SI Units and Significant Figures

Learning Goals

• To recognize and use fundamental SI units;
• To identify and use the correct number of SFs in problem-solving and lab;
• To be able to calculate solution concentrations and interconvert between units;

SI Units

A measurement usually consists of a unit and a number expressing the quantity of that unit. We may express the same physical measurement with different units, which can create confusion. For example, the mass of a sample weighing 1.5 g also may be written as 0.0033 lb or 0.053 oz. To ensure consistency, and to avoid problems, scientists use a common set of fundamental units, several of which are listed in Table 2.1. These units are called SI units after the Système International d’Unités.

Table 2.1: Fundamental Base SI Units

Measurement	Unit	Symbol	Definition (1 unit is...)
mass	kilogram	kg	...the mass of the international prototype, a Pt-Ir object housed at the Bureau International de Poids and Measures at Sèvres, France.^†
distance	meter	m	...the distance light travels in (299 792 458)^–1 seconds.
temperature	Kelvin	K	...equal to (273.16)^–1, where 273.16 K is the triple point of water (where its solid, liquid, and gaseous forms are in equilibrium).
time	second	s	...the time it takes for 9 192 631 770 periods of radiation corresponding to a specific transition of the ^133Cs atom.
current	ampere	A	...the current producing a force of 2 \(\times\) 10^–7 N/m between two straight parallel conductors of infinite length separated by one meter (in a vacuum).
amount of substance	mole	mol	...the amount of a substance containing as many particles as there are atoms in exactly 0.012 kilogram of ^12C.
light	candela	cd	...the luminous intensity of a source with a monochromatic frequency of 540 \(\times\) 10^12 hertz and a radiant power of (683)^–1 watts per steradian.

It is important for scientists to agree upon a common set of units. In 1999 NASA lost a Mar’s Orbiter spacecraft because one engineering team used English units and another engineering team used metric units. As a result, the spacecraft came too close to the planet’s surface, causing its propulsion system to overheat and fail.

Some measurements, such as absorbance, do not have units. Because the meaning of a unitless number may be unclear, some authors include an artificial unit. It is not unusual to see the abbreviation AU, which is short for absorbance unit, following an absorbance value. Including the AU clarifies that the measurement is an absorbance value.

Metric Units

Chemists frequently work with measurements that are very large or very small. A mole contains 602 213 670 000 000 000 000 000 particles and some analytical techniques can detect as little as 0.000 000 000 000 001 g of a compound. For simplicity, we express these measurements using scientific notation; thus, a mole contains 6.022 136 7 × 1023 particles, and the detected mass is 1 × 10^–15 g. Sometimes we wish to express a measurement without the exponential term, replacing it with a prefix (Table 2.3). A mass of 1×10^–15 g, for example, is the same as 1 fg, or femtogram.

Table 2.3: Common Prefixes for Exponential Notation

Prefix	Symbol	Factor	Prefix	Symbol	Factor	Prefix	Symbol	Factor
yotta	Y	10^24	kilo	k	10^3	micro	µ	10^–6
zetta	Z	10^21	hecto	h	10^2	nano	n	10^–9
eta	E	10^18	deka	da	10^1	pico	p	10^–12
peta	P	10^15	—	—	10^0	femto	f	10^–15
tera	T	10^12	deci	d	10^–1	atto	a	10^–18
giga	G	10^9	centi	c	10^–2	zepto	z	10^–21
mega	M	10^6	milli	m	10^–3	yocto	y	10^–24

Exponential Notation Problems

1. How many micrograms are in 1 kg?
2. How many nanograms are in 1 mg?
3. How many picograms are in 1 g?
4. How many kg are equal to 1 microgram?
5. How many micrograms are in 1 nanogram?
6. How many milligrams are in 1 kilogram?

You should be comfortable making metric conversions for units between kilograms and picograms.

Significant Figures

Significant figures are a reflection of a measurement’s magnitude and uncertainty. The number of significant figures in a measurement is the number of digits known exactly plus one digit whose value is uncertain.  A few simple rules can be learned to remember how to determine significant figures. 

1. All non-zero numbers are significant.

   5692                4 SFs

   1.72                 3 SFs

   6                      1 SF

2. All zeros that lie between non-zero digits are significant.

   3904                4 SF

   10.031             5 SF

   16005              5 SF

3. None of the zeros that lie to the left of the first non-zero digit are significant, regardless of the position of the decimal.

   0.0029             2 SF

   0.305               3 SF

   10.03               4 SF

4. Zeros to the right of the last non-zero digit are significant if they lie to the right of the decimal point.

   341.0               4 SF

   3.0000             5 SF

   30.0                 3 SF

5. Zeros to the right of the last non-zero digit, but to the left of a decimal point, are significant if the decimal point is present.

   340.                 3 SF

   340                  2 SF

   3500                2 SF

6. Integers have an infinite number of significant figures if they are:

   1. stoichiometric coefficients

   2. unit conversion factors

7. For measurements using logarithms, such as pH, the number of significant figures is the number of digits to the right of the decimal, including all zeros.

   2.45                2 SF

   2.450              3 SF

8. For addition and subtraction, the answer can only have as many significant figures as the last digit that is significant for all numbers in the calculation.

135.621 + 0.33 + 21.2163 = 157.1673 = 157.17

9. For multiplication and division, the answer contains the same number of significant figures as the number in the calculation having the fewest significant figures.

\[\dfrac{22.91 \times 0.152}{16.302}=0.21361=0.214\nonumber\]

10. Rounding off:

    If the last digit is ≥ 5, round up                     10.015 ⇒ 10.02

    If the last digit is ≤ 5, round down                10.014 ⇒ 10.01

Significant Figures Problems

1. Indicate how many significant figures are in each of the following numbers.

1. 903
2. 0.903
3. 1.0903
4. 0.0903
5. 0.09030
6. 9.03 × 10²

2. Round each of the following to three significant figures.

1. 0.89377
2. 0.89328
3. 0.89350
4. 0.8997
5. 0.08907

3. Round each to the stated number of significant figures.

1. the atomic weight of carbon to 4 significant figures
2. the atomic weight of oxygen to 3 significant figures
3. Avogadro’s number to 4 significant figures
4. Faraday’s constant to 3 significant figures

4. Report results for the following calculations to the correct number of significant figures.

1. 4.591 + 0.2309 + 67.1 =
2. 313 – 273.15 =
3. 712 × 8.6 =
4. 1.43/0.026 =
5. (8.314 × 298)/96485 =
6. log (6.53 × 10^–5) =
7. 10^–7.14 =
8. (6.51 × 10^–5) × (8.14 × 10^–9) =

Concentration Units

Concentration is a general measurement unit that reports the amount of solute present in a known amount of solution.  Although we associate the terms “solute” and “solution” with liquid samples, we can extend their use to gas-phase and solid-phase samples as well. Table 2.4 lists the most common units of concentration.

Table 2.4: Common Units for Reporting Concentration

Name	Units	Symbol
molarity	\(\frac {\text{moles solute}} {\text{liters solution}}\)	M
formality	\(\frac {\text{moles solute}} {\text{liters solution}}\)	F
normality	\(\frac {\text{equivalents solute}} {\text{liters solution}}\)	N
molality	\(\frac {\text{moles solute}} {\text{kilograms solvent}}\)	m
weight percent	\(\frac {\text{grams solute}} {\text{100 grams solution}}\)	% w/w
volume percent	\(\frac {\text{mL solute}} {\text{100 mL solution}}\)	% v/v
weight-to-volume percent	\(\frac {\text{grams solute}} {\text{100 mL solution}}\)	% w/v
parts per million	\(\frac {\text{grams solute}} {10^6 \text{ grams solution}}\)	ppm
parts per billion	\(\frac {\text{grams solute}} {10^9 \text{ grams solution}}\)	ppb

Solution Concentration Problems

1. An analyst wishes to add 256 mg of Cl^– to a reaction mixture. How many mL of 0.217 M BaCl₂ is this?
2. The concentration of lead in an industrial waste stream is 0.28 ppm. What is its molar concentration?
3. Commercially available concentrated hydrochloric acid is 37.0% w/w HCl. Its density is 1.18 g/mL. Using this information calculate (a) the molarity of concentrated HCl, and (b) the mass and volume (in mL) of solution containing 0.315 moles of HCl.
4. The density of concentrated ammonia, which is 28.0% w/w NH₃, is 0.899 g/mL. What volume of this reagent should be diluted to 1.0 × 10³ mL to make a solution that is 0.036 M in NH₃?
5. A 250.0 mL aqueous solution contains 45.1 μg of a pesticide. Express the pesticide’s concentration in weight percent, in parts per million, and in parts per billion.
6. A city’s water supply is fluoridated by adding NaF. The desired concentration of F^– is 1.6 ppm. How many mg of NaF should be added per gallon of treated water if the water supply already is 0.2 ppm in F^–?
7. A solution is made by dissolving 170.1 g of glucose (C₆H₁₂O₆; Molar mass: 180.2 g mol^-1) in enough water to make 1-L of solution. The density of the solution is 1.062 g ml^-1. Express the concentration in a) molality; b) % by mass; c) parts per million (ppm).
8. A solution is 0.396 m in aqueous glucose (C₆H₁₂O₆) at 25°C. What is the concentration of this solution in a) molarity; b) % by mass; c) parts per million (ppm)? The density of the solution = 1.16 g/mL.
9. Rubbing alcohol is a mixture of isopropyl alcohol (C₃H₇OH, MW = 60.06 g/mol) and water that is 70% alcohol by mass. The density of the solution is 0.79 g/mL at 20°C. Express the concentration in a) molarity; and b) molality units.
10. The density of 70.5 wt % aqueous perchloric acid is 1.67 g/mL. Answer the following questions:

• How many grams of solution are in 1.000 L?
• How many grams of HClO₄ are in 1.000 L?
• How many moles of HClO₄ are in 1.000 L?

Contributors and Attributions

• Amy Witter, Dickinson College ([e-mail])
• Tables and problems from Harvey Analytical Chemistry 2.1
• Sourced from the Analytical Sciences Digital Library