Now it's time to examine each step to learn how this process can help you solve
more-complex problems, like those you will encounter in your chemistry course.
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CHEMFILE MINI-GUIDE TO PROBLEM SOLVING CHAPTER 4
Four Steps for Solving Quantitative Problems Maybe you have noticed that the sample problems in the first three chapters of this book are solved in a four-step process. The steps in the process are as follows. 1. 2. 3. 4.
ANALYZE PLAN COMPUTE EVALUATE
Now it’s time to examine each step to learn how this process can help you solve more-complex problems, like those you will encounter in your chemistry course.
1. ANALYZE In this first step, you should read the problem carefully and then reread it. You must determine what specific information and data you are given in the problem and what you need to find. Try to visualize the situation the problem describes. Look closely at the words in the problem statement for clues to understanding the problem. Are you working with elements, compounds, or mixtures? Are they solids, liquids, or gases? What change is taking place? Is it a chemical change? What are the reactants? What are the products? It is always a good idea to collect and organize all of your information in a table, where you can see it at a glance. Include the things you want to find in the table, too. Be sure to include units with both the data and the quantities you must find. Scanning the quantities and their units will often provide clues about how to set up the problem. Remember, you may be given information not needed to solve the problem. You must analyze the data to determine what is useful and what is not.
2. PLAN In the planning step, you develop a method to solve the problem. Always keep in mind what you want to find and its units. Chances are good that an approach that gives an answer with the correct units is the correct one to use. In any case, a setup that gives an answer with the wrong units is
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CHEMFILE MINI-GUIDE TO PROBLEM SOLVING certain to be wrong. You may find it helpful to diagram your solution method. This process helps you organize your thoughts. As you work out your problem-solving method, write down a trial calculation without numbers but with units. When you complete your setup, see if the trial calculation will give you a quantity with the correct units. As stated above, if the setup gives the needed units, it is probably correct. During this planning process you may discover that you need more information, such as atomic masses from the periodic table, the boiling point of alcohol, or the density of tin. You will need to look up such information in the appropriate tables.
3. COMPUTE In this step, you follow your plan, set up a calculation using the data you have assembled, and compute the result. It is a good strategy to write out and check your calculation setup before you start working with your calculator. First reconfirm that the calculation will give a result with the correct units. Go through your calculation and lightly strike through the units that cancel. Be sure the remaining units are those that you want in your answer. Whenever possible, use your calculator in a way that lets you complete the entire problem without writing down numbers and then re-entering them.
4. EVALUATE Everyone makes errors, but good problem solvers always develop strategies to check their work. Confidence in your problem-solving ability will come from knowing how to determine on your own whether your answers are correct. One evaluation strategy is to estimate the numerical value of the answer. In simple problems, you can probably do this in your head. With calculations having several terms, round off each numerical value to the nearest simple value, and then write and compute the estimation. Suppose you had to make the following calculation. 28.8 g ? 6.30 cm 18.9 cm The numerical calculation can be estimated as follows. 30 1 30 0.25 6 20 120 4 Once you have done the actual computation, compare your result with the estimate. In this case, the calculation gives 0.242 g/cm2, which is close to the estimated result. Therefore, it is likely that you made no mistakes in the calculation.
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CHEMFILE MINI-GUIDE TO PROBLEM SOLVING Next, check that your answer is expressed to the correct number of significant figures. Look at the data values you used in the calculation. Usually, significant figures will be limited by the measurement that has the fewest significant figures. Finally, ask yourself the simple question, does this answer make sense based on what you know? If you are calculating the circumference of Earth, an answer of 50 km is obviously much too small. If you are calculating the density of air, a value of 340 g/cm3 is much too large because air density is usually less than 1 g/cm3. You will detect many errors by asking if the answer makes sense.
SAMPLE PROBLEM 1 A 10.0% sodium hydroxide solution has a density of 1.11 g/mL. What volume in liters will 2280 g of the solution have?
SOLUTION 1. ANALYZE • What is given in the problem?
• What are you asked to find?
the density of the sodium hydroxide solution in g/mL, and the mass of the solution whose volume is to be determined the volume of the specified mass of solution
Next, bring together in a table everything you might need in the problem. Include what you want to find in the table. The fact that the solution is 10.0% sodium hydroxide is unimportant in the solution of this problem. This piece of data need not appear in your table. Notice that the problem asks you for a volume in liters and that density is given in grams per milliliter. You will need a factor to convert between these units. Therefore, include in the table any relationships between units that will be helpful.
Items
Data
Density of solution
1.11 g/mL
Mass of solution
2280 g
Volume of solution
?L
Relationship between mL and L
1000 mL 1 L
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CHEMFILE MINI-GUIDE TO PROBLEM SOLVING 2. PLAN m . V Rearrange to solve for V, substitute data, and convert to liters.
• What steps are needed to calculate the volume of 2280 g of the solution?
Apply the relationship D
Mass of solution in g rearrange m D V to solve for V
Volume of solution in L
Volume of solution in mL
convert using the factor 1L 1000 mL
Solve the density equation for V. V
m D
mass of solution in g volume in mL density of solution in g/mL To change the result to liters, multiply by the conversion factor. 1L mass of solution in g volume in L density of solution g/mL 1000 mL 3. COMPUTE 2280 g 1L 2.05 L 1.11 g/mL 1000 mL 4. EVALUATE • Are the units correct? • Is the number of significant figures correct? • Is the answer reasonable?
Yes; units canceled to give liters. Yes; the number of significant figures is correct because data were given to three significant figures. Yes; the calculation can be approximated as 2000/1000 2. Also, considering that 2000 g of water occupies 2 L, you would expect 2 L of a slightly more dense material to have a mass slightly greater than 2000 g.
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CHEMFILE MINI-GUIDE TO PROBLEM SOLVING PRACTICE 1. Gasoline has a density of 0.73 g/cm3. How many liters of gasoline would be required to increase the mass of an automobile from 1271 kg to 1305 kg? 2. A swimming pool measures 9.0 m long by 3.5 m wide by 1.75 m deep. What mass of water in metric tons (1 metric ton = 1000 kg) does the pool contain when filled? The density of the water in the pool is 0.997 g/cm3. 3. A tightly packed box of crackers contains 250 g of crackers and measures 7.0 cm 17.0 cm 19.0 cm. What is the average density in kilograms per liter of the crackers in the package? Assume that the unused volume is negligible.
ans: 47 L
ans: 55 metric tons
ans: 0.11 kg/L
ADDITIONAL PROBLEMS Solve these problems by using the Four Steps for Solving Quantitative Problems. 1. The aluminum foil on a certain roll has a total area of 18.5 m2 and a mass of 1275 g. Using a density of 2.7 g per cubic centimeter for aluminum, determine the thickness in millimeters of the aluminum foil. 2. If a liquid has a density of 1.17 g/cm3, how many liters of the liquid have a mass of 3.75 kg? 3. A stack of 500 sheets of paper measuring 28 cm 21 cm is 44.5 mm high and has a mass of 2090 g. What is the density of the paper in grams per cubic centimeter? 4. A triangular-shaped piece of a metal has a mass of 6.58 g. The triangle is 0.560 mm thick and measures 36.4 mm on the base and 30.1 mm in height. What is the density of the metal in grams per cubic centimeter? 5. A packing crate measures 0.40 m 0.40 m 0.25 m. You must fill the crate with boxes of cookies that each measure 22.0 cm 12.0 cm 5.0 cm. How many boxes of cookies can fit into the crate?
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CHEMFILE MINI-GUIDE TO PROBLEM SOLVING 6. Calculate the unknown quantities in the following table. Use the following relationships for volumes of the various shapes. Volume of a cube l l l Volume of a rectangle l w h Volume of a sphere 4/3r 3 Volume of a cylinder r 2 h D 3
a. 2.27 g/cm
3
m
V
3.93 kg
?L 3
Shape
Dimensions
cube
?m?m?m
rectangle
33 mm 21 mm 7.2 mm
b. 1.85 g/cm
?g
? cm
c. 3.21 g/L
? kg
? dm3 sphere
d. ? g/cm
3
e. 0.92 g/cm3
3
497 g
?m
? kg
? cm3
3.30 m diameter
cylinder
7.5 cm diameter 12 cm
rectangle
3.5 m 1.2 m 0.65 m
7. When a sample of a metal alloy that has a mass of 9.65 g is placed into a graduated cylinder containing water, the volume reading in the cylinder increases from 16.0 mL to 19.5 mL. What is the density of the alloy sample in grams per cubic centimeter? 8. Pure gold can be made into extremely thin sheets called gold leaf. Suppose that 50. kg of gold is made into gold leaf having an area of 3620 m2. The density of gold is 19.3 g/cm3. a. How thick in micrometers is the gold leaf? b. A gold atom has a radius of 1.44 1010 m. How many atoms thick is the gold leaf? 9. A chemical plant process requires that a cylindrical reaction tank be filled with a certain liquid in 238 s. The tank is 1.2 m in diameter and 4.6 m high. What flow rate in liters per minute is required to fill the reaction tank in the specified time? 10. The radioactive decay of 2.8 g of plutonium-238 generates 1.0 joule of heat every second. Plutonium has a density of 19.86 g/cm3. How many calories (1 cal 4.184 J) of heat will a rectangular piece of plutonium that is 4.5 cm 3.05 cm 15 cm generate per hour? 11. The mass of Earth is 5.974 1024 kg. Assume that Earth is a sphere of diameter 1.28 104 km and calculate the average density of Earth in grams per cubic centimeter. 12. What volume of magnesium in cubic centimeters would have the same mass as 1.82 dm3 of platinum? The density of magnesium is 1.74 g/cm3, and the density of platinum is 21.45 g/cm3.
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CHEMFILE MINI-GUIDE TO PROBLEM SOLVING 13. A roll of transparent tape has 66 m of tape on it. If an average of 5.0 cm of tape is needed each time the tape is used, how many uses can you get from a case of tape containing 24 rolls? 14. An automobile can travel 38 km on 4.0 L of gasoline. If the automobile is driven 75% of the days in a year and the average distance traveled each day is 86 km, how many liters of gasoline will be consumed in one year (assume the year has 365 days)? 15. A hose delivers water to a swimming pool that measures 9.0 m long by 3.5 m wide by 1.75 m deep. It requires 97 h to fill the pool. At what rate in liters per minute will the hose fill the pool? 16. Automobile batteries are filled with a solution of sulfuric acid, which has a density of 1.285 g/cm3. The solution used to fill the battery is 38% (by mass) sulfuric acid. How many grams of sulfuric acid are present in 500 mL of battery acid?
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