Chapter 2 Notes: Measurement and Calculations

Measurement and Calculations

Scientific Notation

• A clear way to write large or small numbers in a compact form.

• Clarifies the number of significant digits.

• Expressed as a number multiplied by 10 raised to a power (exponent).

• Example: 2.468 \times 10^8

• 10^8 means multiplying the number by 100 million.

• Moving the decimal place to the right increases the number.

• Moving the decimal place to the left decreases the number.

• 2 \times 10 = 20 (2 with one additional zero)

• 2 \times 10^1 is equivalent to moving the decimal one place to the right.

• Example: convert 0.000423 into scientific notation

  • Count how many times the decimal place must be moved to get 4.23.

  • 4.23 \times 10^{-4}

• Negative exponent indicates a number less than one (a fraction).

• Positive exponent indicates a large number.

• Moving decimal to the right results in a negative exponent.

• Moving decimal to the left results in a positive exponent.

• Example: Write 353,000 to 3 significant figures in scientific notation.

  • 3.53 \times 10^5

  • Move the decimal 5 places to the left, so the exponent is positive 5.

  • 3 significant figures means recording only 3 non-placeholder digits.

Measurement and Uncertainty

• Careful measurement is crucial in both the lab and the kitchen.

• A measurement includes both an amount and a unit.

• Scientists make measurements to confirm or refine hypotheses.

• Measurements always require a unit.

• Estimation is involved in measurement (e.g., reading a thermometer).

• The last digit in a measurement is often a judgment call, leading to uncertainty.

• Example: On a thermometer, you might clearly see 21 degrees, but estimating the next digit is subjective.

• Distinction between exact numbers (counting 10 people) and measurements (temperature) with uncertainty.

• By convention, a measurement includes all certain digits plus one uncertain digit.

• These certain and uncertain digits are called significant figures.

• The number of digits recorded depends on the accuracy of the measuring tool.

• More graduations on a thermometer allow for more significant figures.

• Uncertainty is inherent in measurement because measurement is observation.

Significant Figures

• Rules for Significant Figures:

  • Non-zero digits are always significant.

  • Exact numbers (e.g., counting 10 people) have no uncertainty and do not affect significant figures.

  • Zeros between non-zero digits are significant.

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

  • Zeros at the beginning of a number are not significant (placeholders).

• Using scientific notation clarifies significant figures by removing ambiguity about placeholder zeros.

• Example: 6920 - The zero may or may not be significant.

  • To show the zero is significant: 6.920 \times 10^3 or 6920.

• Practice Examples

  • 20 has 2 sig figs

  • 1.030 has 4 sig figs (zeros after decimal count)

  • 103 people has infinite sig figs (exact number)

  • 0.03 has 1 sig fig (leading zeros don't count)

  • 3.0 \times 10^{-3} km clarifies one significant figure.

Rounding

• Keep many digits while working through a problem, but round the final answer to the correct number of sig figs.

• Rounding Rules:

  • If the digit after the rounding place is less than 5, drop the other digits.

    • Example: 53.2305 rounded to the tenth place is 53.2 (because 3 < 5).

  • If the digit after the rounding place is 5 or greater, round up.

    • Example: 11.789 rounded to the tenth place is 11.8 (because 8 > 5).

• Examples:

  • 79.137 to 4 sig figs: 79.14 (round up because of the 7)

  • 0.4345 to 3 sig figs: 0.435 (round up because of the 5)

  • 136.2 to 3 sig figs: 136 (drop the 2)

  • 0.179 to 2 sig figs: 0.18 (round up because of the 9)

Significant Figures in Calculations

• The result of a calculation is only as precise as the least precise measurement.

• Multiplication and Division:

  • Keep as many digits as possible until the end of the calculation.

  • Round the final answer to the lowest number of sig figs in the problem.

    • Example: 79.2 inches x 1.1 inches = 87.12, round to 87 (2 sig figs).

  • Example: 12.18 x 5.2 x 0.13 Calculation comes out to 0.082216. All values have 2 sig figs.

• Addition and Subtraction:

  • Align numbers based on the decimal point.

  • The answer should have the same number of decimal places as the number with the fewest decimal places.

    • Example: 167.9 + 79 = 246.9. Round to 247 (no digits past the decimal).

    • Example: 142.57 - 13.0 = 129.57. Round to 129.6 (one decimal place).

Metric System (SI - System International)

• Standard system of measurements for physical quantities.

• Each quantity has a standard unit, traceable to a repeatable standard.

• Important standard units:

  • Temperature: Kelvin (K)

    • Reason for using Kelvin: Absolute zero (absence of heat and molecular motion).

  • Length: meter (m)

  • Mass: kilogram (kg)

  • Time: second (s)

  • Amount of substance: mole (mol)

• Prefixes modify metric units to make them bigger or smaller:

  • Examples: centimeter, millimeter, nanometer

  • Important prefixes:

    • Kilo (k): 10^3

    • Centi (c): 10^{-2}

    • Milli (m): 10^{-3}

    • Micro ($\mu$): 10^{-6}

    • Nano (n): 10^{-9}

• Measurements of Length:

  • 1 meter = 100 cm = 1000 mm

  • 1 kilometer (km) = 1000 meters (m)

  • 1 inch = 2.54 cm (standard definition)

Dimensional Analysis

• Stepwise problem solving using conversion factors.

• Conversion factors: Ratios of equivalent quantities used to convert one measurement to another.

• Example: Convert 3 hours into seconds.

  • Conversion factors: 60 minutes = 1 hour, 60 seconds = 1 minute.

  • 3 \text{ hours} \times \frac{60 \text{ min}}{1 \text{ hour}} \times \frac{60 \text{ s}}{1 \text{ min}} = 10,800 \text{ seconds}

• Units cancel out when the conversion factors are set up correctly.

  • Hours and minutes cancel, leaving only seconds.

• Significant figures considered for initial measurement (3 hours).

• Dimensional analysis uses conversion factors or ratios of equivalent quantities to convert one amount into another.

• Use significant figures to avoid increasing the accuracy of the answer beyond the measurements.

• Both conversion factors are equal to one because they have equal quantities on the top and bottom.

• Example: converting meters to kilometers

  • Conversion factor: 1 km = 1000 m
    Conversion:
    20 \text{ m} \times \frac{1 \text{ km}}{1000 \text{ m}} = 0.02 \text{ km}

• Example: Convert centimeters to meters

  • Conversion factor: 1 meter = 100 centimeters.

  • 215 \text{ cm} \times \frac{1 \text{ m}}{100 \text{ cm}} = 2.15 \text{ m}

• Example: Converting Meters to Kilometers

  • Conversion factor: 1 kilometer = 1,000 meters.

  • 125 \text{ m} \times \frac{1 \text{ km}}{1000 \text{ m}} = 0.125 \text{ km}

• Example: How many micrometers are in 0.03 meters.

  • Conversion factor: 1 meter equals 1 \times 10^6 micrometers.

  • 0.03 \text{ m} \times \frac{1 \times 10^6 \mu \text{m}}{1 \text{ m}} = 3 \times 10^4 \mu \text{m}

• Example: Converting Days to Seconds.

• Example: Converting Feet and Centimeters

  • Conversion factor: 1 \text{ inch} = 2.54 \text{ cm}, 1 \text{ foot} = 12 \text{ inches}

  • 250 \text{ cm} \times \frac{1 \text{ inch}}{2.54 \text{ cm}} \times \frac{1 \text{ foot}}{12 \text{ inches}} = 8.20 \text{ feet}

• Example: converting yards to meters

  • Conversion factors: 1 \text{ meter} = 39.37 \text{ inches}, 1 \text{ yard} = 36 \text{ inches}

  • 5 \text{ yd} \times \frac{36 \text{ in}}{1 \text{ yd}} \times \frac{1 \text{ m}}{39.37 \text{ in}} = 4.57 \text{ m}

Measuring Volume

• Converting inches cubed to centimeters cubed

  • To do this we use the same conversion factor between inches and centimeters that we already know, but we cube it, in order to go from inches cubed to centimeters cubed
    V = 2.20 \text{ in} \times 4.00 \text{ in} \times 6.00 \text{ in} = 52.8 \text{ in}^3

• Since we know 1 \text{ in} = 2.54 \text{ cm}, we perform the conversion
  52.8 \text{ in}^3 \times (\frac{2.54 \text{ cm}}{1 \text{ in}})^3 = 865 \text{ cm}^3

Mass vs Weight

• Mass: The absolute amount of matter, measured on a balance and independent of location.

• Weight: The force of gravity on an object, dependent on location.

• SI unit of mass: kilogram (kg).

• Important units of mass:

  • 1 kg = 2.2015 pounds

  • kilogram (kg) = 1000 grams

  • gram (g)

  • milligram (mg)

  • microgram ($\mu$g)

• To convert 343 grams to kilograms:
  343 \text{ g} \times \frac{1 \text{ kg}}{1000 \text{ g}} = 0.343 \text{ kg}

• Conversion Example: How many centigrams are in 0.12 kg.

  • Conversion factor: 1 \text{ kg} = 1000 \text{ g}, 1 \text{ g} = 100 \text{ cg}

  • 0.12 \text{ kg} \times \frac{1000 \text{ g}}{1 \text{ kg}} \times \frac{100 \text{ cg}}{1 \text{ g}} = 1.2 \times 10^4 \text{ cg}

Volume

• Volume: The amount of space occupied by a certain amount of matter.

• SI unit for volume: cubic meter (\text{m}^3). Liter (L) and milliliter (mL) are more commonly used.

• 1 \text{ mL} = 1 \text{ cm}^3

• Common volume relationships:

  • 1 \text{ L} = 1000 \text{ mL}

  • 1 \text{ mL} = 1 \text{ cm}^3

  • 1 \text{ L} = 1.057 \text{ quarts}

• Volume conversion Example

  • 0.3.45 \text{ L} \times \frac{1000 \text{ mL}}{1 \text{ L}} = 345 \text{ mL}

Percent

• Percents is parts per 100

  • \% = \frac{\text{parts}}{\text{total parts}} \times 100\%

Temperature

• Temperature: Measurement of thermal energy.

• Thermal energy: Energy of motion of the small particles that make up matter.

• Heat always flows from regions of high temperature to regions of low temperature.

• Temperature Scales.

  • Kelvin (K): Absolute zero is 0 K (no negative values).

  • Celsius (\textdegree C)

  • Fahrenheit (\textdegree F)

• Temperature Conversion Formulas:

  • Kelvin = Celsius + 273.15

  • Fahrenheit = (9/5 \times Celsius) + 32

Density

• Density: Mass divided by volume (\frac{\text{mass}}{\text{volume}}).

• Physical property.

• Units: grams per milliliter (g/mL) or grams per cubic centimeter (g/cm^3).

  • Density is specific to temperature.

• Density can be used as a conversion factor.

Okay. I have updated Chapter 2 with just the slides on temperature, so we're going to go over those quickly, and then we'll move on so we talked about thermal energy temperature and heat. In the Si unit of temperature. And just briefly, thermal energy is associated with the motion of molecules. So when you get down to 0 in on the Kelvin temperature scale, that's absolute 0 in which there is no longer molecular motion. When you feel something that is hot or cold. That is a measure of the difference in thermal energy between like your hands and what you're touching, and heat always flows from a higher temporary temperature, object to a lower temperature object.
There are different temperature scales. So there's Celsius, Fahrenheit and Kelvin. Celsius and Kelvin are the same scale. But just different different starting points. Celsius and Fahrenheit are degrees, but with the size of the degree is different, because the width or the the amount of degrees measured between freezing point and boiling point of water is different. 0 to 100 versus 32 to 2 12 so there's a the Fahrenheit scale has a hundred 80 degree range between freezing and boiling.
whereas on the Celsius scale it's only a hundred.
first problem. Convert 723\u00b0C to temperature in both Kelvin, which has no degrees, remember, and degrees Fahrenheit.
Karen Welch02:25to Calvin. Okay, so we choose our formula up here. Oops Kelvin equals degree C, plus 2 73.1 5. So
Kelvin temperature equals 7, 23, plus 2 73 point 1 5
So that's 9 96 point 1 5. But remember, when we add with significant figures. we don't take what's passed. The least precise number. So we're gonna say, the temperature is 9, 96.
degrees Fahrenheit degrees Fahrenheit equals 9 fifths times the Celsius temperature plus 32. So that is 9 fifths times 7, 23 plus 32, that is. 1,301.4 plus 32, which is 1,333.4 trees. If but also remember that same thing with Sigfakes. We're gonna get rid of this. So it is 1,333 degrees F and 996. Kelvin. Okay, so those are the worked out problems.
What is the temperature? If 98.6\u00b0F is converted to degrees Celsius.
So first, let's pull our formula in degrees. F equals 9 fifth degree C. Plus 32. So let's write that degree. Write that down degrees F equals 9 fists times degree C, plus 32. So the issue here is that we have the degrees Fahrenheit is known. But we have to find this the degrees Celsius is not known, so what we need to do first is rearrange the equation so that we're solving for the correct quantity.
So first, we're gonna do that. We're gonna move the 32 over so degrees. If minus 32 cause, we're subtracting 32 from both sides to get rid of it equals 9 fifth time degree C,
and then to get degree C by itself. We're going to divide by 9 fifths, which is the same thing as multiplying by its reciprocal. So 5 9 times degrees F. Minus 32 equals degrees C. So now we can solve for degrees C, and get the correct answer. So 5 nights times 98.6 minus 32 equals degree. C,
so we have 98.6 minus