Special note on a particular conversion not mentioned below.

Metric conversions where two units (numerator and denominator) are converted

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Skills you need to perform metric conversions include:

1) memorize the metric prefixes names and symbols

2) determine which of two prefixes represents the larger "size"

3) determine the exponential "distance" between two prefixes

4) significant figure rules

5) scientific notation

Here are two typical metric conversion problems:

Example #1:Convert 2.50 μg to picogram.Example #2:Convert 0.080 cm to km.

The explanation below will focus on the first example. Try to set up the second on as you read through the first example. The answers are provided below.

There is second type of metic conversion, one that involves converting both the numerator and the denominator. You can go to another tutorial which discusses the second type.

Also:

Special note on a particular conversion that is sometimes not mentioned in class.

The note has to do with the fact that 1 mL equals 1 cm^{3}and 1 L equals 1 dm^{3}.

Just below I discuss how to construct a conversion factor. There is an important point about the numerator and the denominator of the conversion factor. Here it is:

The numerator and the denominator of a conversion factor BOTH describe the same amount.In essence, a conversion factor is equal to one. This is because the numerator and the denominator both describe exactly the same amount. Here's an example:

5280 feet / 1 mile

Both 5280 feet and 1 mile describe exactly the same distance.

Here's a metric example:

1 kg / 1000 g

1000 g and 1 kg both have exactly the same amount of mass.

So, when you are done making a given conversion, you haven't changed the amount, you've just changed the way it is written. For example, 1 kg is differently written from 1000 g, but they both describe the same mass.

**Some commentary on one metric prefix being "larger" than another.**

Metric prefixes are associated with a particular exponent. For example, kilo- is associated with 10^{3} or femto- is associated with 10¯^{15}.

It is these exponents that determine the "size" of the unit. Kilo- is larger than femto- because 10^{3} is larger than 10¯^{15}.

The larger the exponent, the larger the size of the unit. Nano- is larger than pico- (because 10¯^{9} is larger than 10¯^{12}). Mega- (10^{6}) is larger than centi- (10¯^{2}).

The key skill in solving these problems is to construct a conversion factor. This conversion factor will make the old unit go away (micrograms and km in the top two examples) and create the new unit (pm and cm) in its place. Along with this change, there will be a change in the value of the number.

**Example #1:** Convert 2.50 μg to picogram.

**STEP ONE:** Write the value (and its unit) from the problem, then in order write: 1) a multiplication sign, 2) a fraction bar, 3) an equals sign, and 4) the unit in the answer. Put a gap between 3 and 4. All that looks like this:

The fraction bar will have the conversion factor. There will be a number and a unit in the numerator and the denominator.

**STEP TWO:** Write the unit from the problem in the denominator of the conversion factor, like this:

**STEP THREE:** Write the unit expected in the answer in the numerator of the conversion factor.

**STEP FOUR:** Examine the two prefixes in the conversion factor. In front of the LARGER one, put a one.

There is a reason for this. I'll get to it in a second.

**STEP FIVE:** Determine the absolute distance between the two prefixes in the conversion unit. Write it as a positive exponent in front of the other prefix.

Now, multiply and put into proper scientific notation format. Don't forget to write the new unit. Sometimes, the exponential number is in the denominator. You must move it to the numerator and when you do so, remember to change the sign. Also, DO NOT move the unit with it. That unit has been cancelled and is no longer there.

**Example #2:** Convert 0.080 cm to km.

Here are all five steps for the second example, put into one image:

Note that the old unit cancels, since it appears in the numerator and denominator of two parts of a multiplication problem.

Why a one in front of the larger unit? I believe it is easier to visualize how many small parts make up one bigger part, like 1000 m making up one km. Going the other way, visualizing what part a larger unit is of one smaller unit, is possible, but requires more sophistication. For example, how many meters are in one nanometer? The answer is 0.000000001 or 10¯^{9}. You may be able to handle a conversion involving what part a larger unit is of a smaller unit and that is just fine.

The conversion factor I have been discussing above is sometimes called a "unitary rate." Unitary in this case simply means that the conversion factor equals one. Look at the conversion factor in the example, the 10^{6} pg / 1 μm. The numerator and the denominator both describe the same amount, as in 10^{6} pg **equals** 1 μm. The word unitary is used to identify the fact that both values describe the same amount.

What I have done is describe a system where the unit with the one (1 μm in my example) can be in either the numerator or denominator of the conversion unit. You may have a teacher that forces the one to be only in the denominator. What that means is that you will have to decide if you are going from a large unit to a small unit (making the numerator exponent positive) or going from a small one to a large one (making the numerator unit negative).

**Three Comments**

1) If you do the conversion correctly, the numerical part and the unit will go in opposite directions. If the unit goes from smaller (mm) to larger (km), then the numerical part goes from larger to smaller. There will never be a correct case where number and unit both go larger or both go smaller.2) A common mistake is to put the one in front of the SMALLER unit. This results in a wrong answer. Put the one in front of the LARGER unit.

3) Sometimes you see this:

10In that case, there is an assumed one in front of the unit with no number. In other words, 10^{6}pg / μm^{6}pg / 1 μm means the same thing as 10^{6}pg / μm. In closing: many times, the 1 will be assumed when it is in the denominator. Very seldom is the 1 assumed when in the numerator. Almost always, the 1 is used explicitly when it is in the numerator.

The answers to the two examples above are 2.50 x 10^{6} pg and 8.0 x 10¯^{7} km.

**Five Conversion Examples**

1) 0.75 kg to milligrams

2) 1500 millimeters to km

3) 2390 g to kg

4) 0.52 km to meters

5) 65 kg to g

Did you read that special note about a conversion sometimes not mentioned in class? Here's the link.

**Example #6:** Convert 1.80 mL to cm^{3}.

1.80 mL times (1 cm^{3}/ 1 mL) = 1.80 cm^{3}

**Example #7:** Convert 1.80 L to dm^{3}

1.80 L times (1 dm^{3}/ 1 L) = 1.80 dm^{3}

Be careful with the next two problems.

**Example #8:** Convert 1.80 L to cm^{3}

1.80 L times (1000 cm^{3}/ 1 L) = 1.80 x 10^{3}cm^{3}

**Example #9:** Convert 1.80 mL to dm^{3}.

1.80 mL times (1 dmOne dm^{3}/ 1000 mL) = 0.00180 dm^{3}

**Example #10:** Convert 4.35 10¯^{9} m^{3} to μL

**Solution:**

The plan is to change cubic meters to cubic centimeters because 1 cm^{3} equals 1 mL. Then, change mL to μL. The key point is that 1 m = 100 cm.

1) Imagine 4.35 10¯^{9} m^{3} like this:

4.35 10¯^{9}m x 1 m x 1 m

2) Now, replace each m with 100 cm:

(4.35 10¯^{9}x 100 cm) x 100 cm x 100 cm

3) Multiply it out to get cm^{3}

0.00435 cm^{3}

4) This equals 0.00435 mL because 1 cm^{3} = 1 mL.

5) Now, change mL to μL:

0.00435 mL times (___ μL / ___ mL)

6) 1 mL is bigger than 1 μL:

0.00435 mL times (___ μL / 1 mL)

7) How many μL in one mL?

The milli prefix means 10¯^{3}and the μ prefix means 10¯^{6}. The absolute exponential distance between the two units is 10^{3}:0.00435 mL times (10

^{3}μL / 1 mL) = 4.35 μL

**Example #11:** The radius of an aluminum atom is 0.143 nm. What is the radius in picometers?

**Solution:**

1) Write the value:

pm <--- the unit that will be in the answer 0.143 nm x ––––––– = the answer nm <--- the unit that will get canceled

2) Put a 1 in front of the prefix that is larger:

pm 0.143 nm x ––––––– = the answer 1 nm Nano- (10¯

^{9}) is larger than pico- (10¯^{12}).

3) There are 1000 (or 10^{3}) of the smaller picometers in 1 of the larger nanometers:

10 ^{3}pm0.143 nm x ––––––– = 143 pm 1 nm

**Example #12:** If a DNA sample has a concentration of 400. mg per mL, how much DNA is in 250. microliters? Use dimensional analysis presentation only.

**Solution going the long way:** converts mg to L first and then from L to μL.

400. mg 10 ^{3}mL1 L 250. μL ––––––– x ––––––– x ––––––– x ––––––– = 100. mg 1 mL 1 L 10 ^{6}μL1

**Solution going one short way:** converts mL to μL directs and uses 1000 mL.

400. mg 10 ^{3}mL250. μL ––––––– x ––––––– x ––––––– = 100. mg 1 mL 10 ^{6}μL1

**Solution going another short way:** converts mL to μL directs and uses 1 mL

400. mg 1 mL 250. μL ––––––– x ––––––– x ––––––– = 100. mg 1 mL 10 ^{3}μL1

**Example #13:** Many chemistry conferences hold a 50. Trillion Ångstrom Run. How long is this run in kilometers and in miles?

**Information about the Ångstrom:**

It is a non-SI unit.1 Å = 10¯

^{8}cm = 10¯^{10}m.Å by itself is pronounced "A ring" and is part of the Swedish alphabet.

**Solution using the cm value:**

50. x 10 ^{12}Å10¯ ^{8}cm10¯ ^{2}m10¯ ^{3}km0.621371 mile ––––––– x ––––––– x ––––––– x ––––––– x ––––––– = 3.1 miles 1 Å 1 cm 1 m 1 km see comments

just below --->(a) (b) (c) (d) (a) This converts Å to cm.

(b) This converts cm to m.

(c) This converts m to km. Stopping here gives the km answer of 5.0 km.

(d) This converts km to miles.

Special note on a particular conversion not mentioned above.

Metric conversions where two units (numerator and denominator) are converted