Measurements

Module 2: Measurements

Introduction to Measurements

  • Measurements are quantities that can be measured (e.g., length, mass, volume, temperature).

  • A number and a unit are required to describe a physical quantity; the measurement's uncertainty should also be indicated.

  • Example: 61.2 kilograms

  • Without units, a number can be meaningless or confusing.

  • Chemistry uses the International System of Units (SI units), an updated version of the metric system (1964).

International System of Units (SI)

  • Base Quantities and Units of the SI System:

    • 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)

Temperature Conversions

  • Temperature measures the "hotness" or "coldness" of an object.

    • Units: Celsius, Fahrenheit, Kelvin

  • Heat is a form of energy (thermal energy) transferred between objects.

    • Units: Joules

  • Key Temperature Points:

    • Boiling point of water:

      • 212 °F

      • 100 °C

      • 373.15 K

    • Freezing point of water:

      • 32 °F

      • 0 °C

      • 273.15 K

    • ToF=95×T°c+32ToF = \frac{9}{5} × T_{°c} + 32

    • T<em>K=273.15+T</em>cT<em>K = 273.15 + T</em>c

  • Example Conversion:

    • Convert 27.60°C to Kelvin.

    • T<em>in Kelvin=T</em>°C+273.15T<em>{in \ Kelvin} = T</em>{°C} + 273.15

    • Tin Kelvin=27.60°C+273.15=300.75KT_{in \ Kelvin} = 27.60°C + 273.15 = 300.75 K

Common Unit Prefixes

  • Prefixes to memorize:

    • femto (f): 10−1510^{-15}

    • pico (p): 10−1210^{-12}

    • nano (n): 10−910^{-9}

    • micro (µ): 10−610^{-6}

    • milli (m): 10−310^{-3}

    • centi (c): 10−210^{-2}

    • deci (d): 10−110^{-1}

    • kilo (k): 10310^{3}

    • mega (M): 10610^{6}

    • giga (G): 10910^{9}

    • tera (T): 101210^{12}

    • Angstrom (Å) = 10−1010^{-10} m

  • How to use the table (using meter as an example):

    • femto (f): 1 fm = 1×10−151 × 10^{-15} m or 1×10151 × 10^{15} fm = 1 m

    • pico (p): 1 pm = 1×10−121 × 10^{-12} m or 1×10121 × 10^{12} pm = 1 m

    • nano (n): 1 nm = 1×10−91 × 10^{-9} m or 1×1091 × 10^{9} nm = 1 m

    • micro (µ): 1 µm = 1×10−61 × 10^{-6} m or 1×1061 × 10^{6} µm = 1 m

    • milli (m): 1 mm = 1×10−31 × 10^{-3} m or 1×1031 × 10^{3} mm = 1 m

    • centi (c): 1 cm = 1×10−21 × 10^{-2} m or 1×1021 × 10^{2} cm = 1 m

    • deci (d): 1 dm = 1×10−11 × 10^{-1} m or 1×1011 × 10^{1} dm = 1 m

    • kilo (k): 1 km = 1×1031 × 10^{3} m or 1×10−31 × 10^{-3} km = 1 m

    • mega (M): 1 Mm = 1×1061 × 10^{6} m or 1×10−61 × 10^{-6} Mm = 1 m

    • giga (G): 1 Gm = 1×1091 × 10^{9} m or 1×10−91 × 10^{-9} Gm = 1 m

    • tera (T): 1 Tm = 1×10121 × 10^{12} m or 1×10−121 × 10^{-12} Tm = 1 m

  • Example: How many meters are in 3.781 micrometers?

    • Know: 1 𝜇𝑚 = 10−610^{-6}m

    • Want: 3.781 𝜇𝑚 = _𝑚

    • 3.781𝜇𝑚×10−6m1𝜇𝑚=3.781×10−6𝑚3.781 𝜇𝑚 × \frac{10^{-6}m}{1 𝜇𝑚} = 3.781 × 10^{-6} 𝑚

  • Example Problem: Change the unit used to report the measurement 4.54 x 10−910^{-9} g by replacing the power of ten with the corresponding unit.

  • Example Problem: Change the unit used to report the measurement 3.76 x 10310^{3} m by replacing the power of ten with the corresponding unit.

Derived Quantities

  • Derived units are derived from other base units.

  • Area = m2m^2

  • Volume = m3m^3

  • SI unit for volume is m3m^3.

  • More common (non-SI) units are: L (dm3dm^3) and mL (cm3cm^3).

  • 1 cm3cm^3 = 1 mL

  • Density of a substance is the ratio of the mass of a sample to its volume.

    • density=massvolumedensity = \frac{mass}{volume}

  • SI unit for density is kilogram per cubic meter (kg/m3kg/m^3).

  • Commonly used density units based on state of matter: g/cm3cm^3 (solids, liquids) and g/L (gases)

Significant Figures

  • Significant Figures in Measurements

    • Certain Digits vs. Estimated Digit

    • Volume = 21.2 mL vs Volume = 47 mL

  • These numbers are always significant.

    • Nonzero digits

    • Captive zeroes (between two nonzero digits)

    • Trailing zeroes to the right of the decimal place or when in scientific notation

  • These numbers are never significant.

    • Leading zeros

    • Trailing zeros before decimal place

    • 0. 008020

    • 3090

    • 1. 30 x 10−310^{-3}

    • leading vs captive vs trailing after decimal vs trailing before decimal

  • How many significant figures are in 0.00004010 kg?

    • Leading zeroes = not significant

    • Captive zero = significant

    • Trailing zero to the right of a decimal place = significant

    • There are 4 significant figures.

    • 4. 010 × 10−510^{-5}

Rounding

  • Rounding (e.g., 3 significant figures)

    • If the first digit to be removed is less than 5, the preceding number is left unchanged.

      • 0. 056432 would be 0.0564

    • If the first digit to be removed is greater than 5, the preceding number is increased by 1.

      • 0. 69174 would be 0.692

    • If the first digit to be removed is exactly 5, round to the closest even number.

      • 4. 7350 would be 4.74 and 4.745 would be 4.74

Accuracy vs. Precision

  • Accuracy – how close a measurement is to the true value

    • Construction/calibration of equipment & user technique determines

    • The average of at least 3 experimental trials is used

  • Precision in experiment – how close a series of replicate measurements of the same thing are to one another

    • Statistical methods used to verify

  • Precision in instrumentation – the degree of confidence in a measurement reflected by significant figures

    • 0°C is less precise than 25.000°C

  • accurate & precise vs precise vs accurate vs neither

  • Precision in measurement – the significant figures in a measurement

    • least precise

      • 20, 21, 22, 23, 24

    • 8

    • 73 most precise

    • 725

Significant Figures Practice

  • How many significant figures are in 20.140?

  • How many significant figures are in 0.00402?

  • How many significant figures are in 6.940 x 10510^5?

  • How many significant figures are in 12.03000?

  • How many significant figures are in 1.0095?

Calculations with Significant Figures

  • To identify a substance, a chemist determined its density. By pouring a sample into a graduated cylinder, they found that the volume was 35.1 mL, and its mass was 30.5 g. Given the data in the table below, what was the substance?

    • d=massvolumed = \frac{mass}{volume}

    • 𝑑 = 30.5g35.1mL\frac{30.5 g}{35.1 mL}= 0.869 g/mL

  • Significant Figures: Addition

    • Volume = 21.2 mL

    • Volume = 47 mL

    • Combined Volume

    • The recorded combined volume can only be as certain as the least certain measurement, 47 mL. Therefore the volume should be recorded as 68 mL.

  • Significant Figures in Addition/Subtraction

    • Addition and Subtraction - Round the result to the same number of decimal places as the number with the least number of decimal places (the least precise value in terms of addition and subtraction).

      • 3482 + 9.34 + 5.832 = ?

  • What is the answer for the following calculation reported to the correct number of significant figures? (1.249 - 0.12) = ?

    • The final answer must be rounded to the same place as the number with the fewest decimal places

    • 1. 129 → 1.13

    • The number of significant figures in the final answer is not determined by the number of sig figs in each addend!

  • Significant Figures in Calculations: Multiplication and Division

    • Multiplication and Division - Round the result to the same number of digits as the number with the least number of significant figures (the least precise value in terms of multiplication and division).

      • 12.45×19.13.2=74\frac{12.45 × 19.1}{3.2} = 74

      • least number of significant figures

  • How many significant figures are in the properly reported answer?

    • The final answer must be rounded to the number with the fewest decimal places

    • 1. 07706879 → 1.1

    • The answer has 2 significant figures.

  • Significant Figures in Calculations: Multi-step calculations

    • In multiple step calculations always retain at least one extra significant figure until the end to prevent rounding errors.

      • (12.3+9.56−13.7)4.15\frac{(12.3 + 9.56 − 13.7)}{4.15}

    • 72.336−72.32272.336×100=?%\frac{72.336-72.322}{72.336} × 100 = ?\%

    • 336-72.322 = 0.014

    • 0.01472.336×100\frac{0.014}{72.336} × 100

    • Round to 2 sf = 0.019%

  • 1.189.7+0.00042.111=?\frac{1.189.7 + 0.0004}{2.111} = ?

    • Addition rules rounds this to the tenths place

    • 10.88042.111=5.1541=5.15\frac{10.8804}{2.111} = 5.1541 = 5.15

Dimensional Analysis

  • A method of calculation utilizing a knowledge of units and equalities.

  • Given units can be multiplied and divided to give desired units.

  • Conversion factors are simple ratios: given unitdesired unit\frac{given \ unit}{desired \ unit}

  • Conversion factor = desired unitgiven unit\frac{desired \ unit}{given \ unit}

  • Don't forget!

    • 1 cm3cm^3 = 1 mL

Conversion Factors to Memorize

  • 1 cm3cm^3 = 1 mL

  • 1 inch = 2.54 cm

  • 1 foot = 12 inches

  • 1 yard = 3 feet

  • 1 angstrom (Å) = 10−1010^{-10} m

  • 1 amu = 1.66 × 10−2410^{-24} g

  • 1 Hz = 1 s−1s^{-1}

Problem Solving Tips

  • Things to think about when problem solving:

    1. What unit should we start with?

    2. What unit do we want to end up with?

    3. What connections can we make between the two?

Dimensional Analysis Examples

  • How many inches are in 37.6 cm?

    • What do we start with? 37.6 cm

    • What do we want to end up with? Inches (in)

    • How do we get there? 1 in = 2.54 cm

    • 37.6cm×1in2.54cm=14.8in37.6 cm × \frac{1 in}{2.54 cm} = 14.8 in

  • Convert 6.88 × 10510^5 ns to s.

    • 6.88×105𝑛𝑠×1s1×109𝑛𝑠=6.88×10−4𝑠6.88 × 10^5 𝑛𝑠 × \frac{1 s}{1 × 10^9 𝑛𝑠} = 6.88 × 10^{−4} 𝑠

  • How many nL are in 2.45 mL?

    • 2.45𝑚𝐿×1𝐿103𝑚𝐿×109𝑛𝐿1𝐿=2.45×106𝑛𝐿2.45 𝑚𝐿 × \frac{1 𝐿}{10^3 𝑚𝐿} × \frac{10^9 𝑛𝐿}{1 𝐿} = 2.45 × 10^6 𝑛𝐿

  • How many mL are in 24.00 oz?

    • 24.00𝑜𝑧×29.57𝑚𝐿1𝑜𝑧=709.7𝑚𝐿24.00 𝑜𝑧 × \frac{29.57 𝑚𝐿}{1 𝑜𝑧} = 709.7 𝑚𝐿

  • How many cubic millimeters are there in a cubic meter?

    • 1𝑚3×(1000𝑚𝑚1𝑚)3=1×109𝑚𝑚31 𝑚^3 × (\frac{1000 𝑚𝑚}{1 𝑚})^3 = 1 × 10^9 𝑚𝑚^3

  • The density of water at 0 C is 1 g/cm3cm^3 exactly. Assuming an ice cube is – in fact – a perfect cube that has an edge length of 3.7 cm, what is the mass of the ice cube?

    • 𝑉 = 𝑙 × 𝑤 × ℎ

    • V = 3.7 cm × 3.7 cm × 3.7 cm = (3.7𝑐𝑚)3(3.7 𝑐𝑚)^3= 50.653 𝑐𝑚3𝑐𝑚^3

    • 50.653𝑐𝑚3×1𝑔1𝑐𝑚3=50.653𝑔=51𝑔50.653 𝑐𝑚^3 × \frac{1 𝑔}{1 𝑐𝑚^3} = 50.653𝑔 = 51 𝑔

  • Convert 1.55 kg/m3m^3 to g/mL.

    • 1.55𝑘𝑔1𝑚3×1000𝑔1𝑘𝑔×(1𝑚)3(100𝑐𝑚)3×1𝑐𝑚31𝑚𝐿=1.55×10−3𝑔/𝑚𝐿1.55 \frac{𝑘𝑔}{1 𝑚^3} × \frac{1000 𝑔}{1 𝑘𝑔} × \frac{(1𝑚)^3}{(100 𝑐𝑚)^3} × \frac{1 𝑐𝑚^3}{1 𝑚𝐿} = 1.55 × 10^{−3} 𝑔/𝑚𝐿

  • The density of water at 0 C is 1 g/cm3cm^3 exactly. Assuming an ice cube is – in fact – a perfect cube that has an edge length of 1.3 in, what is the mass of the ice cube? (1 in = 2.54 cm)

    • 1.3 𝑖𝑛 × 1.3 in × 1.3 in

    • 2. 197 𝑖𝑛3×(2.54𝑐𝑚1𝑖𝑛)3×1𝑔1𝑐𝑚3=36𝑔𝑖𝑛^3 × (2.54 \frac{𝑐𝑚}{1 𝑖𝑛})^3 × \frac{1 𝑔}{1 𝑐𝑚^3} = 36 𝑔

  • How many μL are in 4.39 ft3ft^3?

    • 4.39𝑓𝑡3×(12𝑖𝑛1𝑓𝑡)3×(2.54𝑐𝑚1𝑖𝑛)3×1𝑚𝐿1𝑐𝑚3×1𝐿1000𝑚𝐿×106𝜇𝐿1𝐿=1.24×108𝜇𝐿4.39 𝑓𝑡^3 × (\frac{12 𝑖𝑛}{1 𝑓𝑡})^3 × (\frac{2.54 𝑐𝑚}{1 𝑖𝑛})^3 × \frac{1 𝑚𝐿}{1 𝑐𝑚^3} × \frac{1 𝐿}{1000 𝑚𝐿} × \frac{10^6 𝜇𝐿}{1 𝐿} = 1.24 × 10^8 𝜇𝐿

  • A leaky faucet drips one drop of water every 5.5 seconds. How many gallons of water will this faucet waste as a result of the leak in one year? (1 yr = 365 days, 15 drops = 1 mL, 1 gallon = 3.79 L)

    • 1 𝑦𝑟 × 365𝑑𝑎𝑦𝑠1𝑦𝑟×24h𝑟1𝑑𝑎𝑦×60𝑚𝑖𝑛1h𝑟×60𝑠1𝑚𝑖𝑛×1𝑑𝑟𝑜𝑝5.5𝑠×1𝑚𝐿15𝑑𝑟𝑜𝑝𝑠×1𝐿1000𝑚𝐿×1𝑔𝑎𝑙𝑙𝑜𝑛3.79𝐿=1.0×103𝑔𝑎𝑙\frac{365 𝑑𝑎𝑦𝑠}{1 𝑦𝑟} × \frac{24 ℎ𝑟}{1 𝑑𝑎𝑦} × \frac{60 𝑚𝑖𝑛}{1 ℎ𝑟} × \frac{60 𝑠}{1 𝑚𝑖𝑛} × \frac{1 𝑑𝑟𝑜𝑝}{5.5 𝑠} × \frac{1 𝑚𝐿}{15 𝑑𝑟𝑜𝑝𝑠} × \frac{1 𝐿}{1000 𝑚𝐿} × \frac{1 𝑔𝑎𝑙𝑙𝑜𝑛}{3.79 𝐿} = 1.0 × 10^3 𝑔𝑎𝑙