Dimensional Analysis

Videos on Dimensional Analysis

Steps of Dimensional Analysis

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:

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).

Example 2 – Space Science

 Video  Dimensional Analysis - example from Space Science 12:34

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).

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)?

Example 4 - Chemistry

 Video  Dimensional Analysis - example from Chemistry  16:43

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).

 

 

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).

 

 

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).

Tables 17.11 and 17.12 may be helpful in the activities that follows.

 

 Practice Problems

For answers to these problems, click here

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?