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The solving process is much the same as solving problems with square units, except you have to cube the absolute exponential distance between the two prefixes. I will do the same examples as in the square unit tutorial, except as cubic problems.
Remember that cubic problems deal with volume.
Here are the first two examples:
1.00 m3 = how many cubic centimeters?
1.00 km3 = how many cubic micrometers?
Example #1: 1.00 m3 = how many cubic centimeters?
1) Here's the set up for the conversion:
??? cm3 1.00 m3 x ––––––– = the answer 1 m3
Notice that m3 cancels, leaving cm3 to be the unit on the answer.
2) Some discussion:
The absolute exponential distance between the base unit and centi- is 102. Sometimes a teacher may say the exponential distance is 2. Be aware that the 2 is an exponent, NOT that you will be multiplying something by 2. In a cubic situation, we must cube it to obtain 106. The reason we do this is because there are three centimeter measurements, one for each of the three sides of the cube.
3) Let me anticipate a problem:
You read the above and say "Wait, the cube of 2 is 8, isn't it?" You are right because (2)3 = 2 x 2 x 2 = 8, but we are dealing with an exponential value of 2, not the number 2. What gets cubed is 102, not 2. We would write (102)3 = 102 x 3 = 106.
4) Now, to finish the example:
106 cm3 1.00 m3 x ––––––– = 1.00 x 106 cm3 1 m3
You might also see it like this:
(1.00 m3) (106 cm3 / m3) = 1.00 x 106 cm3
Example #2: 1.00 km3 = how many cubic micrometers?
1) Here's the problem done up as a conversion:
??? μm3 1.00 km3 x ––––––– = the answer 1 km3
Notice that km3 cancels, leaving μm3 to be the unit on the answer.
2) Some discussion:
Between kilo- and micro- is an absolute distance of 109. Our technique calls for the exponential value to be cubed, so (109)3 = 1027. The direction of change is going from the larger prefix to the smaller prefix, so the sign of the exponent in the answer will be positive.
3) Finish the example:
1027 μm3 1.00 km3 x ––––––– = 1.00 x 1027 μm3 1 km3
The two examples above were going from the larger prefix to the smaller. That meant there were lots of the smaller prefix in one big prefix. Examples below will go the opposite direction. Ask yourself: how many of a larger prefix are there in one small prefix? Answer = less than one, so the sign on the exponent will be negative.
Having said that, watch out for a problem where there is an exponent already in the problem and the negative exponent change fails to overwhelm the exponent from the problem. In that case, there will be a positive exponent in the answer.
Example #3: 4310 cubic centimeters. Convert to m3.
Notice how the 1 in the conversion fact is associated with the larger (the m3) of the two units. Larger here means in the sense that 1 m3 is larger than 1 cm3.
Example #4: 86.3 cubic centimeters. What is volume in mm3?
86.3 cm3 x (103 mm3 / 1 cm3) = 8.63 x 104 mm3
There are 10 mm in one cm, so 10 x 10 x 10 yields 103 mm3.
Example #5: 5.94 x 1010 mm3. Convert to dm3. Then, convert to liters.
1 dm3 5.94 x 1010 mm3 x ––––––– = 5.94 x 104 dm3 (100 mm)3
Note how mm3 goes into the denominator of the conversion factor, so as to cancel mm3 on the value to be converted.
1 dm equals 100 mm, so 1 dm3 = (100 mm)3 = 106 mm3
Note how the 1 goes with dm3, the larger of the units when compared to 1 mm3.
5.94 x 104 dm3 = 5.94 x 104 L because 1 dm3 equals 1 L.
Convert L to m3
Example #6: A metal cube is 2.50 cm on a side. Calculate the volume in m3.
1) Convert 2.50 cm to meters:
(2.50 cm) (1 m / 100 cm) = 0.0250 m
2) Multiply to obtain the volume of the cube:
(0.0250 m) (0.0250 m) (0.0250 m) = 1.56 x 10¯5 m3 (to three sig figs)
3) Here's an alternate solution path:
(2.50 cm) (2.50 cm) (2.50 cm) = 15.625 cm3
(15.625 cm3) (1 m / 100 cm)3 = 15.625 x 10¯6 m3
15.625 x 10¯6 m3 = 1.56 x 10¯5 m3 (to three sig figs)
Example #7: Convert 10.6 kg/m3 to g/cm3
1) Do the non-cubic conversion first, convert kg to g:
103 g 10.6 kg/m3 x ––––– = 1.06 x 104 g/m3 1 kg
Note the use of exponential notation in the answer.
2) Do the cubic conversion next, convert meters cubed to centimeters cubed:
(1 m)3 1.06 x 104 g/m3 x ––––––– = 1.06 x 10¯2 g/cm3 (100 cm)3
Note that 100 cubed is 106. You could write the 100 cubed in the conversion as (102)3
You could also write the answer as 0.0106 g/cm3.
3) Be careful to make double-sure you have everything in the right place. When I formatted this problem, I did so about 7:30 AM on a rainy Saturday. I had the conversion factor for the cubic conversion reversed. Needed MOAR COFFEE!!
Example #8: Convert the density of aluminum (2.70 g/cm3) to kg/m3.
1) Here's the plan:
(a) convert g/cm3 to g/m3 (first conversion factor in the step just below)
(b) convert g/m3 to kg/m3 (second conversion factor in the step just below)
2) A little dimensional analysis:
(102 cm)3 1 kg 2.70 g/cm3 x ––––––– x ––––––– = 2.70 x 103 kg/m3 (1 m)3 1000 g
Example #9: Convert 409 cubic inches to liters using dimensional analysis.
1) Here's the plan:
(a) convert in3 to cm3
(b) convert cm3 to mL
(c) convert mL to L
2) Here's the set up:
(2.54 cm)3 1 mL 1 L 409 in3 x ––––––––– x ––––– x ––––––– = 6.70 L (1 in)3 1 cm3 1000 mL
(a) think of 1 in3 being a cube one inch by one inch by one inch. Since 1 inch equals 2.54 cm, one cubic inch equals a cube 2.54 cm by 2.54 cm by 2.54 cm.
(b)1 cm3 = 1 mL is a very handy conversion to remember.
(c) Brian Wilson has a comment concerning the number 409.
Example #10: If a cube is 25.2 mm on a side, what is its volume in liters?
1) Convert 25.2 mm to cm:
(25.2 mm) (1 cm / 10 mm) = 2.52 cm
2) Determine the volume of the cube:
(2.52 cm) (2.52 cm) (2.52 cm) = 16.003 cm3
3) Convert cm3 to mL:
(16.003 cm3) (1 mL / 1 cm3) = 16.003 mL
4) Convert mL to L:
(16.003 mL) (1 L / 1000 mL) = 0.0160 L
Bonus Example: A mole of any ideal gas at STP occupies 22.414 L. How many molecules of hydrogen gas are in 22.414 mm3?
1) I'm going to build a dimensional analysis solution, one factor at a time. First, let's state the information from the problem:
1 mol ––––––– x 22.414 L
I deliberately put the value that way because I know the final answer will be some number of molecules per cubic millimeter. The volume must go in the denominator.
2) We have to get rid of liters and go to some type of cubic measurement:
1 mol 1 L ––––––– x ––––– x 22.414 L 1 dm3
Knowing the 1 L = 1 dm3 is a handy conversion to remember.
3) Now, convert from cubic decimeters to cubic millimeters:
1 mol 1 L 1 dm3 ––––––– x ––––– x –––––––– x 22.414 L 1 dm3 1003 mm3
for this:1003 mm3
Remember that there are 100 mm in 1 dm, so there are:100 mm x 100 mm x 100 mm in one dm3
4) The last conversion is from moles to molecules:
1 mol 1 L 1 dm3 6.022 x 1023 molecules ––––––– x ––––– x –––––––– x ––––––––––––––––––– = 2.687 x 1016 molecules (to 4 sig figs) 22.414 L 1 dm3 1003 mm3 1 mol
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