Unit 23 · Chemistry: Stoichiometry, Solutions and Energy

Pour a teaspoon of water into your palm. It holds about 5 grams. It also holds about 170,000,000,000,000,000,000,000 molecules. No balance can count them, and no one would want to. Chemists needed a way to talk about huge numbers of tiny particles the way a baker talks about a dozen eggs. The answer was a counting unit called the mole.

One mole is 6.022 times 10 to the 23rd power particles. That number is Avogadro's number, named for the Italian scientist Amedeo Avogadro, though he never knew its value. The number is not random. It was chosen so that one mole of carbon-12 atoms has a mass of exactly 12 grams, matching the atomic mass on the periodic table. Since 2019, the mole has been defined as exactly 6.02214076 times 10 to the 23rd power particles, so the link to carbon is now a very close approximation rather than a definition.

The payoff is simple. The periodic table lists the mass of one atom in atomic mass units. Read that same number in grams, and you have the mass of one mole of atoms. One iron atom has a mass of about 55.8 atomic mass units. One mole of iron, a small cube you could hold, has a mass of about 55.8 grams. Weigh 55.8 grams of iron, and you have counted out 602,200,000,000,000,000,000,000 atoms without seeing a single one.

How big is a mole? A mole of pennies, spread across the whole surface of Earth, would make a layer hundreds of kilometers deep. A mole of grains of sand would cover the entire United States far deeper than any building. Yet a mole of water molecules is only about 18 milliliters, a few sips. Atoms are small enough that a number this absurd is exactly the right size for a glass of water.

Water is H2O: two hydrogen atoms and one oxygen atom per molecule. To find the mass of a mole of water, add up the atoms. Hydrogen is 1.008 grams per mole, and there are two, so 2.016 grams. Oxygen is 15.999 grams per mole. The total is about 18.02 grams per mole. That is why a mole of water is such a small puddle: 18 grams is about a tablespoon and a bit.

The same addition works for anything with a formula. Table salt, NaCl, is 22.99 plus 35.45, about 58.44 grams per mole. Carbon dioxide, CO2, is 12.01 plus two times 16.00, about 44.01 grams per mole. Glucose, C6H12O6, is six carbons, twelve hydrogens and six oxygens, about 180.16 grams per mole. Be careful with subscripts and parentheses. Calcium nitrate, Ca(NO3)2, has two nitrate groups, so two nitrogens and six oxygens.

Once you know a molar mass, you can find the percent composition: what share of the mass belongs to each element. In water, hydrogen supplies 2.016 grams of every 18.02, about 11 percent, and oxygen supplies the other 89 percent. That number surprises people. Water is mostly hydrogen by count of atoms but mostly oxygen by mass, because each oxygen atom is about sixteen times heavier than each hydrogen.

Percent composition is how chemists check what they have made. If a lab produces a white powder and claims it is pure sodium chloride, the powder should be 39.3 percent sodium by mass. Burn a sample of a new compound, capture the carbon dioxide and water, weigh them, and you can work backwards to the ratio of atoms in the original. That is how the formulas of thousands of natural substances, from caffeine to vitamin C, were first worked out.

Think of the mole as a bridge with two lanes. On one side is what you can measure, mass in grams. On the other side is what you want to know, number of particles. Molar mass carries you from grams to moles. Avogadro’s number carries you from moles to particles. You can cross in either direction, and every calculation in this chapter uses this bridge.

Take 9.0 grams of water. Divide by the molar mass, 18.02 grams per mole, and you get 0.50 mole. Multiply by 6.022 times 10 to the 23rd power, and you get about 3.0 times 10 to the 23rd power molecules. Going the other way is just as easy. A chemist who needs 2.0 moles of sodium chloride multiplies by 58.44 grams per mole and weighs out 117 grams. The balance does the counting.

The trick is to keep units attached to every number. Grams divided by grams-per-mole gives moles. Moles times particles-per-mole gives particles. If your units do not cancel to what you want, you multiplied when you should have divided. Chemists call this dimensional analysis, and it catches more mistakes than any calculator.

Why does this matter outside a classroom? A pharmacist making a solution needs a certain number of drug molecules per liter, not a certain number of grams, because the body responds to molecules. A fertilizer plant needs the right count of nitrogen atoms per bag. A jeweler testing gold wants the ratio of gold atoms to copper atoms. In every case, the path runs from a balance, through molar mass, to a count.

Strike a match and the wood burns away to almost nothing. It looks like matter vanished. In 1789 the French chemist Antoine Lavoisier showed, with sealed flasks and a very good balance, that it does not. The mass of the ash plus the gases that escaped equals the mass of the match plus the oxygen it used. Atoms rearrange. They do not disappear. This is the law of conservation of mass, and every chemical equation must obey it.

A chemical equation is a sentence written in formulas. Methane burning in oxygen reads CH4 + O2 gives CO2 + H2O. Count the atoms. The left has 1 carbon, 4 hydrogens and 2 oxygens. The right has 1 carbon, 2 hydrogens and 3 oxygens. Hydrogen and oxygen do not match, so the equation is unbalanced. It describes something impossible.

To fix it, you may change only the numbers in front of formulas, called coefficients. You may never change a subscript, because that would turn one substance into a different one. H2O is water; H2O2 is hydrogen peroxide, a bleach. Put a 2 in front of H2O to get 4 hydrogens on the right. Now the right has 4 oxygens, so put a 2 in front of O2. The balanced equation is CH4 + 2 O2 gives CO2 + 2 H2O. One carbon, four hydrogens and four oxygens on each side.

A useful order is to balance metals first, then nonmetals, then hydrogen, then oxygen, saving any element that appears alone, like O2, for last, since changing its coefficient affects nothing else. Check every atom at the end. A balanced equation is more than tidy bookkeeping. Its coefficients are the recipe, the exact ratio in which molecules react, and the rest of this chapter depends on them.

A balanced equation is a recipe written in moles. Read 2 H2 + O2 gives 2 H2O as: two moles of hydrogen react with one mole of oxygen to make two moles of water. The numbers 2, 1 and 2 are the mole ratio. Double the hydrogen and you need double the oxygen and get double the water. Everything scales together. Using these ratios to predict amounts is called stoichiometry, from Greek words meaning to measure elements.

Here is the standard path. Start with a mass of one substance. Convert it to moles using molar mass. Use the mole ratio from the equation to find moles of the substance you want. Convert those moles back to grams. Three steps: grams to moles, moles to moles, moles to grams. Try it: how much water forms when 4.0 grams of hydrogen burns completely? 4.0 grams of H2 is 2.0 moles. The ratio of H2 to H2O is 2 to 2, so 2.0 moles of water form. That is 2.0 times 18.02, about 36 grams of water.

Notice that 4 grams of hydrogen gave 36 grams of water. The extra 32 grams came from oxygen. Mass is conserved, but the mass ratio, 4 to 32 to 36, is not the mole ratio, 2 to 1 to 2. Never take a ratio straight from the grams. You must pass through moles, because the equation counts molecules, and molecules of different substances have different masses.

Industry runs on these calculations. A steel plant knows how many tons of coke it must burn per ton of iron ore. A bakery knows how much baking soda releases the right volume of carbon dioxide for a batch of dough. The airbag engineer knows that 2 moles of sodium azide, about 130 grams, release 3 moles of nitrogen. The ratio in the equation is the whole design.

You have 10 slices of bread and 3 slices of cheese. Each sandwich needs 2 bread and 1 cheese. You can make 3 sandwiches, not 5, and 4 slices of bread are left over. Cheese is the limiting reactant: it runs out first and sets the amount of product. Bread is the excess reactant. Chemistry works the same way. Reactants are almost never mixed in the exact ratio of the equation, so one of them always limits.

To find the limiting reactant, convert each reactant to moles, then ask how much product each could make on its own. The smaller answer wins. Suppose 4.0 grams of hydrogen (2.0 moles) meets 16 grams of oxygen (0.50 mole). Hydrogen alone could make 2.0 moles of water. Oxygen, in the ratio 1 O2 to 2 H2O, could make only 1.0 mole. Oxygen limits. Only 1.0 mole of water forms, and 1.0 mole of hydrogen sits unreacted.

The amount of product the limiting reactant allows is the theoretical yield, the most you could possibly get. Real reactions rarely reach it. Some product stays stuck to the glassware. Some reactant takes a side path to something else. Some reactions simply stop before finishing. The amount you actually collect is the actual yield, and the percent yield is actual divided by theoretical times 100.

Percent yield is a report card on a procedure. A drug company making a medicine in ten steps, each at 80 percent yield, ends with only about 11 percent of what it started with, because 0.8 multiplied by itself ten times is about 0.11. That is why chemists spend careers pushing each step toward 100 percent. In an airbag, an excess of azide is deliberate, so the igniter is never the limiting factor and the bag always fills.

Squeeze a sealed plastic bottle and it pushes back. Leave a balloon in a hot car and it swells. Take a bag of chips onto a plane and it puffs up as the cabin pressure drops. Gases respond to squeezing and heating in ways that liquids and solids barely do, and those responses follow simple rules that were among the first quantitative laws in chemistry.

In 1662 Robert Boyle trapped air in a J-shaped glass tube with mercury and measured how the air shrank as he added more mercury. Doubling the pressure halved the volume. Tripling it cut the volume to a third. Boyle's law says that for a fixed amount of gas at constant temperature, pressure times volume is constant. Gas at 1 atmosphere in 6 liters, squeezed to 3 atmospheres, occupies 2 liters. This is why a scuba diver’s lungs, full of air at depth, would burst if the diver held that breath while rising.

Heat a gas and it expands. Jacques Charles in the 1780s and Joseph Gay-Lussac around 1802 measured how much. Volume grows in a straight line with temperature. Extend that line downward and every gas shrinks toward zero volume at the same temperature, about minus 273 degrees Celsius. That point is absolute zero, and a temperature scale starting there, the kelvin scale, makes the law simple. Charles's law: at constant pressure, volume divided by kelvin temperature is constant. Warm a gas from 300 K to 600 K and its volume doubles.

Hold the volume fixed instead, as in a sealed steel tank, and pressure rises in proportion to kelvin temperature. That is why aerosol cans warn against heat, and why tire pressure climbs after a long drive. Always convert to kelvins: add 273.15 to the Celsius reading. Going from 20 degrees to 40 degrees Celsius is not a doubling. It is 293 K to 313 K, a rise of about 7 percent.

Boyle’s law, Charles’s law and Avogadro’s idea that equal volumes hold equal numbers of molecules all describe one gas from different angles. Combine them and you get a single equation: PV = nRT. P is pressure, V is volume, n is the number of moles, T is kelvin temperature, and R is a constant that makes the units agree. This is the ideal gas law. With any three of the four variables, it gives you the fourth.

R, the gas constant, is 0.08206 liter-atmospheres per mole-kelvin when pressure is in atmospheres and volume in liters, or 8.314 joules per mole-kelvin in SI units. Plug in 1 mole at 1 atmosphere and 273 K, the conditions chemists call standard temperature and pressure or STP, and V comes out to 22.4 liters. Warm that mole to 25 degrees Celsius, 298 K, and it fills about 24.5 liters. Every gas, hydrogen or carbon dioxide or nitrogen, takes up the same volume per mole, because the volume depends on the number of molecules, not their size.

This is the link between the balance and the bag. Stoichiometry gives moles of gas. The ideal gas law turns moles into liters. Two moles of sodium azide, 130 grams, release 3 moles of nitrogen. At 25 degrees Celsius and 1 atmosphere, that is 3 times 24.5, about 74 liters. Hot gas fresh from the reaction takes up even more, and it pushes harder, which is exactly what the designers want in the first milliseconds.

The law also works backwards. Weigh a flask of unknown gas, measure its volume, temperature and pressure, solve for n, and divide mass by moles to get the molar mass. That is how nineteenth-century chemists identified gases and, following Stanislao Cannizzaro’s argument in 1858, finally settled the atomic masses of the elements. Avogadro’s ignored idea became the key to the whole periodic table.

Why does a gas obey such simple laws? The answer is the kinetic-molecular theory, built up in the 1800s. Its picture has a few parts. A gas is mostly empty space, with molecules far apart compared to their size. The molecules move in straight lines until they collide, with each other or the walls, and the collisions lose no energy. And the average kinetic energy of the molecules depends only on the kelvin temperature. Higher temperature means faster molecules.

Pressure, in this picture, is molecules hitting the walls. Squeeze the volume in half and each molecule reaches a wall twice as often, so pressure doubles. That is Boyle’s law. Heat the gas and the molecules hit harder and more often, so pressure rises. That is Gay-Lussac’s law. Because equal numbers of molecules at the same temperature deliver the same average kick, equal volumes at the same T and P hold equal numbers. That is Avogadro’s law. One model explains all of them.

The numbers are startling. At room temperature, nitrogen molecules move at roughly 500 meters per second on average, faster than sound. Lighter molecules move faster: hydrogen at about 1,900 meters per second. Yet a bottle of perfume opened across a room takes minutes to smell, because each molecule collides billions of times per second and staggers rather than flies. The molecules do not all share one speed. Some crawl, some race, and the spread widens as the gas warms.

Real molecules are not points, and they do attract one another slightly. At ordinary pressures this hardly matters, and the ideal gas law works to within a percent or so. Crush a gas to hundreds of atmospheres, or cool it toward its boiling point, and the molecules’ own volume and their mutual attraction start to show. Pressure comes out lower than predicted, then the gas condenses. Johannes van der Waals added two correction terms in 1873 to handle exactly this, and his equation still guides the design of tanks that hold liquefied gases.

Now put every tool together on one problem. A driver-side airbag must fill a bag of about 60 liters with nitrogen. Suppose the gas in the bag ends up near 1 atmosphere and, for a first estimate, near room temperature, 298 K. How much sodium azide is needed? Start with the ideal gas law. n = PV divided by RT = 1 times 60 divided by 0.08206 times 298, about 2.5 moles of nitrogen.

Next, the mole ratio. The decomposition is 2 NaN3 gives 2 Na + 3 N2, so 2 moles of azide give 3 moles of nitrogen. To get 2.5 moles of N2 you need 2.5 times 2 divided by 3, about 1.6 moles of NaN3. Finally, molar mass. Sodium azide is 22.99 plus 3 times 14.01, about 65.0 grams per mole. So 1.6 moles is about 105 grams. Real modules use less, because the gas is very hot when the bag fills and hot gas takes more space per mole. The estimate lands in the right range, which is what a first calculation is for.

The design has to handle more than the ideal case. The bag must vent so the driver’s head meets a cushion that is already deflating, not a hard wall of gas. The temperature of the gas must stay safe. The solid must not decompose on its own in a car parked in the Arizona sun. And the leftover sodium must be turned into something harmless, which is why the potassium nitrate and silica are there. Each of those is another stoichiometry problem.

Since the early 2000s most manufacturers have moved to other propellants that produce less toxic residue and are safer to handle in factories and scrapyards. The chemistry changed; the calculation did not. Whatever the solid, the engineer measures the bag, converts liters to moles with PV = nRT, walks the mole ratio back to the reactant, and converts to grams. That chain of reasoning is the whole chapter in one problem.

Stoichiometry tells you how much. It says nothing about how fast. A log in a fireplace and a cloud of sawdust in a grain elevator hold the same chemistry, wood plus oxygen, yet one burns for an hour and the other can explode. Speed is a separate question, answered by reaction rate: how much reactant is used up, or product made, per second.

Molecules react when they collide with enough energy and the right orientation. Anything that makes collisions more frequent or more energetic speeds a reaction. Raise the temperature, and molecules move faster and hit harder. Raise the concentration, or the pressure of a gas, and collisions come more often. Grind a solid into powder, and far more of its surface is exposed. That last one is why the sodium azide in an airbag is a fine powder, why grain dust explodes, and why a sugar cube resists a flame that ignites powdered sugar instantly.

The energy a collision needs is the activation energy, a hill the reactants must climb before they can roll down to products. Sodium azide sits stable for years because room temperature almost never gives a collision that energy. The igniter supplies it, and once the first bit decomposes, the heat it releases carries the rest over the hill in a few milliseconds. A catalyst offers a lower hill and speeds a reaction without being used up. Your body runs on thousands of them, the enzymes.

In the lab, the rules of rate are the rules of safety. Keep reactive powders in small amounts and away from heat. Add a concentrated acid to water, never water to acid, because the heat released near the surface can flash the water to steam and spray acid. Use a fume hood for anything that gives off gas, since a mole of gas is 24 liters. Wear goggles every time. A chemist who can calculate how much gas a reaction makes, and how quickly, is a chemist who does not get surprised.

Stir a spoon of sugar into tea and it disappears. Stir in a spoon of sand and it sits on the bottom. Both are solids in water, yet one forms a solution, a uniform mixture of one substance spread through another, and the other does not. The dissolved substance is the solute; the substance doing the dissolving is the solvent. Water is the solvent in tea, in blood, in lakes and in nearly all of the chemistry of life.

Dissolving is a tug of war. Water molecules are polar, with a slightly negative oxygen end and slightly positive hydrogen ends. Table salt is a crystal of sodium ions and chloride ions. When water meets the crystal, the oxygen ends crowd around each sodium ion and the hydrogen ends around each chloride ion, and the pulls are strong enough to lift ions off the crystal one by one. Sugar molecules are not ions, but they carry polar groups that water grabs the same way. Oil has no such handles, so water leaves it alone. The rule of thumb is that like dissolves like: polar solvents dissolve polar and ionic solutes, and nonpolar solvents dissolve nonpolar ones.

Every solute has a limit. Keep adding salt to a glass of water and eventually the crystals stop disappearing. The solution is saturated. The maximum amount that dissolves in a given amount of solvent at a given temperature is the solubility. For most solids, solubility rises with temperature, which is why hot tea holds more sugar than iced tea and why a cooling solution can suddenly drop crystals. Gases behave the other way. A warm soda goes flat faster because carbon dioxide is less soluble in warm water than in cold.

Pressure matters for gases too. A soda is bottled under several atmospheres of carbon dioxide, which forces far more gas into the liquid than air pressure could. Open the cap, the pressure drops, and the excess escapes as bubbles. The same law explains why a diver who rises too fast gets the bends: nitrogen dissolved in the blood at depth comes out of solution as the pressure falls. And it explains why a warming lake or ocean holds less dissolved oxygen, a problem for fish long before the water feels hot.

A recipe that says add some salt is useless to a chemist. So is one that says add 5 grams, because 5 grams of sodium chloride and 5 grams of sugar contain very different numbers of particles. Chemists describe concentration as molarity: moles of solute per liter of solution, written with a capital M. A 1.0 M salt solution has 1.0 mole, 58.44 grams, of sodium chloride in each liter. A 0.10 M solution has one tenth as much.

Making a solution of known molarity takes care. To prepare 1.0 liter of 0.50 M NaCl, weigh out 0.50 mole, about 29.2 grams. Dissolve it in less than a liter of water, then add water until the total volume reaches exactly 1.0 liter. The order matters. Dissolving salt changes the volume slightly, so you cannot simply add 29.2 grams to a full liter and expect 1.0 liter of solution. Chemists use a flask with a single line etched at the exact volume for this job.

Often a lab keeps one concentrated stock solution and dilutes it as needed. Adding water changes the volume but not the number of moles, so the moles before and after are equal. Since moles equal molarity times volume, M1 times V1 equals M2 times V2. To make 500 milliliters of 0.10 M acid from a 1.0 M stock, solve for V1: 0.10 times 500 divided by 1.0 equals 50 milliliters. Measure 50 milliliters of stock, add water to 500 milliliters, done.

Molarity connects solutions to stoichiometry. If you know the molarity and the volume, you know the moles, and moles are what the balanced equation counts. A nurse preparing a saline drip, a water plant adding chlorine, and a chemist neutralizing an acid spill all reason the same way: how many moles do I need, and what volume of this solution holds them?

Every winter, Chicago spreads tens of thousands of tons of salt on its streets. The salt does not melt ice by being warm. It works because a solution freezes at a lower temperature than the pure solvent. Dissolved particles get in the way when water molecules try to lock into the orderly pattern of ice, so the water must be colder before ice can form. This is freezing point depression, and it is one of a family of colligative properties, effects that depend on the number of dissolved particles rather than their identity.

The size of the effect is easy to estimate. For water, each mole of dissolved particles in a kilogram of water lowers the freezing point by about 1.86 degrees Celsius. Sodium chloride splits into two ions, so one mole of salt gives two moles of particles and about 3.7 degrees of depression. Calcium chloride, CaCl2, splits into three ions and works harder per mole, which is why road crews use it in the coldest weather. Seawater, with about 35 grams of dissolved salts per kilogram, freezes at about minus 1.9 degrees Celsius.

The same crowding raises the boiling point. Pure water boils at 100 degrees Celsius at sea level; each mole of particles per kilogram raises that by about 0.51 degrees. A pot of salted pasta water boils only a fraction of a degree hotter, so the salt is for flavor, not speed. Antifreeze in a car radiator does both jobs at once: it keeps the coolant from freezing in January and from boiling in July.

Colligative properties gave nineteenth-century chemists a way to weigh molecules. Dissolve a known mass of an unknown substance, measure how far the freezing point drops, and you can compute how many moles you added and therefore the molar mass. When salt lowered the freezing point about twice as much as expected, it was a clue that salt breaks into two particles in water. Svante Arrhenius built his theory of ions on exactly such puzzles.

Lemon juice, vinegar, and the fluid in your stomach share a sharp taste, a sting on a cut and the power to fizz on marble. They are acids. Soap, baking soda solution and household ammonia feel slippery and taste bitter. They are bases. For centuries chemists sorted substances this way by their effects. The modern definition explains the effects. An acid is a substance that donates a hydrogen ion, H+, which is just a bare proton. A base is a substance that accepts one.

Put hydrogen chloride gas in water and every HCl molecule hands its proton to a water molecule, forming a hydronium ion, H3O+, and a chloride ion. Chemists usually write H3O+ simply as H+ for short. Because HCl gives up essentially all of its protons, it is a strong acid. Sulfuric and nitric acids, the acids in acid rain, are strong too. Acetic acid in vinegar is different. Only about one molecule in a hundred donates its proton at any moment; the rest hold on. It is a weak acid. Strong and weak describe how completely an acid ionizes, not how concentrated it is. A dilute strong acid and a concentrated weak acid can be equally dangerous.

Bases work in reverse. Sodium hydroxide, lye, splits completely into sodium ions and hydroxide ions, OH-, and hydroxide grabs any proton it meets. That makes it a strong base. Ammonia, NH3, is a weak base: it pulls a proton off water to form NH4+ and OH-, but only a small fraction does so. When an acid meets a base, the proton moves from one to the other. H+ and OH- combine to make water, and the leftover ions form a salt. Acid plus base gives salt plus water: neutralization.

Water itself does a little of both. In pure water, a tiny fraction of molecules pass protons among themselves, so there is always a trace of H+ and OH-, one of each for every 550 million water molecules at room temperature. Add acid and H+ rises while OH- falls. Add base and the reverse. Their product stays fixed. That balance is the reason a single scale, pH, can describe both acids and bases.

Hydrogen ion concentrations in everyday solutions range over a trillion-fold, from stomach acid to lye. Writing those numbers out is clumsy, so in 1909 the Danish chemist Soren Sorensen, working at a brewery laboratory, proposed a shortcut. Take the hydrogen ion concentration in moles per liter and find the power of ten. Flip the sign. That is pH. A solution with H+ at 0.01 M, which is 10 to the minus 2, has pH 2. Pure water, with H+ at 10 to the minus 7 M, has pH 7.

The scale runs backward from what people expect: lower pH means more acid. And because it is logarithmic, each step of one unit is a factor of ten. Rain at pH 4.6 holds ten times the hydrogen ions of rain at pH 5.6, and rain at pH 3.6 holds a hundred times as much. A drop of pH from 8.2 to 8.1 sounds tiny, but it is a 26 percent increase in hydrogen ions. Chemists say a solution is acidic below 7, basic above 7 and neutral at 7, at least at room temperature.

You can measure pH in two ways. An indicator is a dye that changes color at a particular pH. Litmus turns red in acid and blue in base; phenolphthalein is colorless in acid and pink in base; the juice of red cabbage runs through a whole rainbow. A pH meter uses a glass electrode whose voltage depends on the hydrogen ion concentration, and it reads to a hundredth of a unit. Field scientists carry both.

The scale is not limited to 0 through 14. Very concentrated strong acids can drop below 0 and very strong bases can rise above 14, though ordinary life stays inside that range. Your blood holds close to pH 7.4, your stomach near pH 2, and a swimming pool near 7.5. A change of a few tenths in your blood would be a medical emergency, which is why the next lesson matters.

Your blood receives acid all day long. Every cell dumps carbon dioxide into it, and carbon dioxide dissolved in water is an acid. Exercise floods it with lactic acid. Yet blood pH stays between about 7.35 and 7.45, and drifting outside that range by a few tenths is fatal. Something is absorbing the acid. That something is a buffer, and buffers are among the most important ideas in biology and environmental chemistry.

A buffer is a solution containing a weak acid together with its conjugate base, the particle left when the acid gives up its proton. Carbonic acid, H2CO3, and bicarbonate ion, HCO3-, form the main buffer in blood. Add acid, and bicarbonate grabs the extra protons, turning into carbonic acid. Add base, and carbonic acid hands over protons to neutralize it, turning into bicarbonate. Either way, the hydrogen ion concentration barely moves, because the added acid or base is converted into more of one buffer partner and less of the other.

A buffer has a capacity. It works only as long as both partners remain. Pour in enough acid to use up all the bicarbonate and the pH crashes. That is what happened to the northern lakes. Dissolved bicarbonate from the small amount of carbonate in their watersheds buffered the first years of acid rain, so the lakes seemed fine. When the bicarbonate ran out, the pH fell fast. Limestone-bedded lakes like Lake Michigan never ran out, because the rock itself keeps supplying carbonate.

Chemists design buffers for a chosen pH by picking a weak acid whose natural tendency to ionize lands near that value, then mixing it with roughly equal amounts of its conjugate base. Aquariums, swimming pools, shampoo, intravenous fluids and the enzyme reactions in every test tube in a biology lab all rely on a buffer chosen this way. The ocean is buffered by the same carbonate system as blood, and the last section of this chapter is about what happens when that buffer is pushed.

A bottle of vinegar claims to be 5 percent acetic acid. How would you check? You cannot see acid molecules, but you can count them by neutralizing them. Measure out a known volume of vinegar. Add a few drops of phenolphthalein. Then, from a long graduated tube called a burette, add sodium hydroxide solution of known molarity, drop by drop. Each hydroxide ion neutralizes one acetic acid molecule. The instant the last acid molecule is gone, the next drop of base makes the solution basic and the indicator flashes pink. This is titration.

The moment when the moles of base added exactly equal the moles of acid present is the equivalence point. The color change that signals it is the endpoint, and a good indicator is chosen so the two nearly coincide. Now the arithmetic is stoichiometry. Suppose 25.0 milliliters of vinegar needed 20.9 milliliters of 1.00 M NaOH. Moles of base equals 1.00 times 0.0209 liters, 0.0209 mole. Acetic acid and NaOH react one to one, so the vinegar held 0.0209 mole of acid in 25.0 milliliters, which is 0.836 M. Multiply by 60.05 grams per mole and you get about 50 grams of acetic acid per liter, close to 5 percent.

The shape of a titration matters as much as the endpoint. Plot pH against volume of base added and you get a curve that rises slowly, then leaps several units in a single drop near the equivalence point, then levels off. For a strong acid and strong base, the leap passes through pH 7. For a weak acid like acetic acid, the equivalence point sits above 7, because the conjugate base left behind is itself slightly basic. That is why phenolphthalein, which changes around pH 8 to 10, is the right choice for vinegar.

Titration is how water plants check alkalinity, how food chemists measure the acid in juice, and how environmental scientists measured the buffer capacity of lakes during the acid rain years. A lake sample titrated with acid tells you exactly how much more acid rain it can absorb before its pH falls. Those numbers, lake by lake, drew the map of which waters were in danger.

Crack a chemical hand warmer and it grows hot for hours. Squeeze an instant cold pack and it turns icy in seconds. Both are reactions, one that releases heat and one that absorbs it. A reaction that releases heat to its surroundings is exothermic; one that draws heat in is endothermic. Chemists measure the heat of a reaction at constant pressure as its enthalpy change, written delta H. Exothermic reactions have negative delta H, because the system loses energy; endothermic reactions have positive delta H.

To measure it, you let the reaction warm or cool a known amount of water and watch the thermometer. Water needs 4.18 joules to raise one gram by one degree Celsius; that number is its specific heat. Heat equals mass times specific heat times temperature change, q = m c delta T. If a reaction warms 100 grams of water by 5.0 degrees, it released 100 times 4.18 times 5.0, about 2,090 joules, or 2.09 kilojoules. A foam cup with a lid and a thermometer is a perfectly good calorimeter for a school lab.

Divide the heat by the moles that reacted and you have delta H per mole, a number that belongs to the reaction itself. Burning one mole of methane, 16 grams, releases about 890 kilojoules. Burning one mole of glucose, the sugar your cells use, releases about 2,800 kilojoules whether it burns in a flame or is broken down slowly in your body. Dissolving one mole of ammonium nitrate, the salt in many cold packs, absorbs about 26 kilojoules, which is why the pack turns cold.

The first careful calorimeter was built in 1783 by Antoine Lavoisier and Pierre-Simon Laplace. They put a guinea pig in a chamber surrounded by ice and weighed the meltwater. The animal’s body heat melted ice at about the rate that burning charcoal did for the same carbon dioxide produced. It was the first quantitative evidence that respiration is a slow form of combustion, and it was done with nothing more than ice, a balance and a good idea.

Some reactions cannot be measured directly. Carbon burning to carbon monoxide, C + half O2 gives CO, always makes some carbon dioxide too, so no calorimeter can isolate it. In 1840 the Swiss-Russian chemist Germain Hess found the way around. The heat of a reaction, he showed, is the same whether the reaction happens in one step or in several. Only the start and end matter. This is Hess's law, and it follows from the fact that enthalpy is a state function, a property that depends on the current state and not on the path taken to reach it.

Altitude is a state function. Climb a mountain by the steep trail or the winding road and your gain in height is the same. Distance walked is not a state function; the road is longer. Enthalpy is like altitude. So if you can measure two reactions that together add up to the one you want, add their heats. Carbon to carbon dioxide releases 393.5 kilojoules per mole. Carbon monoxide to carbon dioxide releases 283.0. Subtract, and carbon to carbon monoxide must release 393.5 minus 283.0, which is 110.5 kilojoules per mole. No one has ever measured it directly, and every chemist trusts the number.

The rules for adding equations are simple. Reverse an equation and flip the sign of delta H. Multiply an equation by a number and multiply delta H by the same number. Then add the equations, canceling anything that appears on both sides. Chemists have tabulated the enthalpy of forming each compound from its elements, the standard enthalpy of formation, so that the heat of any reaction can be computed as the formation enthalpies of products minus those of reactants.

Hess’s law is the reason a food label can state calories without burning your lunch. The energy in a sandwich is the sum of the energies of its carbohydrates, fats and proteins, each measured once in a calorimeter. It is also why engineers can predict the heat of a rocket fuel that has never been tested, and why the total energy of a hand warmer can be worked out from a table before it is built.

Not every reaction that releases heat happens, and some reactions that absorb heat happen anyway. Ice melts in a warm room even though melting absorbs heat. A cold pack works even though it makes itself colder. Heat alone cannot be the whole story. The missing piece is entropy, a measure of how spread out energy and matter are, roughly, how many ways the particles can be arranged. Gas has more entropy than liquid, liquid more than solid, and a mixture more than the pure substances separated.

Nature favors two things: lower energy and higher entropy. Sometimes they agree. Burning methane releases heat and turns one gas molecule plus two into one plus two, so both push forward. Sometimes they conflict. Melting ice raises energy but also raises entropy, because liquid water molecules can arrange themselves in far more ways than ice. Which one wins depends on temperature. Above 0 degrees Celsius entropy wins and ice melts. Below it, energy wins and water freezes.

In the 1870s the American physicist Josiah Willard Gibbs put the two together into a single number, the free energy change: delta G equals delta H minus T times delta S, where T is kelvin temperature and delta S is the entropy change. If delta G is negative, the reaction is spontaneous: it can go forward on its own. If delta G is positive, it cannot, though the reverse can. Multiplying entropy by temperature is what makes heat matter more in the cold and disorder matter more in the heat. For melting ice, delta H is about 6.0 kilojoules per mole and delta S about 22 joules per mole-kelvin, and delta G crosses zero at 6,000 divided by 22, about 273 K. That is 0 degrees Celsius, exactly where it should be.

Spontaneous does not mean fast. Diamond turning to graphite is spontaneous and takes longer than the age of Earth. Gasoline reacting with air is spontaneous and waits for a spark. Free energy says whether a reaction can happen and how far it will go. Rate, from the last chapter, says how long it takes. Living cells run reactions with positive delta G all the time, but only by coupling them to reactions with larger negative delta G, chiefly the breakdown of glucose, so that the total is still downhill.

The ocean is the largest buffered solution on Earth, and it is being titrated. Every year the burning of coal, oil and gas adds carbon dioxide to the atmosphere, and the ocean absorbs roughly a quarter to a third of it. When carbon dioxide dissolves in water, some of it becomes carbonic acid, H2CO3, which gives up protons to form bicarbonate and carbonate ions. More carbon dioxide means more protons. Measurements since the late 1980s, along with estimates from before industry, show the average pH of surface seawater has fallen from about 8.2 to about 8.1.

One tenth of a pH unit sounds trivial. Because the scale is logarithmic, it is roughly a 30 percent increase in hydrogen ion concentration, and it has happened faster than any change in the ocean’s chemistry that geologists can find for millions of years. The ocean is still basic, with a pH above 7, so scientists call the process ocean acidification: the water is becoming less basic, moving toward acid, not becoming acid.

The chemistry that matters most to living things is the carbonate ion. Corals, clams, oysters, sea snails and the tiny drifting organisms at the base of ocean food webs build shells of calcium carbonate, CaCO3, and they need dissolved carbonate ions to do it. Added protons grab carbonate ions to form bicarbonate, so acidification lowers the carbonate available. Laboratory studies and field measurements show shells forming more slowly, and in some cold waters, where carbon dioxide is most soluble, thin shells beginning to dissolve. Oyster hatcheries on the U.S. Pacific coast were among the first businesses to notice, when larvae began failing in the 2000s.

This is the same chemistry as the acid lakes, run at a different scale. Both involve a carbonate buffer being consumed by added acid. The lakes got sulfuric acid from smokestacks; the ocean is getting carbonic acid from the air. And both have the same kind of remedy: reduce the source. The ocean’s enormous buffer capacity means it will not become truly acidic, but the organisms that depend on carbonate live at the margin of that buffer, and the margin is where the change is felt.

The name acid rain is older than the problem people remember. In 1872 the Scottish chemist Robert Angus Smith, studying the air of industrial Manchester, England, wrote about rain made acid by the sulfur in coal smoke and coined the term. For a century the damage stayed local: blackened stone, corroded iron. Then power plants built tall stacks to send the smoke away from cities, and the acid traveled hundreds of miles instead. Coal from the Illinois Basin is high in sulfur, and burning it makes sulfur dioxide, SO2. In the air, SO2 reacts with oxygen and water to become sulfuric acid, H2SO4, a strong acid.

The chemical fix is neutralization on an industrial scale. A scrubber sprays a slurry of ground limestone, calcium carbonate, into the flue gas. Sulfur dioxide, an acidic gas, reacts with the carbonate to form calcium sulfite, and oxygen in the gas converts most of it to calcium sulfate, gypsum. The same reaction that makes lemon juice fizz on marble turns poison into wallboard. A large scrubber removes more than 90 percent of the sulfur dioxide from a plant’s smoke. Nitrogen oxides, the other source of acid rain, are cut with catalysts and with burners designed to run cooler.

The policy fix was a cap. The 1990 Clean Air Act Amendments set a national ceiling on sulfur dioxide from power plants and issued permits, one per ton, that could be bought and sold. A plant that cut emissions cheaply could sell its spare permits to one that could not. The cap fell over time. Companies chose among scrubbers, low-sulfur coal from Wyoming and Montana, and eventually natural gas. By 2020, sulfur dioxide from those plants was down more than 90 percent from 1990, at a cost far below the early forecasts. Chicago’s last two coal plants, Fisk and Crawford, closed in 2012.

Recovery has been slower than the cleanup. Rain over the Midwest and Northeast is far less acidic than it was, but many lakes lost so much buffering that their carbonate is still being rebuilt from the slow weathering of rock. Some have been limed by hand, tons of crushed limestone dumped from boats or helicopters, which is a titration in reverse. Fish have returned to many waters. The lesson for the ocean is the same as for the lakes: buffers can be spent, and the cheapest cure is to stop adding the acid.