### Moles, Molarity, and Molality

Over the last few sections, you have learned how to calculate the weight of atoms and molecules. Until now, we haven't given units for these measurements. Obviously, a carbon atom cannot weigh 12 grams, or else we all would be very much heavier than we are now! The unit of mass for atoms and molecules is the atomic mass unit, or a.m.u. However, in terms of chemical reactions and equations, the a.m.u. is far too small, while 1-gram samples of two reactants can have different amounts of molecules, making reaction calculations difficult.

In light of these difficulties, scientists created a new unit: the mole. One mole of a substance is defined as the quantity that will give a number of grams equal to that substance's unit weight. For example, one mole of carbon atoms (which have an atomic weight of 12) will weigh 12 grams; one mole of water molecules, with a molecular weight of 18, will weigh 18 grams. It turns out that the quantity needed for both examples above is the same: 6.02 x 1023 units! One mole of anything will have a weight in grams equal to the its number of a.m.u.'s. For reference, the mole is often written as "mol" when used in calculations.

To convert a quantity given in grams to moles, divide the gram weight by the unit weight (atomic weight, molecular weight, etc.). In our first-year Chemistry course, this was referred to as "getting on the mole train." For instance, 5 grams of carbon divided by 12 grams per mole is equal to .41667 moles. 20 grams of methanol (with a molecular weight of 32) is 20 g/(32 g/mol) = 0.625 moles. To convert from moles to grams, multiply the number of moles by the grams per mole of the substance. For instance, .5 mol of acetic acid (CH3COOH, 48 g/mol) weighs .5 * 48 = 24 grams. Generally, these weights can use rounded-off values (one decimal place) unless extremely precise work is being done.

 Example Problem 1 Convert the following amounts from grams to moles or moles to grams. Use a periodic table if necessary to look up atomic or molecular weights. A. 39.8 grams of ammonia (NH3) B. 7.02 moles of cupric chloride (CuCl2) C. 365.4 grams of hydrogen peroxide (H2O2) D. 6.0 kilograms of uranium E. 91.2 milligrams of water (H2O) F. .00127 moles of potassium hydroxide (KOH) Answers: A. Ammonia weighs 17 grams per mole. Therefore, 39.8 g / (17 g/mol) = 2.34 mol B. Cupric chloride has a molecular weight of 134.5. Multiplying this by 7.02 moles gives 944.19 grams, or .94419 kilograms C. Hydrogen peroxide weighs 34 grams per mole, so 365.4 g / (34 g/mol) = 10.75 mol D. First, convert kilograms to grams (6.0 kilograms = 6000 grams). Then, divide by 238 g/mol for the atomic weight of uranium: 6000 g / (238 g/mol) = 25.21 mol E. First, convert milligrams to grams (91.2 milligrams = .0912 grams). Then, divide by 18 g/mol for the molecular weight of water: .0912 g / (18 g/mol) = .005067 mol F. Potassium hydroxide has a molecular weight of 56. Multiplying this weight by .00127 gives 0.07112 grams, or 71.12 milligrams

Moles are also a convenient way to express quantities of dissolved or diluted substances. Because many reactions involve aqueous (dissolved) products or reactants, a standard unit was also developed from the mole to use for these solutions. This unit, molarity (or molar, when used as an adjective), is based on concentration in moles of substance per liter of total volume. The symbol for molarity is a capital M. For example, 2 moles of KOH dissolved in 2 liters of water will give a molarity of 1 (2 moles / 2 liters = 2 moles/liter), or a 1-molar solution. Often, amounts of solute are provided in grams and must be converted to moles before molarity can be determined.

Converting from molarity to moles is also a common occurrence. Say you need to know how many moles of acid are in 300 milliliters of 5M hydrochloric acid. To do this, multiply the molarity by the volume in liters: (5 mol/l)(300 ml / 1000 ml/l) = 5 mol/l * 3/10 l = 1.5 moles. This technique can be used when diluting a solution as well. Let's assume a chemist has 1 liter of a .5M solution of KNO3, but adds 3 more liters of water. What is the molarity of the resulting solution? First, find the number of moles in the original solution: .5 moles/liter * 1 liter = .5 moles. Now, find the new volume: 1 liter + 3 liters = 4 liters. Finally, divide the number of moles by the new volume to find the molarity: .5 moles / 4 liters = 0.125 molar.

 Example Problem 2 Find the molarity of the following solutions. A. .5 moles of NaCl dissolved in 600 milliliters of water B. 7.02 moles of cupric chloride (CuCl2) C. 6.3 grams of AgNO3 in .22 liters of water D. 25 ml of .6M solution of HCl mixed with an 80 ml sample of 2M HCl Answers: A. Molarity is calculated in moles per liter, so first convert 600 milliliters to .6 liters. Next, divide: .5 moles / .6 liter = 0.83 molar. B. Cupric chloride has a molecular weight of 134.5. Multiplying this by 7.02 moles gives 944.19 grams, or .94419 kilograms C. First, convert grams of silver nitrate to moles. Since silver nitrate has a molecular weight of 169.9, 6.3 grams is .0371 moles. Then, to find molarity, divide .0371 moles of solute by .22 liters of water to get a concentration of .1685 moles per liter. D. First, find the number of moles in each solution: (.6M)(25/1000 l) = 0.015 moles; (2M)(80/1000 l) = 0.16 moles. The total number of moles is therefore 0.175. Finally, divide by total volume (25 ml + 80 ml = 105 ml or .105 l): .175 mol / .105 l = 1.67M. If you got that, give yourself a pat on the back!

Notice that on all the calculations we do, the units always cancel to give the desired amount. In some calculations, moles per liter are multiplied by liters, giving simply moles. If you're lost in a calculation, sometimes stopping and deciding which units you have, which units you want, and what conversion factors you need to get there can set you on the right track. This technique is called dimensional analysis. For instance, converting 13 miles per hour to centimeters per second would require several steps of dimensional analysis. We won't run through the calculations, but feel free to try yourself. By the way, the answer is 581.152 cm/sec!

Another unit of concentration is the molal (symbol "m"). It is not commonly used except for a few specific calculations, but is helpful to know just in case. To find concentration in molals, divide the number of moles of solute by kilograms of solvent. For instance, the molality of 5 moles of KOH dissolved in 2 kilograms of water is 5 moles / 2 kilograms = 2.5 molals. Since molality is so rarely used, use molality unless otherwise stated in a problem. To convert from molality to molarity, change the kilograms of solvent into liters by using its density (for water, the density is one gram per milliliter, or one kilogram per liter). Molality is therefore about the same as molarity for water solutions, but the weight of the solute is subtracted from the total weight to give kilograms of solvent.

 Example Problem 3 Find the molality of the following solutions. A. 3.4 moles of NH3 dissolved in 4500 grams of water B. 0.50 moles of HNO3 dissolved in 1.2 kilograms of water C. 3.00 liters of 1.50M KBr solution D. 17.0 grams of LiF in 34.0 moles of water Answers: A. To find molality, divide moles by kilograms of solvent: 3.4 mol / 4.5 kg = 0.76 molal B. Using the same method, the solution has a molality of .50 mol / 1.2 kg = 0.4167 C. First, find the moles of solute: 3.00 liters * 1.50 moles / liter = 4.50 moles. Since water has a weight of 1 kilogram per liter, a 3-liter solution weighs 3 kilograms. Next, find the weight of the solute (KBr has a molecular weight of 118.9, so 4.50 moles weighs 535 grams) and subtract this from the mass of solution to get a solvent weight of 2460 grams. Finally, divide to get a molality of 4.50 mol / 2.46 kg = 1.83 molal. Note the difference between the molality and molarity of this solution. D. This problem is a little less work! First, find out how many moles are in 17 grams of LiF: the molecular weight is 25.9, so 17 g / (25.9 g/mol) = 0.656 moles. Next, find out how much 34 moles of water weighs: multiplying 18.0 grams per mole for water by 34.0 moles gives 612 grams, or .612 kilograms. Finally, find molality: .656 mol / .612 kg = 1.07 molal.

This is the last lesson page in this chapter. The next page, the Review section, will recap the important concepts you've learned and present quiz questions, letting you make sure you understand the contents of this chapter.

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