Key Dimensional Analysis Worksheet

Dimensional Analysis Worksheet Dimensional Analysis is the use of unit (dimension) relationships to analyze a problem mathematically. This approach is logical and can be applied to many mathematical problems when relationships between units are known. The skill is seeing the relationships and interpreting how to use them; a skill that grows with practice. A unit relationship can be a rate, a definition, or a ratio that relates two or more units. Example Set 1: Below each statement is mathematical restating of the unit relationship described. Rate: A cheetah can run for short bursts of time at 60. miles per hour. 60. 𝑚𝑖 ℎ𝑟 Definition: One mile is equal to 1.609 kilometers. 1 𝑚𝑖 = 1.609 𝑘𝑚 𝑜𝑟 1 = 1.609 𝑘𝑚 𝑚𝑖 Ratio: For every 100 meters a rested cheetah runs, a tired cheetah runs 25 meters. 100. 𝑚𝑒𝑡𝑒𝑟𝑠 𝑜𝑓 𝑎 𝑟𝑒𝑠𝑡𝑒𝑑 𝑐ℎ𝑒𝑒𝑡𝑎ℎ = 25. 𝑚𝑒𝑡𝑒𝑟𝑠 𝑜𝑓 𝑎 𝑡𝑖𝑟𝑒𝑑 𝑐ℎ𝑒𝑒𝑡𝑎ℎ or 1 = 25.𝑚𝑒𝑡𝑒𝑟𝑠 𝑜𝑓 𝑎 𝑡𝑖𝑟𝑒𝑑 𝑐ℎ𝑒𝑒𝑡𝑎ℎ 100. 𝑚𝑒𝑡𝑒𝑟𝑠 𝑜𝑓 𝑎 𝑟𝑒𝑠𝑡𝑒𝑑 𝑐ℎ𝑒𝑒𝑡𝑎ℎ Example Set 2: Write mathematical statements that reflect each unit relationship described. Rate: Mr. Hungry eats donut holes at a rate of 9.5 donut holes per minute. 9.5 𝑑ℎ 𝑚𝑖𝑛 Definition: A donut hole weighs 1.0 ounce. 1 𝑑ℎ = 1.0 𝑜𝑧 Ratio: A whole donut weighs about four times as much as a donut hole 1 𝑑𝑜𝑛𝑢𝑡 = 4.0 × (𝑚𝑎𝑠𝑠 𝑑ℎ ) Here we use 4.0 rather than 4 as a single sig.fig. would suggest 3-5 donuts, where we just mean 4, but not exactly 4 (3.9-4.1) when we say about as much. Example Set 3: Write mathematical statements that reflect each unit relationship described. Rate: A candle burns 0.100 mole of wax molecules per hour. 𝑚𝑜𝑙𝑒 0.100 ℎ𝑟 Definition: One mole of a substance is 6.022 x1023 molecules of that substance. 1 𝑚𝑜𝑙𝑒 6.022 𝑥 1023 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒𝑠 𝑎𝑠 𝑏𝑜𝑡ℎ 𝑟𝑎𝑡𝑖𝑜𝑠 𝑎𝑟𝑒 𝑒𝑞𝑢𝑎𝑙 𝑜𝑟 1 𝑚𝑜𝑙𝑒 6.022 𝑥 1023 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒𝑠 Ratio: For every mole of paraffin wax (C31H64) consumed 48 moles of oxygen are also consumed, yielding 31 moles of CO2 and 32 moles of H20. Chemical Equation: 𝐶31𝐻64 + 48𝑂2 → 31𝐶𝑂2 + 32𝐻2 𝑂 Stoichiometry states that ratios can be used to predict amount of product Dimensional Analysis Worksheet Dimensional analysis uses multiplication and division to convert a starting value or relationship to the units needed for the answer. Remember to apply your significant figure rules! Example Question: How many miles could a hiker travel if she were to keep a constant pace of 3 miles per hour? Approach: This is accomplished because units cancel each other out such that multiplying 3 hours by 3 miles per hour would cancel out the hours unit giving an answer of 9 miles. 3 ℎ𝑟 × 3 𝑚𝑖 ℎ𝑟 = 9 𝑚𝑖 OR in my shorthand format Cheetahs: 1. What is the maximum speed of a cheetah in kilometers per hour? 60. 𝑚𝑖 1.609 𝑘𝑚 𝑘𝑚 𝑘𝑚 = 96.54 𝑆𝑖𝑔𝐹𝑖𝑔𝑠 → 97 × 1 𝑚𝑖 ℎ𝑟 ℎ𝑟 ℎ𝑟 2. Given the distance of a tired cheetah runs compared to a rested cheetah, what is the speed of that tired cheetah in kilometers per hour? 𝑘𝑚 𝑟𝑒𝑠𝑡𝑒𝑑 𝑐ℎ𝑒𝑒𝑡𝑎ℎ 𝑘𝑚 𝑘𝑚 25. 𝑚𝑒𝑡𝑒𝑟𝑠 𝑜𝑓 𝑎 𝑡𝑖𝑟𝑒𝑑 𝑐ℎ𝑒𝑒𝑡𝑎ℎ 96.54 = 24.135 𝑆𝑖𝑔𝐹𝑖𝑔𝑠 → 24 × 100. 𝑚𝑒𝑡𝑒𝑟𝑠 𝑜𝑓 𝑎 𝑟𝑒𝑠𝑡𝑒𝑑 𝑐ℎ𝑒𝑒𝑡𝑎ℎ ℎ𝑟 ℎ𝑟 ℎ𝑟 3. How far can a tired cheetah run in 15 minutes? Give the answer in kilometers and meters. 24.135 𝑘𝑚 1 ℎ𝑟 15 𝑚𝑖𝑛 = 6.03375 𝑘𝑚 𝑆𝑖𝑔𝐹𝑖𝑔𝑠 → 6.0 𝑘𝑚 × × ℎ𝑟 60 𝑚𝑖𝑛 1 Donuts: 4. How many ounces of donut holes can Mr. Hungry eat in 30 minutes? 9.5 𝑑ℎ 1.0 𝑜𝑧 30 𝑚𝑖𝑛 = 285 𝑜𝑧 𝑆𝑖𝑔𝐹𝑖𝑔𝑠 → 290 𝑜𝑧 × × 𝑚𝑖𝑛 1 𝑑ℎ 1 5. How many donut holes would be equal to 5 donuts? 5 𝑑𝑜𝑛𝑢𝑡𝑠 4.0 𝑑ℎ × = 20 𝑑ℎ 𝑆𝑖𝑔𝐹𝑖𝑔𝑠 → 20. 𝑑𝑜𝑛𝑢𝑡 ℎ𝑜𝑙𝑒𝑠 1 1 𝑑𝑜𝑛𝑢𝑡 Five donuts is a counted number, so it would not limit the SigFigs of your answer. 6. How many donuts could he eat in 15 minutes assuming he consumed them at the same rate in ounces per minute as he consumes donut holes? 9.5 𝑑ℎ 1 𝑑𝑜𝑛𝑢𝑡 15 𝑚𝑖𝑛 = 35.625 𝑑𝑜𝑛𝑢𝑡𝑠 𝑆𝑖𝑔𝐹𝑖𝑔𝑠 → 36 𝑑𝑜𝑛𝑢𝑡𝑠 × × 1 𝑚𝑖𝑛 4.0 𝑑ℎ Note that this answer tells us that the values and calculations used limit us only know approximately 36 donuts, but not exact. We would know that this value of 36 has a level of uncertainty in it such that we would expect 35-37 donuts. Candles: 7. How many molecules of wax are burnt in 30. minutes? 0.100 𝑚𝑜𝑙𝑒 𝑤𝑎𝑥 6.022 × 1023 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒𝑠 30. 𝑚𝑖𝑛 1 ℎ𝑟 = 3.011 × 1022 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒𝑠 × × × 60 𝑚𝑖𝑛 1 ℎ𝑟 1 𝑚𝑜𝑙𝑒 𝑆𝑖𝑔𝐹𝑖𝑔𝑠 → 3.0 × 1022 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒𝑠 8. How many moles of Oxygen (O2) are consumed in that time? 0.100 𝑚𝑜𝑙𝑒 𝑤𝑎𝑥 1 ℎ𝑟 48 𝑚𝑜𝑙𝑒 𝑂2 30. 𝑚𝑖𝑛 × × × = 2.4 𝑚𝑜𝑙𝑒 𝑂2 ℎ𝑟 60 𝑚𝑖𝑛 1 𝑚𝑜𝑙𝑒 𝑤𝑎𝑥 1 𝑆𝑖𝑔𝐹𝑖𝑔𝑠 → 2.4 𝑚𝑜𝑙𝑒 𝑂2 9. How many moles of CO2 are generated? 30. 𝑚𝑖𝑛 1 ℎ𝑟 0.100 𝑚𝑜𝑙𝑒 𝑤𝑎𝑥 31 𝑚𝑜𝑙𝑒 𝐶𝑂2 = 1.55 𝑚𝑜𝑙𝑒 𝐶𝑂2 × × × 1 60 𝑚𝑖𝑛 ℎ𝑟 1 𝑚𝑜𝑙𝑒 𝑤𝑎𝑥 𝑆𝑖𝑔𝐹𝑖𝑔𝑠 → 1.6 𝑚𝑜𝑙𝑒 𝑂2 10. CO2 has a mass of 30.01 grams per mole. How many grams of CO 2 are produced? 1.55 𝑚𝑜𝑙𝑒 𝐶𝑂2 30.01 𝑔 𝐶𝑂2 = 46.5155 𝑔 𝐶𝑂2 × 1 1 𝑚𝑜𝑙𝑒 𝐶𝑂2 𝑆𝑖𝑔𝐹𝑖𝑔𝑠 → 47 𝑚𝑜𝑙𝑒 𝑂2 Two significant figures due previsions limitation of 30. min in problem 9. The unrounded number 1.55 was used in this calculation to avoid rounding error.