# Dimensional (Unit) Analysis and Problem Solving

A convenient check on your work, or even a way to determine the best approach to a problem, is to use dimensional analysis. This simply means to include all of the units for each factor in an equation, and then to check to see that the units on both sides of the equation are equal.

Example: How many centimeters are there in one foot?

Known: 1 ft = 12 inches (defined, therefore exactly); 2.54 cm = 1 inch (defined).

Set up: (1 ft)(12 inches/ft)(2.54 cm/inch)

note that ft cancels ft and inches cancels inches to give cm!

Solve: (1 ft)(12 inches/ft)(2.54 cm/inch) = 30.48 cm.

How about sig figs? In this problem there are no significant figures the way its set up, because there are no measurements! That is, all of the numbers are part of definitions, so they are exact, and that means the answer is exact as well.

 Extra Example: Assume it is 293 miles to your destination in the Bay Area. If you are driving a car that gets 27 miles per gallon and gas costs an average of \$3.45/gallon for the trip, how much will it cost you to drive there and back? Again, look at units. Want answer in dollars, so \$3.45/gallon ?? = \$ So we have \$ on both sides, but we need to get rid of (/gallon), so let's try dividing by 27 miles/gallon: (\$3.45/gallon) / (27 miles/gallon) ? = \$ We can now cancel gallons, as shown, and all we need to do now is multiply by miles to cancel miles: (293 miles)(\$3.45) / (27 miles) = \$ We now have the same units on each side, and can do the math, which I will leave to you.

# SI Units (metric system)

SI Units: The metric system originated around the French Revolution as a rational system of measurements to rescue France from the chaos of pre-revolutionary measurements and thus prevent tax collectors from cheating.

Wanted to base system on "natural" universal standards. Thus for length they chose the size of the Earth: specifically the meter was defined as one ten-millionth (10-7) of the Earth's meridian (line from the S to the N pole) through Paris. For mass the Kilogram was defined as the mass of a cube of water 0.1 meter on a side. Of course these are not convenient, so standards were quickly created: the meter became the distance between two lines on a platinum-iridium bar stored in a vault in Paris, while the kilogram became a cylindrical mass of platinum-iridium stored in the same vault.

Today the various units are defined by international agreement to give the SI (Systéme International) units:

• Length: the meter (m) is defined as the distance light travels in a vacuum in 1/299,792,458 sec (note that this is truly universal: in principle it can be determined by anyone, anytime, anywhere in the Universe).
• Mass: the kilogram (kg) however is still based on the International Prototype Kilogram in Paris and the derived standard kilogram standards held by governments around the world.
• Time: the second (s) is defined today as the duration of 9,192,631,770 periods of the radiation of two hyperfine levels of the ground state of the cesium 133 atom.
• Amount of substance: the mole (mol) is defined as the number of atoms in 0.012 kg (defined, so sig figs not restricted) of carbon 12 atoms.
• Temperature: the kelvin (K) is defined as 1/273.15 of the thermodynamic temperature of the triple point of water.
• Electric current: the ampere is defined as the the current which carries one coulomb (6.24146 x 1018 times the charge on an electron or proton) of charge through a conductor in one second.

Prefixes: Note Table 1.2 in your text (p 10). You should know (memorize) and be able to interconvert the prefixes in the table below:

 Prefix Symbol Magnitude tera- T 1012 giga- G 109 mega- M 106 kilo- k 103 base 100 deci- d 10-1 centi- c 10-2 milli- m 10-3 micro- (or mc) 10-6 nano- n 10-9 pico- p 10-12 fempto- f 10-15

Memorize: 1 mL = 1 cm3; 1 inch = 2.54 cm (defined); 1 liter is about 1 quart; density of water = 1 g/mL; 0° C = 32 °F, 100°C = 212 °F, -40 °C = -40 °F.

## Temperature

Look in your text for conversions between °C and °F and example problems

# Density

Density is defined as the mass of a given volume of a substance: Density = mass/volume. Note that this weeks laboratory exercise give practice in Density, significant figures etc.

Let's try some density problems. First note that the units of density are g/cm3 or g.cm-3.

• A student found that 20.0 mL of a liquid weighed 35.987 g. What is its density?

Known: Density = mass/volume, generally expressed as g/mL = g/cm3

Solve: (35.987 g) / (20.0 mL) = 1.79935 g/mL

note that the units are those of density so we are confident we set it up correctly.

How about sig figs? Use multiplication/division rules, so count: 3 for 20.0 and 5 for 35.987, therefore should have three sig figs:

1.79935 g/mL = 1.80 g/mL

 Extra Example: Using a jewelers balance a student found that a coin weighed 2.34 carats in air. By weighing it again submerged in water she found it had a volume of 0.034 mL. What is its density? (1 carat = 200 mg, defined)* Known: 1 carat = 200 mg (defined), density is g/mL Solve: (2.34 carats)(200 mg/carat)(1 g/1,000 mg) / 0.034 mL = 13.764706 g/mL How about sig figs? Both conversion factors are defined, so exact. Two measurements: 2.34 and 0.034 = 3.4 x 10-2. Thus the answer will have only two sig figs since using counting rule - least number of sig figs. 13.764706 g/mL = 14 g/mL

# Matter

What is matter? Stuff. Has mass and occupies space.

## Mass

Mass is the measure of quantity for matter. Mass is the property of matter resulting in its inertia and and attraction via gravity.

Do not confuse mass and weight. Weight is the force acting on an object due to gravity. We often interchange these terms in conversation, but they are quite different - you have the same mass whether you are weightless in space on here on Earth (taking a shuttle flight is no substitute for a diet!). To confuse us further we call the determination of mass "weighing"!

### NEXT

 Syllabus / Schedule C109 Home

© R A Paselk