Ch.2 Measurement and Problem Solving – Study Notes

Measurement and Problem Solving - CH. 2

Scientific Notation

Scientific notation is a method used to express very large or very small numbers in a compact form, making them easier to read and work with in calculations.

• Coefficient: The number at the front, between 1 and 10 (but not including 10).

• Base: Always 10 in scientific notation.

• Exponent: Indicates how many times the coefficient is multiplied or divided by 10.

Example: $6.58 \times 10^{12}$

• Correct Format: The coefficient must be between 1 and 10, and the exponent is an integer.

Scientific Notation to Standard Notation: Move the decimal point to the right (positive exponent) or left (negative exponent) as indicated by the exponent.

Standard Notation to Scientific Notation: Move the decimal point to create a coefficient between 1 and 10, and count the number of places moved to determine the exponent.

SI Units (Simplified)

The International System of Units (SI) is the standard system for measurement in science, based on seven base units.

Physical Quantity	Name	Symbol
Length	meter	m
Mass	kilogram	kg
Time	second	s
Temperature	kelvin	K
Amount of substance	mole	mol
Electric current	ampere	A
Luminous intensity	candela	cd

Example: 1.25 kg is given in SI units (mass).

Metric Prefixes

Metric prefixes are modifiers that indicate multiples or fractions of base units, making it easier to express very large or small quantities.

Prefix	Symbol	Multiplier
kilo	k	$10^3$
centi	c	$10^{-2}$
milli	m	$10^{-3}$
micro	\mu	$10^{-6}$
nano	n	$10^{-9}$
pico	p	$10^{-12}$

Memory Tool: "The Great Monarch King Henry's Daughter Barbara drinks chocolate milk until nine pm." (Mnemonic for kilo, hecto, deca, base, deci, centi, milli, micro, nano, pico)

Example: Convert 654 kg to g: $654 \text{ kg} \times \frac{1000 \text{ g}}{1 \text{ kg}} = 654,000 \text{ g}$

Significant Figures

Significant figures (sig figs) are the digits in a measurement that are known with certainty plus one estimated digit. They reflect the precision of a measurement.

• Exact Numbers: Have an infinite number of significant figures (e.g., 12 eggs, 1 dozen).

• Measured Numbers: Have a limited number of significant figures, determined by the measuring instrument.

Rules for Significant Figures:

• All nonzero digits are significant.

• Zeros between nonzero digits are significant.

• Leading zeros are not significant.

• Trailing zeros are significant only if there is a decimal point.

Example: 0.003840 has 4 significant figures.

Precision in Measurements

Precision refers to how close repeated measurements are to each other. When recording a measurement, always estimate one digit beyond the smallest marked unit.

• Eyeball Test: Estimate the last digit based on the instrument's scale.

Significant Figures in Calculations

• Multiplication/Division: The result should have the same number of significant figures as the value with the fewest significant figures.

• Addition/Subtraction: The result should have the same number of decimal places as the value with the fewest decimal places.

Example: $3.0 \times 0.00567 = 0.017$ (rounded to 2 significant figures)

Conversion Factors

A conversion factor is a ratio that expresses how many of one unit are equal to another unit. It is used to convert measurements from one unit to another.

• Example: 1 day = 24 hours, so the conversion factor is $\frac{24 \text{ hr}}{1 \text{ day}}$ or $\frac{1 \text{ day}}{24 \text{ hr}}$.

Length	Volume	Mass
1 in = 2.54 cm	1 L = 1000 mL	1 kg = 1000 g
1 ft = 12 in	1 mL = 1 cm3	1 g = 1000 mg

Dimensional Analysis

Dimensional analysis is a systematic approach to problem solving that uses conversion factors to move from one unit to another.

• Write the given amount.

• Multiply by conversion factors so that units cancel appropriately.

• Continue until the desired unit is reached.

Example: To convert 115 min to years:

• 115 min × $\frac{1 \text{ hr}}{60 \text{ min}}$ × $\frac{1 \text{ day}}{24 \text{ hr}}$ × $\frac{1 \text{ year}}{365 \text{ days}}$

Density

Density is a physical property defined as the mass of a substance per unit volume.

Formula:

$\text{Density} = \frac{\text{Mass}}{\text{Volume}}$

• For solids and liquids, units are typically g/cm3 or g/mL.

• For gases, units are often g/L.

Example: If the density of a metal is 2.14 g/cm3, express it in kg/m3:

$2.14 \text{ g/cm}^3 \times \frac{1000 \text{ kg}}{1,000,000 \text{ g}} \times \frac{1,000,000 \text{ cm}^3}{1 \text{ m}^3} = 2,140 \text{ kg/m}^3$

Density of Non-Geometric Objects: Water Displacement

Water displacement is a method used to determine the volume of an irregularly shaped object by measuring how much water it displaces when submerged.

• Example: If the water level rises from 200 mL to 265 mL after submerging an object, the object's volume is 65 mL.