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 = \frac{9}{5} × T_{°c} + 32
TK = 273.15 + Tc
Example Conversion:
Convert 27.60°C to Kelvin.
T{in \ Kelvin} = T{°C} + 273.15
T_{in \ Kelvin} = 27.60°C + 273.15 = 300.75 K
Common Unit Prefixes
Prefixes to memorize:
femto (f): 10^{-15}
pico (p): 10^{-12}
nano (n): 10^{-9}
micro (µ): 10^{-6}
milli (m): 10^{-3}
centi (c): 10^{-2}
deci (d): 10^{-1}
kilo (k): 10^{3}
mega (M): 10^{6}
giga (G): 10^{9}
tera (T): 10^{12}
Angstrom (Å) = 10^{-10} m
How to use the table (using meter as an example):
femto (f): 1 fm = 1 × 10^{-15} m or 1 × 10^{15} fm = 1 m
pico (p): 1 pm = 1 × 10^{-12} m or 1 × 10^{12} pm = 1 m
nano (n): 1 nm = 1 × 10^{-9} m or 1 × 10^{9} nm = 1 m
micro (µ): 1 µm = 1 × 10^{-6} m or 1 × 10^{6} µm = 1 m
milli (m): 1 mm = 1 × 10^{-3} m or 1 × 10^{3} mm = 1 m
centi (c): 1 cm = 1 × 10^{-2} m or 1 × 10^{2} cm = 1 m
deci (d): 1 dm = 1 × 10^{-1} m or 1 × 10^{1} dm = 1 m
kilo (k): 1 km = 1 × 10^{3} m or 1 × 10^{-3} km = 1 m
mega (M): 1 Mm = 1 × 10^{6} m or 1 × 10^{-6} Mm = 1 m
giga (G): 1 Gm = 1 × 10^{9} m or 1 × 10^{-9} Gm = 1 m
tera (T): 1 Tm = 1 × 10^{12} m or 1 × 10^{-12} Tm = 1 m
Example: How many meters are in 3.781 micrometers?
Know: 1 𝜇𝑚 = 10^{-6}m
Want: 3.781 𝜇𝑚 = _𝑚
3.781 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^{-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 10^{3} m by replacing the power of ten with the corresponding unit.
Derived Quantities
Derived units are derived from other base units.
Area = m^2
Volume = m^3
SI unit for volume is m^3.
More common (non-SI) units are: L (dm^3) and mL (cm^3).
1 cm^3 = 1 mL
Density of a substance is the ratio of the mass of a sample to its volume.
density = \frac{mass}{volume}
SI unit for density is kilogram per cubic meter (kg/m^3).
Commonly used density units based on state of matter: g/cm^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^{-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^{-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 10^5?
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 = \frac{mass}{volume}
𝑑 = \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).
\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.
\frac{(12.3 + 9.56 − 13.7)}{4.15}
\frac{72.336-72.322}{72.336} × 100 = ?\%
336-72.322 = 0.014
\frac{0.014}{72.336} × 100
Round to 2 sf = 0.019%
\frac{1.189.7 + 0.0004}{2.111} = ?
Addition rules rounds this to the tenths place
\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: \frac{given \ unit}{desired \ unit}
Conversion factor = \frac{desired \ unit}{given \ unit}
Don't forget!
1 cm^3 = 1 mL
Conversion Factors to Memorize
1 cm^3 = 1 mL
1 inch = 2.54 cm
1 foot = 12 inches
1 yard = 3 feet
1 angstrom (Å) = 10^{-10} m
1 amu = 1.66 × 10^{-24} g
1 Hz = 1 s^{-1}
Problem Solving Tips
Things to think about when problem solving:
What unit should we start with?
What unit do we want to end up with?
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
37.6 cm × \frac{1 in}{2.54 cm} = 14.8 in
Convert 6.88 × 10^5 ns to s.
6.88 × 10^5 𝑛𝑠 × \frac{1 s}{1 × 10^9 𝑛𝑠} = 6.88 × 10^{−4} 𝑠
How many nL are in 2.45 mL?
2.45 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?
24.00 𝑜𝑧 × \frac{29.57 𝑚𝐿}{1 𝑜𝑧} = 709.7 𝑚𝐿
How many cubic millimeters are there in a cubic meter?
1 𝑚^3 × (\frac{1000 𝑚𝑚}{1 𝑚})^3 = 1 × 10^9 𝑚𝑚^3
The density of water at 0 C is 1 g/cm^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 = (3.7 𝑐𝑚)^3= 50.653 𝑐𝑚^3
50.653 𝑐𝑚^3 × \frac{1 𝑔}{1 𝑐𝑚^3} = 50.653𝑔 = 51 𝑔
Convert 1.55 kg/m^3 to g/mL.
1.55 \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/cm^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 𝑖𝑛^3 × (2.54 \frac{𝑐𝑚}{1 𝑖𝑛})^3 × \frac{1 𝑔}{1 𝑐𝑚^3} = 36 𝑔
How many μL are in 4.39 ft^3?
4.39 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 × \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