The factor label method

1. The factor label method • A way to solve math problems in chemistry • Used to convert km to miles, m to km, mol to g, g to mol, etc. • To use this we need: 1) desired quantity, 2) given quantity, 3) conversion factors • Conversion factors are valid relationships or equities expressed as a fraction E.g. for 1 km=0.6 miles the conversion factor is Q. write conversion factors for 1 foot =12 inches Q. what conversion factors can you think of that involve meters?

2. Conversion factors Conversion factors for 1 ft = 12 in There are almost an infinite number of conversion factors that include meters:

3. Conversion factors • We have looked at conversion factors that are always true. There are conversion factors that are only true for specific questions • E.g. A recipe calls for 2 eggs, 1 cup of flour and 0.5 cups of sugar • We can use these conversion factors

4. The steps to follow Now we are ready to solve problems using the factor label method. The steps involved are: • Write down the desired quantity/units • Equate the desired quantity to given quantity • Determine what conversion factors you can use (both universal and question specific) • Multiply given quantity by the appropriate conversion factors to eliminate units you don’t want and leave units you do want • Complete the math

5. Factor label example Q - How many kilometers are in 47 miles? (note: 1 km = 0.621 miles) # km First write down the desired quantity

6. Factor label example Q - How many kilometers are in 47 miles? (note: 1 km = 0.621 miles) # km = 47 mi Next, equate desired quantity to the given quantity

7. Factor label example Q - How many kilometers are in 47 miles? (note: 1 km = 0.621 miles) # km = 47 mi Now we have to choose a conversion factor

8. 1 km 0.621 mi 0.621 mi 1 km Factor label example Q - How many kilometers are in 47 miles? (note: 1 km = 0.621 miles) # km = 47 mi What conversion factors are possible?

9. 1 km 0.621 mi 0.621 mi 1 km Factor label example Q - How many kilometers are in 47 miles? (note: 1 km = 0.621 miles) # km = 47 mi Pick the one that will allow you to cancel out miles

10. 1 km 0.621 mi 0.621 mi 1 km Factor label example Q - How many kilometers are in 47 miles? (note: 1 km = 0.621 miles) # km = 47mi Pick the one that will allow you to cancel out miles

11. 1 km 0.621 mi 0.621 mi 1 km Factor label example Q - How many kilometers are in 47 miles? (note: 1 km = 0.621 miles) # km = 47mi Multiply given quantity by chosen conversion factor

12. x 1 km 0.621mi Factor label example Q - How many kilometers are in 47 miles? (note: 1 km = 0.621 miles) # km = 47mi Multiply given quantity by chosen conversion factor

13. x 1 km 0.621mi Factor label example Q - How many kilometers are in 47 miles? (note: 1 km = 0.621 miles) # km = 47mi Cross out common factors

14. x 1 km 0.621 Factor label example Q - How many kilometers are in 47 miles? (note: 1 km = 0.621 miles) # km = 47 Cross out common factors

15. x 1 km 0.621 Factor label example Q - How many kilometers are in 47 miles? (note: 1 km = 0.621 miles) # km = 47 Are the units now correct?

16. x 1 km 0.621 Factor label example Q - How many kilometers are in 47 miles? (note: 1 km = 0.621 miles) # km = 47 Yes. Both sides have km as units.

17. x 1km 0.621 Factor label example Q - How many kilometers are in 47 miles? (note: 1 km = 0.621 miles) # km = 47 Yes. Both sides have km as units. #km

18. x 1 km 0.621 Factor label example Q - How many kilometers are in 47 miles? (note: 1 km = 0.621 miles) = 75.7 km # km = 47 Now finish the math.

19. x 1 km 0.621 Factor label example Q - How many kilometers are in 47 miles? (note: 1 km = 0.621 miles) = 75.7 km # km = 47 The final answer is 75.7 km

20. Summary The previous problem was not that hard In other words, you probably could have done it faster using a different method However, for harder problems the factor label method is easiest

21. x 1 Can$ 0.65 US$ More examples • You want to buy 100 U.S. dollars. If the exchange rate is 1 Can$ = 0.65 US$, how much will it cost? # Can$ = 100 US$ = 153.85 Can$

22. x 3 ft x 12 in x 1 cm 1 yd 1 ft 0.394 in More examples 2. There are 12 inches in a foot, 0.394 inches in a centimeter, and 3 feet in a yard. How many cm are in one yard? # cm = 1 yd = 91.37 cm

23. The Chemist's Dozen: Avogadro’s Number • Chemists count atoms, molecules and ions in groups called moles. (abbreviated mol). • The number 6.02 x 1023 is called Avogadro’s number (NA) • 1 mol = 6.02 x 1023

24. What 2 factors can you make from that relationship? 1mol and 6.02 x 1023 6.02 x 1023 1 mol The unit with Avogadro’s number could be atoms, ions or molecules.

25. Let’s do a calculation A sample contains 1.25 mol of nitrogen dioxide molecules. • How many molecules are in the sample? b. How many atoms are in the sample?

26. a. How many molecules? • x molecules NO2 = 1.25 mol x conversion factor Which factor is appropriate? 1mol or 6.02 x 1023molecules 6.02 x 1023 molecules 1 mol

27. How many molecules in 1.25 moles? • x molecules = 1.25 mol x 6.02 x 1023molecules 1 mol = 7.525 x 1023 molecules The unit mol cancels out and we have the unit molecules in our answer.

28. How many atoms in 1.25 moles of NO2? • There are 2 relationships we need. • 1 mol = 6.02 x 1023 molecules • 1 molecule of NO2 contains 3 atoms of • 1 molecule = 3 atoms. • Note this last factor is specific to NO2. It would not be the same for CH4 or other molecules.

29. How many atoms in 1.25 moles NO2? x atoms in NO2 = 1.25 mol NO2 x6.02 x 1023 molecules NO2x 3 atoms in NO2 1 mol NO2 1 molecule NO2 Use the units to guide the calculation. NO2 is a compound so its particles are molecules. We used that in part a. In part b we want to know how many atoms are in the 1.25 mol sample so we add a conversion factor that shows that there are 3 atoms (1 N and 2 O) atoms in each molecule of NO2. After doing the math x atoms in NO2 = 1.25 x 6.02 x 1023 x 3 atoms = 2.58 x 1023 atoms This could also be done as a 2 step problem using your answer from part a followed by the 3 atom/1 molecules conversion factor.