# 17.7 Solving Problems with Dimensional Analysis

Chess is one of the most popular board games in the world, and is also one of the most complex. White opens the game by selecting any of 20 possible opening moves. Black counters with 20 possible moves, bringing the number of variants to 400 at the end of two moves (1 complete turn). The number grows rapidly as the game progresses: 3 moves - 8,902, 4 moves (2 complete turns) - 197,281, 5 moves - 4,865,609, 6 moves (3 complete turns) - 119,060,324. Thus, at the end of just 6 moves, there are nearly 120 million possible variants of the game! Expert chess players often plan three or four moves ahead, but certainly they cannot ponder all of the options! Instead of considering all scenarios, they focus on logical ones, dramatically increasing their efficiency and effectiveness. Chess masters develop heuristics, methods for directing their attention to reasonable options, while ignoring unreasonable ones. Rather than considering all options, they concentrate only on those that are more likely to produce favorable results. Chess players will not advance in the game until they develop good heuristics that allow them to discard unlikely variants and focus only on the likely ones.

In a similar manner, scientists use problem-solving strategies that allow them to focus their attention on likely possibilities rather than all possibilities. Perhaps the most widely used problem-solving strategy is dimensional analysis (also know as unit analysis and factor-label method). Dimensional analysis allows you to set up the problem and check for logic errors before performing calculations, and allows you to determine intermediate answers in route to the solution. A student or scientist who does not use dimensional analysis is like a chess-player that has not learned key strategies of the game. He or she will spend an inordinate amount of time checking illogical possibilities, with no assurance that the steps taken are correct. Dimensional analysis involves five basic steps:

• Unknown: Clearly specify the units (dimensions) of the desired product (the unknown). These will become the target units for your equation.
• Knowns: Specify all known values with their associated units. It is often a good idea to draw a diagram of what you know about the situation, placing values with their units on the diagram.
• Conversion factors and formulas: Specify relevant formulas and all conversion factors (with their units).
• Equation : Develop an equation (using appropriate formulas and conversion factors) such that the units of the left side (the side containing the known values) are equivalent to the units of the right side (the side containing the unknown). If the units are not equal, then the problem has not been set up correctly and further changes in the setup must be made.
• Calculation: Perform the calculation only after you have analyzed all dimensions and are certain that both sides of your equation have equivalent units.

## Example 1 – Medicine

The label on a stock drug container gives the concentration of a solution as 1200mg/ mL. Determine the volume of the medication that must be given to fill a physician’s order of 1600 mg of the drug (figure 17.8).

• Unknown An analysis of the problem shows that the unknown (the volume of solution to be given) must have units of volume (mL medicine).
• Knowns: We know that the solution has a concentration of 1200 mg drug/mL medicine and that we must obtain 1600 mg of the drug. Figure 17.8 illustrates what must be done.
• Conversion factors and formulas: None necessary.
• Equation: You must divide by concentration and multiply by mass of the drug in order to get the desired units (mg medicine).
• Calculation: After the units are canceled, the equation yields 1.333 mL of medicine. ## Example 2 – Space Science

On June 19, 1976, the United States successfully landed Viking1 on the surface of the planet Mars. Twenty years later, on July 4, 1997, NASA landed another robotic probe named the Mars Pathfinder at a distance of 520 miles from the Viking 1 landing site. Unlike the Viking mission, the Pathfinder mission included a surface rover known as Sojourner, a six-wheeled vehicle that was controlled by an Earth-based operator. Knowing that the distance between the landing site of the Mars Pathfinder and the Viking 1 craft is 520 miles, what would be the minimum number of hours required to drive Sojourner to the Viking site assuming a top speed of 0.70 centimeters per second, and no obstacles (figure 17.9).

• Unknown: The number of hours to reach the Viking 1 site.
• Knowns: The distance is 520 miles, and the speed is 0.7 centimeters per second.
• Conversion factors and formulas: The distance is measured in customary units (miles) while the speed of Sojourner is measured in metric units (centimeters/second). We will therefore need to use the following conversion ratios to obtain units with the correct dimensions: 2.54 centimeters/inch, 12 inches/foot, 5280 feet/mile.
• Equation: The answer must have units of time. The only known factor that includes units of time is the speed of the rover (distance/time). It is therefore evident that we must divide by speed to get units of time in the numerator where they are needed. To arrive at the desired units of time, it is necessary to cancel the units of distance by multiplying by the distance that must be traveled. It is now necessary to multiply or divide by the appropriate conversion ratios to insure that all units of distance are canceled.
• Calculation: After the units are canceled, the equation yields the answer in hours, as desired. The number is changed to two significant figures since one of the factors has only two significant figures, and you can have no greater accuracy than your least accurate factor. ## Example 3 - Physics

A 2.00-L tank of helium gas contains 1.785 g at a pressure of 202 kPa. What is the temperature of the gas in the tank in kelvin given that the molecular weight of helium is 4.002 g/mol and the universal gas constant is 8.29 x 103 L·Pa/mol·K (figure 17.10)?

• Unknown : The unknown is the temperature of the gas, expressed in kelvin.
• Knowns: Volume of helium container (2.00 L), mass of helium (1.785 g), molecular weight (MW) of helium (He; 4.002 g/mol), pressure of helium (202 kPa), universal gas constant (8.29 x 103 L·Pa/mol·K). In addition, we know the number of moles (n = 0.446 mol) of helium since n = m/MW.
• Conversion factors and formulas: This problem requires the use of the ideal gas law equation (PV=nRT) which must be expressed in terms of temperature: T=PV/nR.
• Equation: The equation must be set up so all units cancel except the desired units, kelvin (K).
• Calculation: Once the equation is set up so that the units cancel to leave only the target units of K, then calculations can be performed. ## Example 4 - Chemistry

Calculate the mass of silver metal that can be deposited if a 5.12 ampere current is passed through a silver nitrate solution for 2.00 hours. Note: there are 96,500 C per mole of electrons, and the gram atomic weight of silver is 107.9g/mol (figure 17.11).

• Unknown An analysis of the problem shows that the unknown (the mass of silver metal deposited) must have units of grams silver.
• Knowns: We know that the current is 5.12 amps for a period of 2.00 hours. We also know that 1 mole of silver is deposited per mole of electrons from the fact that silver is a plus one cation (Ag+ + e - ®Ag). From the problem description we know the experimental setup and can therefore draw a diagram. We also can acquire the gram-atomic weight of silver from the periodic table.
• Conversion factors and formulas: This problem will require a number of conversion factors in order to get the appropriate units. One coulomb is one amp second. One mole of electrons is 96,500 coulombs. One hour is 60 minutes. One minute is 60 seconds. Because these are equalities, they can be represented as conversion factors, each of which is equal to one.
• Equation: The units of the unknown become the “target units” and are set up on the right side of the equation. The left side of the equation is assembled so that units will cancel and leave only the target units.
• Calculation: Once the equation is set up so that the units cancel to leave only the target units, calculations can be performed. ## Example 5 – Earth Science

The island of Greenland is approximately 840,000 mi2, 85 percent of which is covered by ice with an average thickness of 1500 meters. Estimate the mass of the ice in Greenland in kg (assume two significant figures). The density of ice is 0.917 g/mL, and 1 cm3 = 1 mL (figure 17.12).

• Unknown An analysis of the problem shows that the unknown must have units of mass. Since the specific units of mass are not specified, we will use the MKS unit of kilograms.
• Knowns: Since we know that the area of Greenland is 840,000 miles2, and 85% of it is covered by ice, then 714,000 miles2 must be covered by ice. We also know that the density of ice = 0.917 g/mL and the ice has an average depth (height) of 1500 m.
• Conversion factors and formulas: Some measurements are in customary units, while others are in metric. We should always convert all units to metric unless otherwise specified. To do so, we will need to convert miles to meters using the following conversion factors: 5280 ft/mile, and 0.3048 m/ft. We also need to use conversion factors to obtain consistent metric units for mass and volume. Knowing that 1 cm = 0.01 m, then 1 cm3 = 0.000001 m3. We also know that 1 kg= 1000 g.
• Equation: The units of the unknown become the “target units” and are set up on the right side of the equation (kg ice). The equation mass = (height x area) density is the basic equation, and the conversion factors are inserted to make certain all units are consistent.
• Calculation: Once the equation is set up so units cancel to leave only the target units of kilograms of ice, calculations can be performed. ## Example 6 - Biology

The rate of photosynthesis is often measured in the number of micromoles of CO2 fixed per square meter of leaf tissue, per second (µmol CO2/m2s). What is the rate of photosynthesis in a leaf with an area of 10 cm2 if it assimilates 0.00005 grams of carbon dioxide each minute (MW of CO2= 48 g/mol)? (figure 17.13).

• Unknown We are trying to determine the rate of photosynthesis in units of µmol CO2/m2s.
• Knowns: The rate of carbon dioxide assimilation by the leaf is 0.00005 grams of carbon dioxide per minute. We also know that the leaf area responsible for this is 10 cm2.
• Conversion factors and formulas: The gram molecular weight of CO2= 48 g/mol. This will be essential in determining the number of micromoles of carbon dioxide. We also know that there are 106 micromoles/mole, 100cm/m, and 60s/min. We may multiply or divide by these unit factors because each one is an identity (equal to 1).
• Equation: Since the rate of photosynthesis is defined as the number of moles of carbon dioxide absorbed per square meter of tissue per second, the equation becomes: Rate=quantity of CO2 per square area of tissue, per second.
• Calculation: Once the equation is set up so that the units cancel to leave only the target units of µmol CO2/m2s, then calculations can be performed. Tables 17.11 and 17.12 may be helpful in the activities that follows.

## Problems

Activity 17.7.1 – Solving problems in everyday life with dimensional analysis

(1) Convert the following

(a) 61.0 kilometers to miles

(b) 2.7 quarts to liters

(c) 56 grams to pounds

(d) 17 pounds to kilograms

(e) 1 million seconds to days

(f) 21 ft/minute to miles/hour

(g) 0.391 grams/ml to kg/liter

(h) 85.5 meters/day to cm/minute

(2) The food that the average American eats in one day provides 2000 Calories of energy. How many Calories per second is this?

(3) Three people estimate the height of the Washington monument in Washington DC: tourist, 555 feet; congressman, 158 yards; lobbyist, 0.173 km. Which is closest to the true height of 169.3 meters?

(4) The EPA sticker on a car states that it will obtain 30 miles per gallon on the highway. How many liters of gasoline must the driver have to insure that he/she can get home from college if there are 300 miles of highway driving between the college and home?

(5) You want to earn \$600 to buy a new bicycle. You have a job that pays \$6.75/hour, but you can work only 3 hours/ day. How many days before you will have enough to buy the bike?

(6) Corn sells for \$8.00/bushel. Your land’s yearly yield is 25 bushels/acre. How many acres should you put in corn to make \$1000 each year?

(7) A landscaper charges \$7 per square meter to plant sod. How much will it cost to plant a one-acre lawn? (1 hectare = 10,000 m2)

(8) A chemist was traveling 2850 cm/s on his way to work. Could he be cited for speeding if the speed limit was 60 miles/h ?

(9) Two of fastest cars ever built were the Lingenfelter Corvette 427 biturbo (which would accelerate from 0 km/s to 100 km/h in 1.97 seconds), and the Hennessey Dodge Viper Venom 800 ( 0 miles/hr to 62 miles/hr in 2.40 seconds). Which car demonstrated greater acceleration?

Activity 17.7.2 – Solving problems in biology with dimensional analysis

(1) What is the net primary productivity (kg/m2.y) of a field of wheat if an average of 2500 kg is harvested each year in a plot that is 10m x 10m.

(2) Cheetahs are the fastest land mammals and are capable of sprinting at 27.8 m/s in short bursts. How long would it take a cheetah to run the length of a 100 yd football field running at this top speed (1 yd = 0.914 m).

(3) If an artificial heart is capable of pumping at least 57,000,000 pints of blood before failure, how long will it probably last in a patient if their average heart rate is 72 beats per minutes, and average stroke volume (the amount of blood pumped with each stroke) is 70 mL?

(4) The rate of respiration in is often measured in the number of micromoles of CO2 fixed per gram of tissue, per second (µmol CO2/g·s). What is the rate of respiration in an organism with a mass of 1 g if it produces 0.002 grams of carbon dioxide each minute (MW of CO2= 48 g/mol)?

(5) A 125 pound patient is to receive a drug at a rate of 0.300 mg per 1.00 kilogram of body weight. If the drug is supplied as a solution containing 5.00 mg/mL, how many mL of drug solution should he/she receive?

(6) A calcium report indicated 8.00 mg/dL of calcium ions in the blood. If we assume that the patient has 6.00 quarts of blood, how many grams of calcium ions are in his/her blood? (1 dL = 0.1 L)

(7) A large dose of an antileukemia drug is to be administered to a 190 lb patient by intravenous (IV) injection. The recommended dosage is 50.0 mg per kilogram of body weight, and the drug is supplied as a solution that contains 20.0 mg per milliliter. The IV has a flow rate of 3.00 mL/minute. How long will it take to administer the recommended dose?

Activity 17.7. – Solving problems in earth science with dimensional analysis

(1) Earth has an orbital velocity of 1.0 km/s. How far will it travel in one year?

(2) When a 4.13 g chunk of rock is dropped into a graduated cylinder containing 8.3 mL of water, the water level rises to 9.8 mL. What is the density of the rock in grams per cubic centimeter? Is this rock more likely granite (2.7 – 2.8 g/cm3) or basalt (2.9 g/cm3)?

(3) The Atlantic Ocean is growing wider by about 1 inch/year. There are 12 inches/ft. and 5280 ft/ mile. How long will it take for the Atlantic to grow 1 meter in width?

(4) The average distance between the Earth and Sun is approximately 93,000,000 miles. Express this distance in centimeters.

(5) A chunk of the mineral galena (lead sulfide) has a mass of 12.4 g and a volume of 1.64 cm3. What is its density? Will it float or sink in a pool of mercury (densityHg =13600 kg/m3)?

(6) A solid concrete dam measures 50 GL. How many cubic meters of concrete are in this structure? 1 GL = 1 ´ 109 L; 1 L = 1000 cm3.

(7) The mass of Earth is 5.97 x 10 24 kilograms. What is its average density in g/mL if it has a radius of 6378 km? (Vsphere=4/3pr3)

Activity 17.7.4 – Solving problems in chemistry with dimensional analysis

(1) Platinum has a density of 21.4 g/mL. What is the mass of 5.90 mL of this metal?

(2) The mass of a proton is 1.67272 ´ 10-27 kg. What is its mass in mg?

(3) What mass of silver nitrate must be used to make 2.00 cubic decimeters of a 1.00M solution? 1 dm = 0.1m, 1 dm3 = 1L.

(4) Calculate the mass of solute required to make 750 mL of a 2.50M sodium chloride solution.

(5) Calculate the molarity of a 1.50 ´ 103 cubic centimeter solution that contains 200.0 grams of MgCl2

(6) What is the mass of solute in 300.0 mL of a solution, if the solution is 85% water and has a density of 1.60 g/cm3.

(7) A copper refinery produces a copper ingot weighing 150 lb. If the copper is drawn into wire of diameter 8.25 mm, how many feet of copper can be obtained from the ingot? The density of copper is 8.94 g/cm3.

Activity 17.7.5 – Solving problems in physics with dimensional analysis

(1) A beta particle travels at a speed of 112,000 miles per second. What is its speed expressed in centimeters per second? (Give three digits in your answer and use scientific notation). 5280 ft = 1.00 mile; 12 inch = 1 ft; 2.54 cm = 1.00 inch.

(2) An object is traveling at a speed of 7.5 ´ 103 cm/s. Convert the value to kilometers per minute.

(3) Traffic accident investigators often discuss reaction time when trying to determine liability for an accident. If a person’s reaction time is 1.5 seconds, how many meters will his or her car travel before the brakes are activated if the car is traveling at 70 miles per hour?

(4) The wavelength of visible light is 706 nm. What is its frequency in sec-1 (Hz) ? The speed of light, c = 3.00 ´ 108 m/sec. 1 nm = 1 ´ 10-9 m.

(5) The acceleration due to gravity on Earth is 9.8 m/s2 while it is 3.7 m/s2 on the surface of Mars. If you weigh 700 N on Earth, how many newtons would you weigh on Mars? 1 N = 1 kg·m·s-2.

(6) A light-year is the distance light (c= 3.0 ´ 108 m/s) travels in one Earth-year. Alpha Centauri C, the star closest to our Sun, is 4.22 light-years away. How far is this expressed in meters?

(7) The escape velocity for earth is 11.2 km/s. How far will a spacecraft travel in an hour if it is traveling at 1.6 times the escape velocity?

## The Hazards of not using Units

Discuss class data.