We've spent much of our first two chapters talking about intensive properties of substances, compounds, and atoms: that is, properties which depend directly on the character of the substance, and not on how much of the substance we have. Intensive properties are often used to identify a given substance. But we need extensive properties as well, and the mole gives us one way of measuring the amount of a given substance.
The two types of properties are not necessarily independent of one another. Both volume and mass are extensive properties of a substance: they both depend on the amount of the substance. But divide mass by volume and you get density--a property which remains the same regardless of how much of the substance you have.
The mole is a convenient way of talking about large groups of atoms or particles. We use it in the same way we use "dozen" to talk about 12 bagels, except that a mole is roughly 6.022 * 1023 of whatever we are counting. The exact definition is "the number of atoms in 12.0g of carbon-12". Since carbon-12 has a an atomic weight of 12, this gives us an easy way to move back and forth between mass and number in atoms. A gram of protons (e.g., hydrogen atoms, since the mass of electrons is negligible compared to the proton) contains a mole of protons. A mole's worth of atoms of an element with atomic weight 3 will weigh 3 grams: this is the element's molar mass. Now we have a way to move back and forth easily between mass measurements that we can make in the lab, and counts of atoms that will be taking part in reactions.
There are several ways to determine the number of atoms in a gram-weight of an element [this number is called the mole]. One common way is to use electrolysis. In this process (which we will do later in the year), we use a source of electricity like a battery to run current through an acid solution containing a dissolved metal ion. The ion is deposited on one of the battery electrodes. After measuring the amount of current used and the mass of the metal deposited in a given period of time, we can determine the number of atoms of the metal involved:
First calculate total charge from the amount of current used: current * time = charge. Then multiply all the involved quantities so the units cancel:
Averaging the results of many such experiments gives 6.022 * 10 ** 23 atoms/mole. One mole of hydrogen has 6.022 * 10 ** 23 atoms of hydrogen, and weighs 1.008 grams.
To honor of Avogadro's work in determining mass ratios and formula, chemists call this "Avogadro's number".
Let's look at the real problems chemists ran into in trying to determine compound proportions. Assume that we don't have access to the periodic table, and we haven't yet discovered that we can figure out the number of atoms by balancing the valence electrons involved.
In the late 1700s, a number of chemical researches led to the realization that atoms combine to form compounds according the the law of constant proportionality, which became of the three major hypotheses of Dalton's atomic theory. Hydrogen is the lightest element, and combines with oxygen in the mass proportion 1:8 to form water. Other compounds can also be analyzed in terms of their relative masses. Note that this doesn't tell us how many atoms are involved in the compound. The early chemists, John Dalton among them, believed that water was made of one hydrogen and one oxygen atom.
How can we determine how many atoms are involved? In the early 1800s, Avogadro suggested that at the same pressure, temperature, and volume, different gasses will have the same number of atoms or molecules. Since (keeping temperature and pressure constant) a volume of hydrogen gas combines with a volume of chlorine gas to produce two volumes of hydrogen chloride, this would mean either that the hydrogen and chlorine atoms split, or that (as Avogadro proposed) hydrogen and chlorine existed as 2-atom molecules in their gaseous form. Most chemists followed Dalton's assumption [wrong, as it turns out] that like atoms repelled one another and could not form molecules like H2. They rejected Avogadro's suggestion for over 50 years. By 1861, however, there was enough evidence to support it, and chemists begin to use Avogadro's hypothesis use it to determine the proportions of compounds by atom count as well as by weight.
Experiments like sparking a mixture of hydrogen gas and oxygen gas to produce water led to the determination that while the mass proportions of water are 1 mass of hydrogen to 8 masses of oxygen, the atomic proportions are 2 hydrogen atoms to 1 oxygen atom. With this as a scale, chemists were able to determine a atomic weight of 16 "hydrogen masses" for oxygen.
The atomic weights for non-gasses were determined using other techniques. In the early 1800s, research in metals yielded some peculiar information about the way metals absorb heat energy. Specific heat is the amount of heat, in calories, required to raise the temperature of 1 gram of a substance 1 degree centigrade. [By definition, a calorie is the amount of heat required to raise the temperature of 1 gram of water from 20 degrees centigrade to 21 degrees centrade at sea level, so specific heats are always done in comparison to water.] The specific heat of each element and compound is one of the physical characteristics of that compound. In 1819, Pierre Dulong and Alexis Petit discovered that the atomic weight of an element multiplied by the specific heat of the element was a constant of approximately 6.3. This is an empirical relationship: while Dulong and Petit didn't understand the reason for the relationship, they were able to apply it to many, but not all, metals successfully.
The calculations we are learning now are basic to the activities of the analytic chemist, who is concerned with how much of what is in any given sample of matter. This can be an important question to a geologist looking for particular mineral deposits, to a pharmaceutical company preparing a new drug, to industries trying to create new paint colors or new plastics. While the calculations may look complicated, if you keep in mind the purpose of each process, you will quickly sort out what approach to take for a given sample and set of data.
There is a box of epsom salts on my laundry shelf, with the formula MgSO4 * 7 H2O on the side. If I heat the epsom salt in a test tube and boil off all the water, what change in mass can I expect? If it isn't at least 10%, I probably won't do the experiment, because I wouldn't be able to measure enough difference on my not-very-accurate balance to know whether I was getting an error or a real difference in mass.
One way to determine this is by computing the fraction of mass in the total sample which is contributed by each element. I need a chart! After adding up the masses contributed by each element, the total amount of mass in a single molecule, including its water molecule, is 241.37.
|Element||Atomic Mass (g/mole)||Number moles in gram
|Total mass||Percent of total (241.37)|
|Magnesium Mg||24.30||1||24.30||24.30/241.37 = .10067 = 10%|
|Sulfer S||32.07||1||32.07||32.07/241.37 = .13286 = 13%|
|Oxygen (in sulfate)||16.00||4||64||64.00/241.37 = .2652 = 27%|
|Oxygen (in water)||16.00||7||112||112/241.37 = .4640 = 46%|
|Hydrogen||1.008||7*2||14.00||14/241.37 = .0410 = 4%|
So I should expect about 50% of the weight of my sample (the part in the water) to evaporate. Since that is half my sample weight, I will probably do the experiment (observing proper safety precautions, of course!). If I weigh out 10g of epsom salts on my scale, I can expect 50% of 10g or 5g of it to evaporate as I heat the sample.
A geologist enters our chemical consulting firm. He has a gorgeous rock sample, covered with colorless crystals, which dissolves easily in acids, allowing us to separate out the components. These are calcium, boron, silicon, oxygen, and hydrogen. A single crystal yields
|Element||Sample mass||Atomic mass/mole||# moles present||#moles/smallest #moles = # atoms/molecule|
|Ca||4.028 g||40.08||.35||.35/.35 = 1|
|B||37.84 g||10.81||.35||35/.35 = 1|
|S||9.83 g||28.09||35||.35/.35 = 1|
|O||8.00 g||16.00||1.75||35/.175 = 5|
|H||.35 g||1.008||35||.35/.35 = 1|
The ratios are Ca:B:S:5 O:H, so the simplest forumla is CaBSiO5H. Further analysis shows the molar mass is 149.18g, which is the total of the simplest formala, calculated in gram weights. So the simple formula is the molecular formula, in this case. Even more analysis determines that the H is paired with one of the oxygens to form a hydroxide ion, so the actual formula is CaBSiO4(OH), but there is no way to determine this internal structure from the masses.
The actual rock involved, by the way, is datolite, a nesosilicate crystalline form found in parts of Germany, Norway, Italy, around Lake Superior, in Massachusetts and in New Jersey. When deposits are large enough, it is a commercial source of boron.
Here's another example, this time from masses in a reaction. A 1.000g mass of acetic acid burns to give 1.466 g of carbon dioxide and .6001 g of water. What are the ratios of C, O, and H in the acetic acid sample (this is equivalent to asking for the simplest formula)?
First calculate the gram/mole of the products:
CO2 = 1 carbon at 12.01 plus 2 oxygens at 16.00 each = 12.01 + 32.00 = 44.01 g/mole
H2O = 2 hydrogen at 1.008 each plus 1 oxygen at 16.00 = 2.016 + 16.00 = 18.02 g/mole
Now calculate the amount of each element in the sample. Since oxygen is in both products, we save it until last and calculate it from the differences.
There is 1 mole of carbon in 1 mole of CO2, so there are 12.01 g of C in 44.01g of CO2 (note how the units cancel to give grams of C):
C in product is 1.466 g CO2 * (12.01g C)/(44.01g CO2) = .4001 g C
Do the same with the hydrogen in the water:
H in product is .6001 g H2O * (1.008g H)/(18.02g H2O) = .0673 g H
Since no mass was lost in the reaction, if .4001g and .0673g are accounted for by the C and H respectively, the remaining mass of the 1.000g we started out with must belong to the oxygen:
1.000g - .4001g - .0673g = .533g oxygen.
Now we calculate the moles involved from the moles/gram-weights, in order to get the element ratios in a mole of acetic acid:
C = .4001 g C * 1 mole C/ 12.01g C = .0333 moles of carbon
H = .0673 g H * 1 mole H/1.008g H = .0666 moles of hydrogen
O = .533 g O * 1 mole O/16.00g O = .0333 moles of oxygen
The ratios are .0333 : .0666 : .0333 is 1:2:1, or 1 carbon to 2 hydrogen to 1 oxygen. The simplest formula is CH2O.
By the way, acetic acid is just white vinegar, so now you know the simplest formula for part of your salad dressing!
Since atom have have constant masses and combine in constant proportions, we can determine compound mass ratios from the atomic weights of the components and the compound formula. Two amus of hydrogen (2 hydrogen atoms at amu 1 each) combine with 16 amus of oxygen (1 oxygen atom at 16 amus) to form one water molecule (18 amu). Therefore 2 grams of hydrogen combine with 16 grams of oxygen to form 18 grams of water, since both amus and grams are units of mass. The gram-weight of any element is just its atomic number in grams!
Compound formulas are a kind of algebraic equation for the mass of the compound, in amus or in grams:
Epsom salt is magnesium phosphate with 7 water molecules attached:
MgSO4 . 7 H2O.
To get the atomic weight: follow the rules above. Magnesium is 24, sulfur is 32, oxygen 16, and hydrogen 1:
24 + 32 + 4*16 + 7 * (1*2 + 16) = 246
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