CHM Ch 2

Measurement: A Foundation of Good Science

• Measurement is fundamental to good science and essential for various activities beyond science, such as graphic design, cooking, and construction.

• In chemistry, it is crucial to make accurate measurements, understand their limitations, and communicate findings clearly.

• The adage "Measure twice, cut once" emphasizes the importance of accurate measurement to avoid wasting time and materials.

Scientific Notation: Working with Very Large and Very Small Numbers

• Challenge: Science often deals with extremely large or small numbers, such as the distance from Earth to the Sun ($149,600,000,000$ meters) or the mass of a hydrogen atom ($0.000000000000000000000001672$ grams).

• Solution: Scientific notation is a concise way to express these numbers.

• Format: Numbers are expressed as the product of two values: a coefficient and a multiplier.

  • Coefficient: A measured value with only one non-zero digit before the decimal point.

  • Multiplier: $10$ raised to an exponent ($10^x$). This term shifts the decimal point left or right to yield the value in standard form.

• Example: $5,100$ expressed in scientific notation is $5.1 imes 10^3$.

• Powers of Ten (Table 2.1):

  • $10^3 = 1,000$

  • $10^2 = 100$

  • $10^1 = 10$

  • $10^0 = 1$

  • $10^{-1} = 0.1$

  • $10^{-2} = 0.01$

  • $10^{-3} = 0.001$

• Relationship: Each increase of one in the exponent means the value is ten times greater, shifting the decimal one place.

• The exponent in scientific notation indicates the number of spaces the decimal point must be moved.

Converting from Scientific Notation to Standard Notation

• Move the decimal the number of units indicated by the exponent.

• Positive exponents: Move the decimal point to the right to create a large number.

  • Example: $5.1 imes 10^1 = 51.$ (1 space right)

  • Example: $5.1 imes 10^4 = 51,000.$ (4 spaces right)

• Negative exponents: Move the decimal point to the left to create a small number.

  • Example: $5.1 imes 10^{-1} = 0.51$ (1 space left)

  • Example: $5.1 imes 10^{-4} = 0.00051$ (4 spaces left)

• Summary (Table 2.2):

  • Large numbers have a positive exponent.

  • Small numbers have a negative exponent.

Converting from Standard Notation to Scientific Notation

• Move the decimal point until only one nonzero digit remains before it in the coefficient.

• The number of places the decimal is moved becomes the exponent of the $10^x$ multiplier.

• Use negative exponents for small values (decimal moved right) and positive exponents for large values (decimal moved left).

• Example: $0.0000234$ becomes $2.34 imes 10^{-5}$.

Calculations Involving Scientific Notation

• Multiplication: Multiply the coefficients and add the exponents in the multipliers.

  • $(A imes 10^x) imes (B imes 10^y) = (A imes B) imes 10^{(x+y)}$

• Division: Divide the coefficients and subtract the exponent in the denominator from the exponent in the numerator.

  • $(A imes 10^x) / (B imes 10^y) = (A / B) imes 10^{(x-y)}$

• Example 2.1: Multiplication with Scientific Notation

  • Calculate $(3.95 imes 10^{-22}) imes (4.00 imes 10^{24})$.

  • Multiply coefficients: $3.95 imes 4.00 = 15.8$.

  • Add exponents: $-22 + 24 = 2$.

  • Initial answer: $15.8 imes 10^2$.

  • Proper Scientific Notation: Adjust the coefficient to have only one digit before the decimal point.

    • Move the decimal one place left: $1.58$.

    • Add one unit to the exponent to compensate: $2 + 1 = 3$.

    • Final Answer: $1.58 imes 10^3$.

• Calculator Caution: Different calculators handle scientific notation differently. Practice is encouraged.

Units of Measurement

• Importance: Units are quantities with accepted values that facilitate communication. Without standard units, measurements are meaningless (e.g., $1/4$ cup for a recipe needs a consistent standard).

• Variety: Many units exist for a single measurement (e.g., inches, meters, yards for length).

• Systems: Two main systems:

  • English Units: Commonly used in the United States (inches, feet, gallons, pounds).

  • Metric System: Used by most of the world.

• Relationships (Table 2.3):

  • Length: $1 ext{ m} = 3.281 ext{ ft}$; $1 ext{ km} = 0.621 ext{ mi}$

  • Mass/Weight: $1 ext{ kg} = 2.205 ext{ lb}$

  • Volume: $1 ext{ L} = 0.264 ext{ gal}$

• SI Units (International System of Units): An internationally accepted set of fundamental units, based on the metric system, for easier scientific communication.

• Fundamental (SI) Units (Table 2.4):

  • Mass: kilogram (kg)

  • Length: meter (m)

  • Time: second (s)

  • Temperature: kelvin (K)

  • Light intensity: candela (cd)

  • Electric current: ampere (A)

  • Amount: mole (mol)

• Derived Units: Units formed from combinations of fundamental units.

  • Liter (L) for volume (corresponds to $0.001$ cubic meters, $ext{m}^3$).

  • Meters per second (m/s) for velocity.

  • Kilograms per liter (kg/L) for density.

• Metric Prefixes: Used to describe larger or smaller amounts concisely, similar to scientific notation.

• Common Metric Prefixes (Table 2.5):

  • tera- (T): $10^{12}$ ($1,000,000,000,000$)

  • giga- (G): $10^9$ ($1,000,000,000$)

  • mega- (M): $10^6$ ($1,000,000$)

  • kilo- (k): $10^3$ ($1,000$)

  • deci- (d): $10^{-1}$ ($0.1$)

  • centi- (c): $10^{-2}$ ($0.01$)

  • milli- (m): $10^{-3}$ ($0.001$)

  • micro- ($\mu$): $10^{-6}$ ($0.000001$)

  • nano- (n): $10^{-9}$ ($0.000000001$)

  • pico- (p): $10^{-12}$ ($0.000000000001$)

• Example: $160,000,000$ bits can be expressed as $160$ megabits ($160$ Mb).

• Working with Small Units: $1 ext{ mg} = 1/1,000 ext{ g}$, which means $1,000 ext{ mg} = 1 ext{ g}$.

Describing the Quality of Measurements

• Uncertainty in Measurement: All measurements have some degree of uncertainty. It is acceptable to estimate the last number in a measurement, and other scientists understand this digit is an estimated value.

  • Example: Estimating a ball's diameter as $2.7$ cm, with the last digit being an estimate (could be $2.6$ or $2.8$ cm).

• Precision Tools: For meticulous work, more precise tools like a caliper (measuring to $0.01$ mm) are needed instead of rulers.

• Accuracy vs. Precision:

  • Accuracy: Refers to how reliable measurements are; how close a measurement (or average of measurements) is to the true value.

    • Example: A measurement of the ball might be accurate if its average is close to true value, even if individual readings vary slightly.

  • Precision: Refers to how finely a measurement is made, or how close groups of measurements are to each other (reproducibility).

    • Example: A laboratory balance measuring mass to $\pm 0.001$ gram is precise, showing fine measurement.

• Distinctions:

  • It is possible to be accurate without being precise (e.g., the initial ball measurement).

  • It is possible to be precise without being accurate (e.g., an uncalibrated balance gives consistent but incorrect readings).

• Visualization: Target analogy (Figure 2.5):

  • Accurate: Average hits the bullseye.

  • Precise: Shots are tightly grouped together.

• Calibration: The process of testing and adjusting instruments (e.g., laboratory balances) to ensure accurate readings.

Describing Precision: Significant Digits

• Significant Digits (Significant Figures): The digits contained in a measured value that indicate how precisely a measurement is made.

• Estimating Digits: When using a tool with marked units, estimate one digit between the marked values.

  • Example: A beaker marked to the tens place allows estimation to the ones place (e.g., $48$ mL, two significant digits).

  • Example: A graduated cylinder marked to the ones place allows estimation to the tenths place (e.g., $48.1$ mL, three significant digits).

  • More significant digits imply a more precise measurement.

• Example 2.2: Measuring Volume in a Graduated Cylinder

  • Meniscus: The curved surface of water in narrow glass cylinders due to adhesion.

  • Reading Rule: Report the volume at the bottom of the meniscus.

  • Estimation: If markings are at the ones place (e.g., $23$ mL, $24$ mL), estimate to the nearest tenth of a milliliter (e.g., $23.3$ mL, $23.2$ mL, or $23.4$ mL are all acceptable estimates).

Determining the Number of Significant Digits in a Measurement

1. Nonzero digits and Zeros between nonzero digits: Always significant.

   • Example: $25.006$ has five significant digits. $80004$ has five significant digits.

2. Zeros to the right of the last nonzero digit (if a decimal point is present): Always significant.

   • Example: $25.000$ has five significant digits. $4.50$ has three significant digits.

3. Zeros to the left of the nonzero numbers (and to the left of the decimal point): Never significant (placeholders).

   • Example: $0.0045$ has two significant digits. The zeros just show where the decimal point lies.

   • Converting units (e.g., $4.5$ millimeters to $0.0045$ meters) does not change the number of significant digits.

4. Zeros to the right of the last nonzero digit (if no decimal point is present): May or may not be significant; ambiguous.

   • Example: $10,000,000$ dollars – unclear if measured to the nearest million, thousand, etc.

   • To clarify:

     • Specify uncertainty: 10,000 { ext{ extpm }} 100 ext{ kg}.

     • Use scientific notation: $1.00 imes 10^4 ext{ kg}$ (this notation indicates three significant digits).

Working with Exact Numbers

• Exact numbers: Values with no uncertainty; significant digit rules do not apply.

• Types of Exact Numbers:

  1. Counted values: (e.g., $7$ pennies, $3$ tacos, $13$ floats).

  2. Defined relationships: (e.g., $12$ items in a dozen, $1,000$ milligrams in a gram, metric prefixes like $10^9$ bytes in a gigabyte).

Using Significant Digits in Calculations

• Calculators often display many digits, but it's the scientist's responsibility to report answers with the correct number of significant digits, reflecting the precision of the measurements.

• Rounding Rule:

  • If the last dropped digit is $5$ or greater, round up.

  • If the last dropped digit is $4$ or less, round down.

Rules for Reporting Calculations with Significant Digits

1. Multiplication or Division: Report the same number of significant digits as in the least precise (fewest significant digits) starting measurement.

   • Example 2.3: $283.1 ext{ Calories} / 55.023 ext{ grams} = 5.145121131… ext{ Calories/gram}$. Since $283.1$ has 4 sig figs and $55.023$ has 5 sig figs, the answer is rounded to 4 sig figs: $5.145 ext{ Calories/gram}$.

   • Example 2.4 (Multiplying by an Exact Number): Mass of 8 quarters ($5.670 ext{ g/quarter} imes 8 ext{ quarters}$). The integer $8$ is an exact number and does not limit significant digits. The measured quantity $5.670$ has 4 sig figs, so the answer is $45.36 ext{ grams}$.

2. Addition or Subtraction: Round the answer to the same last decimal place as the least precise starting measurement.

   • Example 2.5: Adding $3.62 ext{ L}$, $0.255 ext{ L}$, and $21.2 ext{ L}$. The least precise value ( $21.2 ext{ L}$) is known only to the tenths place. The sum $(25.075 ext{ L})$ is rounded to the tenths place: $25.1 ext{ L}$.

• Multiple Calculations (Example 2.6):

  • When performing a series of calculations, carry extra digits through intermediate steps to minimize rounding errors.

  • Round only the final answer to the correct number of significant digits based on the rules for the final operation.

• Example 2.6 (Cont.): Adding $12.83 ext{ mg}$, $11.2 ext{ mg}$, $14.391 ext{ mg}$.

  • Sum: $38.421 ext{ mg}$.

  • $11.2 ext{ mg}$ is least precise (tenths place). Rounded sum: $38.4 ext{ mg}$. (This has 3 sig figs).

  • Then, to find average mass per liter if total volume is $1.514 ext{ L}$. We divide $38.421 ext{ mg}$ (unrounded intermediate) by $1.514 ext{ L}$.

  • $(38.421 ext{ mg}) / (1.514 ext{ L}) = 25.377$.

  • $38.421$ has 5 sig figs (from sum of multiple measurements), $1.514$ has 4 sig figs. Since we are dividing, we round to the least number of sig figs, which is 4.

  • Final Answer: $25.38 ext{ mg/L}$. (Note: The provided example text implies using the 3 sig figs from 38.4, leading to 25.4 m/L but the reasoning for 3 sig figs for 38.4 g is also based on the least precise value for sum. The overall principle is to not round until the very end, hence 38.421 is used for calculation. Then after rounding for significant digits for 38.421, it is 3 sig figs as 38.4. And the volume is 4 sig figs. Thus the lowest is 3 sig figs for the quotient.)

Unit Conversion

• Necessity: Often required for international travel, scientific experiments, and various other situations.

• Dimensional Analysis (Factor-Label Method): A systematic approach to unit conversions.

  • Core Principle: Whatever mathematical operation is applied to the number, it must also be applied to the units.

  • Example: Area = length $imes$ width. If length and width are in feet, area is in $ext{ft} imes ext{ft} = ext{ft}^2$.

• Conversion Factors: Fractions equal to one, with equivalent amounts of different units in the numerator and denominator (e.g., $\frac{1 ext{ kg}}{1,000 ext{ g}}$ or $\frac{1,000 ext{ g}}{1 ext{ kg}}$).

  • Multiplying by a conversion factor changes only the units, not the underlying value.

  • Selection: Choose the conversion factor that allows the old units to cancel out and leaves the new unit.

• Checking Work: If units do not cancel out correctly, the problem was likely set up incorrectly.

• Example 2.7: Converting between Units

  • Convert $150.0$ calories to joules (given $1 ext{ calorie} = 4.184 ext{ joules}$).

  • $150.0 ext{ calories} imes \frac{4.184 ext{ joules}}{1 ext{ calorie}} = 627.6 ext{ joules}$.

• Example 2.8: Converting between Units

  • Convert $815.2$ grams to kilograms (given $1 ext{ kg} = 1,000 ext{ g}$).

  • $815.2 ext{ g} imes \frac{1 ext{ kg}}{1,000 ext{ g}} = 0.8152 ext{ kg}$.

Problems Involving Multiple Conversions

• When multiple conversions are needed, conversion factors can be placed adjacent to each other in a single calculation.

• Example 2.9: Multiple Conversions

  • Convert $0.0129$ inches to micrometers (given $2.54 ext{ cm} = 1 ext{ inch}$ and $10,000 \mu \text{m} = 1 ext{ cm}$).

  • $0.0129 ext{ inches} imes \frac{2.54 ext{ cm}}{1 ext{ inch}} imes \frac{10,000 \mu \text{m}}{1 ext{ cm}} = 328 \mu \text{m}$.

  • Rounded to three significant digits (from $0.0129$ inches).

• Example 2.10: Converting Units in the Denominator

  • Convert $768$ miles/hour to meters/second (given $1 ext{ mile} = 1,609.3 ext{ meters}$ and $1 ext{ hour} = 3,600 ext{ seconds}$).

  • $\frac{768 ext{ miles}}{ ext{hour}} imes \frac{1,609.3 ext{ meters}}{1 ext{ mile}} imes \frac{1 ext{ hour}}{3,600 ext{ seconds}} = 343 ext{ m/s}$.

  • Rounded to three significant digits (from $768$ miles/hour).

Converting between Volume Units

• Volume Measurement: Uses units of length raised to the third power (e.g., $ext{cm}^3$ = cubic centimeters, $ext{m}^3$ = cubic meters).

• Liter (L): A common derived unit of volume. Defined as $1$ cubic decimeter ($1 ext{ dm}^3$), roughly the size of a tissue box.

  • $1 ext{ L} = 1 ext{ dm}^3$

• Cubic Centimeter ($ ext{cm}^3$): Another common unit, equivalent to a milliliter (mL). Roughly the size of a standard game die.

  • $1 ext{ mL} = 1 ext{ cm}^3$

• Converting Units Raised to a Power: When converting between units raised to a power, both the numbers and the units must be raised to that power.

  • Example: $1 ext{ m} = 10 ext{ dm}$. Cubing both sides: $(1 ext{ m})^3 = (10 ext{ dm})^3 \implies 1 ext{ m}^3 = 1000 ext{ dm}^3$.

• Example 2.11: Conversions with Cubic Units

  • How many cubic centimeters ($ext{cm}^3$) are in $1$ cubic meter ($ext{m}^3$)?

  • $1 ext{ m} = 100 ext{ cm}$.

  • $(1 ext{ m})^3 = (100 ext{ cm})^3 \implies 1 ext{ m}^3 = 1,000,000 ext{ cm}^3$.

• Example 2.12: Conversions with Volume Units

  • Convert $15.3 ext{ cm}^3$ to liters.

  • $15.3 ext{ cm}^3 imes \frac{1 ext{ mL}}{1 ext{ cm}^3} imes \frac{1 ext{ L}}{1,000 ext{ mL}} = 0.0153 ext{ L}$.

• Example 2.13: Conversions with Volume Units (Multiple)

  • An IV drip is set at $125 ext{ mL/h}$. Convert this to liters per day.

  • $\frac{125 ext{ mL}}{ ext{hour}} imes \frac{1 ext{ L}}{1,000 ext{ mL}} imes \frac{24 ext{ hours}}{1 ext{ day}} = 3.00 ext{ L/day}$.

  • The conversion factors are exact, so the answer retains the three significant digits of the measured drip rate.

Density: Relating Mass to Volume

• Definition: Density ($d$) is the mass ($m$) of a substance per unit volume ($V$).

  • Formula: $d = \frac{m}{V}$.

• Importance: Density is a crucial physical property. Materials with low densities (e.g., aluminum, titanium) are valued for lightweight applications (e.g., prosthetic limbs, aircraft).

• Densities of Common Materials (Table 2.6):

  • Aluminum: $2.70 ext{ g/cm}^3$

  • Titanium: $4.51 ext{ g/cm}^3$

  • Iron: $7.87 ext{ g/cm}^3$

  • Copper: $8.96 ext{ g/cm}^3$

  • Lead: $11.34 ext{ g/cm}^3$

  • Gold: $19.31 ext{ g/cm}^3$

  • Water: $1.00 ext{ g/cm}^3$

  • Seawater: $1.02 ext{ g/cm}^3$

  • Air: $0.001 ext{ g/cm}^3$

• Measuring Density: Determine mass using a balance and volume using equipment like a graduated cylinder.

• Calculation Example: For a sample with mass $11.29 ext{ g}$ and volume $10.4 ext{ mL}$, the density is $\frac{11.29 ext{ g}}{10.4 ext{ mL}} = 1.09 ext{ g/mL}$ (rounded to 3 sig figs).

Converting between Mass and Volume

• Density serves as a conversion factor between mass and volume.

• Example 2.14: Conversions Using Density I (Volume to Mass)

  • An iron sample has a volume of $8.83 ext{ cm}^3$. Density of iron is $7.87 ext{ g/cm}^3$. Find its mass.

  • Rearrange formula: $m = dV$

  • $m = (7.87 ext{ g/cm}^3)(8.83 ext{ cm}^3) = 69.5 ext{ g}$ (units of $ext{cm}^3$ cancel).

• Example 2.15: Conversions Using Density II (Mass to Volume)

  • A $1.213 ext{ kg}$ sample of mercury (density $13.534 ext{ g/mL}$). Find its volume.

  • First, ensure consistent mass units: Convert $1.213 ext{ kg}$ to $1,213 ext{ g}$.

  • Rearrange formula: $V = \frac{m}{d}$

  • $V = \frac{1,213 ext{ g}}{13.534 ext{ g/mL}} = 89.63 ext{ mL}$ (grams cancel, rounded to 4 sig figs).

Will It Float?

• An object's density determines if it floats or sinks in a liquid.

• Density of pure water: $1 ext{ g/cm}^3$ or $1 ext{ g/mL}$.

• Floating: Objects less dense than water ($d < 1 ext{ g/cm}^3$) float.

• Sinking: Objects more dense than water ($d > 1 ext{ g/cm}^3$) sink.

• Example 2.16: Will It Float?

  • A toy submarine has a volume of $14.3 ext{ mL}$ and a mass of $15.2 ext{ g}$.

  • Calculate density: $d = \frac{15.2 ext{ g}}{14.3 ext{ mL}} = 1.06 ext{ g/mL}$.

  • Since $1.06 ext{ g/mL} > 1 ext{ g/mL}$ (density of water), the toy will sink.

Measuring Temperature

• Three Common Temperature Scales:

  1. Fahrenheit ($^{\circ}\text{F}$): Most commonly used in the United States.

  2. Celsius ($^{\circ}\text{C}$): Most of the world uses this scale. Freezing point of water is $0^{\circ}\text{C}$, boiling point is $100^{\circ}\text{C}$.

  3. Kelvin (K): Used by scientists for very low temperatures and for describing gas behavior.

• Scale Comparison (Figure 2.19):

  • The difference between freezing and boiling points of water is $180^{\circ}\text{F}$ vs. $100^{\circ}\text{C}$. This means a degree Fahrenheit is a smaller unit than a Celsius degree.

  • A unit on the Kelvin scale is the same size as a degree Celsius.

• Conversion Formulas:

  • $^{\circ}\text{F} = \frac{9}{5} ^{\circ}\text{C} + 32$

  • $^{\circ}\text{C} = \frac{5}{9} (^{\circ}\text{F} - 32)$

  • $\text{K} = ^{\circ}\text{C} + 273.15$

• Kelvin Scale Specifics:

  • Zero Point: $0^{\circ}\text{C}$ is the freezing point of water. $0\text{ K}$ (absolute zero) is the lowest possible temperature, where particles have zero kinetic energy.

  • Reporting: Use