($y = mx + b$).

• Slope ($m$): For line graphs, the slope indicates the linear relationship between variables, acting like a conversion factor.

Data Interpretation:

• Interpolation: Estimating a value between known data points along the function line.

• Extrapolation: Extending the line on a graph to predict values beyond the range of measured data.