LearnChem · Part III — Why do some things react and others just sit there?
Module 14 used the word temperature in every line. It never said what temperature is. This module opens a sealed box and watches. Inside are particles that never stop. What they are doing is the temperature. And it answers a question about washing.
“Why do wet clothes dry on a cold day, when water boils at 100 °C?”
Last time a referee with two whistles priced every change. One whistle blew for heat let go. The other blew for freedom gained, and temperature was its volume knob. The referee said which way is downhill. It never said what temperature is. Module 13 counted a gas's arrangements without watching a single particle move. So the word temperature has done a great deal of work on trust. This module opens the box and looks. What the particles are doing turns out to be the temperature.
Picture a sealed box. Inside it is nothing but marbles. They fly across the box. They bounce off the walls. They bounce off each other. Nothing slows them down, and nothing ever will. That is the picture this module keeps coming back to. Now ask a plain question about the box. The wall is being held out. Something inside is pushing on it. There is nothing in the box but the marbles. So what is the push? Say what it is before the bench counts it.
A sealed box of air sits on the table. Commit — what is holding the wall out?
Commit before you look. First answers are counted anonymously, never named.
One hit hands the wall a little momentum. Momentum is mass times speed. It is what a moving thing carries into a collision. The wall feels the sum of a huge number of hits. Pressure is the momentum handed over, divided by the wall's area, divided by the time. Nothing else is in that sentence. No spring. No leaning. That is a thing you can measure by counting. Bench 1 counts it.
Move one slider at a time, and let the meter settle between moves. Three shapes appear on the graph. Nobody drew them. Each one was measured by counting hits.
| you change | the measured pressure | and |
|---|---|---|
| more particles, same box, same temperature | rises in step, a straight line | twice the particles, twice the hits |
| smaller box, same particles | rises as one over the volume | P × V stays put |
| hotter, same box | rises in step with temperature | the mean speed rises only as the square root |
The third row hides a gap. Double the temperature and the pressure doubles. The mean speed does not double. It rises only as the square root of the temperature. So where does the rest of the factor come from? A faster particle hits harder, because it carries more momentum. A faster particle also hits more often, because it crosses the box sooner. Each of those rises as the square root. Multiply the two together and you have the whole factor. Now load the room-air preset. It puts 200 nitrogen molecules in a box 20 nm on a side at 300 K. The meter settles near 1.04 bar. That is ordinary room air.
Quick check — a sealed box of gas reads 1 bar. Suppose you could halve the box side, from 20 nm to 10 nm, and change nothing else. What does the pressure become?
Chemists write the three shapes as one line.
PV = nRT
P is the pressure and V is the volume. n is the amount of gas in moles. Module 03 taught you to count moles by weighing. T is the temperature in kelvin. R is the gas constant, 8.314 J/(K mol). Module 13 used the same R when it counted arrangements. This line is the ideal gas law. An ideal gas is one whose particles take up no room and never pull on each other. Section 4 asks whether any real gas is like that.
Here is something the bench cannot claim. The bench's panel shows P × V divided by N × T. N is the number of particles in the box. The ratio comes back as Boltzmann's constant, to within a fraction of a per cent at the default settings. Boltzmann's constant is the gas constant for one molecule instead of one mole. That is not a discovery. We set the box's temperature from the particles' energy, using that same constant. So it had to come back out. What the bench genuinely shows is the three shapes. Nobody put those in. A real experiment works the other way round. It measures pressure, volume, amount and temperature with instruments that never mention molecules. The constant that falls out of those measurements is R.
Now ask how empty the box is. One mole of any gas fills 22.711 litres at 0 °C and 1 bar. Divide that by Avogadro's number, from module 03. Each molecule has a cube about 3.35 nm on a side to itself. Its own size is a small fraction of one per cent of that cube. A gas is mostly empty space. That is the first half of the sentence you keep, and now it is earned. The second half was earned at the gate above. Pressure is just particles hitting the wall.
Module 13 met this same gas and counted its arrangements. It said a gas spreads into an empty space for one reason. There are far more ways to be spread than to be bunched. This module says what the particles were doing while that happened. They were flying at hundreds of metres a second. They were bouncing. They had nowhere else to go until the tap opened. Module 13 also measured something odd. The gas expanded into a vacuum and the thermometer did not move. Here is the reason in the marble picture. A bounce off a wall changes a direction. It does not change a speed. A bounce off another particle can hand speed from one to the other. It never changes the total. So the gas that has spread is moving exactly as it was. These are two views of one world. This module does not replace module 13.
A popcorn kernel is a sealed box with water in it. Heat it and the marbles inside hit harder and hit more often. The pressure climbs. In 2015 Virot and Ponomarenko measured bursting kernels. Their paper is in the Journal of the Royal Society Interface. They found the temperature at which a kernel bursts. It is 180 °C. One number about the inside of the kernel is weaker. The pressure at bursting is often quoted as about 10 bar. That figure comes from one science-press report quoting the lead author. We could not reach the paper's own text to confirm it. This course marks such a number as flagged. A flagged number has a weak source, or a source we could not reach, and it is treated as an estimate. Section 3 shows where a number near 10 bar comes from.
Virot & Ponomarenko, J. R. Soc. Interface 12(104), 2015, DOI 10.1098/rsif.2014.1247. The 180 °C figure is corroborated by two independent science-press reports. The 10 bar figure is carried by one of them only.
— Pressure is not a substance in the box. It is how hard the particles hit, and how often, spread over the wall's area.
Bench 1 reported one number for speed. It called it the mean speed. Now look at the marbles one at a time. The box is at 300 K. Every marble in it is at 300 K, because temperature belongs to the whole box. Does that mean every marble has the same speed? Bench 2 will measure it. Commit first.
Every molecule in this box is at 300 K. Commit — which is true?
Commit before you look. First answers are counted anonymously, never named.
There is a spread, and it is permanent. A mean is an average, and an average hides whatever it averages. Collisions do not smooth the spread away. Collisions make it. The bench below shows that in the plainest way there is. It starts with no spread at all.
Watch the first seconds. Every particle set off at one speed, so the histogram was a single bar. A histogram is a chart that counts how many fall into each range of speed. After enough collisions the bar collapses into a hump with a long right-hand tail. Nobody drew the hump. Press the restart button and watch. The bench runs the first couple of seconds in slow motion, and those seconds are the whole point. The single spike is there for a moment. Then the hump grows out of it. Now switch the curve on. The curve is called the Maxwell–Boltzmann distribution. It is the shape the spread settles into, for any gas at any temperature.
One number on the panel could not have come out any other way. That is the rms speed. Rms is short for root mean square. You square every speed, take the mean of the squares, then take the square root. Squaring is what ties it to energy, because energy of motion goes as speed squared. Now, the collisions in the box are elastic. An elastic collision is one in which no energy of motion is lost. So the total energy is fixed from the first frame. The rms speed is set by the total energy alone. It was settled the moment you chose the temperature. It is not a prediction. What the collisions decide is the shape. The mean falls to 0.9213 of the rms speed, and the peak lands within a bar or so of 0.8165 of it. Nobody put those two numbers in. That is the measured result.
The three speeds have names and formulas. Each one is a different way of picking one number out of the hump. M is the molar mass in kilograms per mole, from module 03. π is the circle constant from your maths class.
| the speed | its formula | what it picks out |
|---|---|---|
| the most probable speed, vp | √(2RT ⁄ M) | the top of the hump, where the most molecules are |
| the mean speed, v̄ | √(8RT ⁄ πM) | the plain average, which the tail drags to the right of the peak |
| the root-mean-square speed, vrms | √(3RT ⁄ M) | the one fixed by the energy, and the largest of the three |
For nitrogen at 298 K they are 421, 475 and 515 m/s. Those are the textbook figures. On the bench the rms speed will read 515 at 298 K, because that is what you set. The other two are measured, and they should land close to the textbook figures once the frames have settled.
Quick check — you switch the bench from nitrogen to xenon at the same temperature. Xenon's molar mass is 131.293 g/mol and nitrogen's is 28.014. Which reading stays put, and what happens to the rms speed?
Now look at the right-hand end of the hump. It falls steeply. Far out, it lies nearly flat on the axis. Leave the cut-off slider at 1000 m/s. The panel reports the share of molecules moving faster than that. It is about one in a hundred. Chemists call that end the tail. The next question is about the tail. It is a question about the whole shape, not about any one molecule. Warm the box a little. The average barely moves. What does the tail do?
Raise the temperature from 300 K to 310 K. The mean speed rises by about 1.7 per cent. Commit — what happens to the share of molecules moving faster than 1500 m/s?
Commit before you look. First answers are counted anonymously, never named.
At 300 K the share of nitrogen above 1500 m/s is about fourteen in a million. At 310 K it is 1.48 times bigger. A move of 1.7 per cent in the average is a move of 48 per cent in the tail. The reason is the shape. Out in the tail the curve falls steeply. Shift it a little to the right and the part beyond the cut-off grows by a lot. Check it on the bench at 1000 m/s. At 300 K the share reads about 1.05 in a hundred. At 310 K it reads about 1.25. That is a fifth more, against 1.7 per cent in the mean. A fifth is not a half, and that is not a contradiction. The further out the cut-off sits, the bigger the jump for the same warming. That is why 1000 m/s gives a fifth and 1500 m/s gives a half.
Now slide the cut-off to 1500 m/s. The panel says none seen yet, and it keeps saying so for a while. The share there is fourteen in a million. With 260 particles the bench catches one about once in every three hundred frames. Wait, and a number does appear. It takes about half a minute to come near the right one. Even then it wanders by half its own size as you watch. Push the cut-off further out and the wait runs past anything you would sit through. Counting does not fail out here. It gets slow and noisy exactly where the interesting molecules are. That is where a formula earns its keep. It reaches where counting stops being worth it.
Think of a class and its exam marks. Every student has a mark, the way every molecule has a speed. The class average stands for the temperature. The spread is real, and nobody is exactly average. Now draw a cut-off line. Raise the class average a little. The group above a low cut-off grows by a few in a hundred. The group above a high cut-off can grow by half again. Fewer students cross the high line. But there were far fewer above it to start with, so that small group is the one that changes most. That is what warming does to the fastest few. The next module runs on this picture. The energy a reaction needs is the cut-off mark.
Here the image breaks, in three places. A student's mark is fixed once the exam is over. A molecule's speed changes at every collision, about eight thousand million times a second. It is fast one moment and slow the next. So the group above the line is not the same molecules from one instant to the next. The fast group is refilled all the time. Only the shape of the spread holds still. Second, marks come from ability. Speeds come from nothing but chance. No molecule is better at being fast than any other. Third, the class average is not quite the temperature. Temperature follows the squares of the speeds, which is the rms speed from earlier in this section.
Nitrogen at room temperature averages 475 m/s. That is faster than a passenger jet. Yet a smell takes minutes to cross a room. The reason is the crowd. A molecule in air at 1 atm and 25 °C travels only about 60 nanometres before it hits another. Chemists call that distance the mean free path. Divide the mean speed, 475 m/s, by that 60 nanometres. You get about eight thousand million collisions every second. Each collision sends the molecule off in a new direction. The trip across the room is a stagger, not a sprint. The 60 nanometres is a textbook figure. Other sources, working from assumed molecule sizes, give anything from about half of it to three times it. So this page says about 60 nanometres, and no more.
Steel is the other half of the question. Do not say a steel atom never leaves. A steel atom's neighbours hold it far harder than a perfume molecule's do. So the share of steel atoms fast enough to break away is not zero. It is so small that nothing reaches a nose in a lifetime. That is the honest sentence. It is also the better one, because it is the same tail doing the same job.
Quick check — pump half of the air out of the room, keeping the temperature the same. What happens to a nitrogen molecule's mean free path, and to how often it collides?
LPG has almost no smell of its own. Indian Oil's own customer FAQ says so. A smell is added on purpose, so that escaping gas can be noticed. If you can smell it, gas is escaping. The added substance is ethyl mercaptan. Numaligarh Refinery's LPG specification names that substance. The specification sets the amount by IS 4576. The amount must make a leak detectable at one fifth of the lower flammability limit. That limit is the smallest share of gas in air that can burn. LPG is heavier than air. A leak pools low. It does not rise.
Indian Oil Corporation, FAQ — https://iocl.com/pages/faq. Numaligarh Refinery Ltd, LPG Specifications — https://www.nrl.co.in/upload/nrlLPG-Specifications.pdf.
— One temperature does not mean one speed. It means one average, and a tail that matters more than the average.
Start with the hook. Wet clothes hang on a line at 20 °C. By evening they are dry. Water boils at 100 °C, and the clothes never came near it. So something left the cloth at 20 °C, and it left in quantity. A whole bucket of water went into the air. Nobody heated it. The marbles in a box have been the answer to everything so far. Here the box is the open air, and the wet cloth is one wall of it. Say what left, and how.
Wet clothes on a line at 20 °C dry in a few hours. Water boils at 100 °C. Commit — what is leaving the cloth?
Commit before you look. First answers are counted anonymously, never named.
Section 2's tail is the whole answer. A water molecule at the surface is pulled by its neighbours. To break away it needs enough energy to beat that pull. The tail always reaches that far. So at every temperature some molecules leave. Raise the temperature and the tail grows. That is why warm clothes dry faster. It is not why they dry at all.
Molecules also come back. A vapour molecule that hits the surface can be caught by it. Leaving and returning reach a balance. The pressure the vapour settles at is the vapour pressure of the liquid at that temperature.
NOAA's weather glossary defines it the same way. At a given temperature, the saturation vapour pressure is the pressure of the vapour in balance with a flat surface of the pure liquid. Air that holds that much vapour is called saturated. Saturated air holds all the vapour it can at that temperature. Here is water's vapour pressure, from 0 °C to 100 °C.
| temperature | vapour pressure of water (kPa) |
|---|---|
| 0 °C | 0.611 |
| 10 °C | 1.227 |
| 20 °C | 2.337 |
| 25 °C | 3.167 |
| 30 °C | 4.244 |
| 40 °C | 7.378 |
| 60 °C | 19.928 |
| 80 °C | 47.371 |
| 100 °C | 101.328 |
Every value is computed here from the Antoine coefficients for water in the NIST Chemistry WebBook (webbook.nist.gov), which are fitted to measured vapour pressures. NOAA/NWS glossary, forecast.weather.gov/glossary.php, defines saturation vapour pressure verbatim as “the vapor pressure of a system, at a given temperature, wherein the vapor of a substance is in equilibrium with a plane surface of that substance's pure liquid or solid phase.”
Two units are on this page, so fix them now. One bar is 100 kilopascals. One atmosphere is 101.325 kilopascals.
Read the 20 °C row. Water's vapour pressure there is 2.337 kPa. That is 2.31 per cent of one atmosphere. The air around the clothes holds less water vapour than that, on any day that is not fog. So more molecules leave the cloth than return to it. Water keeps leaving until the cloth is dry. That is the hook's answer.
The hook said a cold day. Read the 10 °C row. Water's vapour pressure there is 1.227 kPa, about half the 20 °C value. So the push out of the cloth is about half as strong. The clothes still dry. They take longer. There is one case where they do not dry at all. Cold air is often close to saturated, and when it is saturated the push shrinks to nothing. On that day the clothes stay wet, and section 2's tail is not the reason. The air is.
This is the paragraph the module exists for. Evaporation happens at any temperature, because the tail always reaches escape. Boiling happens at one temperature. That is the temperature where the vapour pressure catches up with the whole pressure pushing down from above. Only then can a bubble of vapour hold itself open inside the liquid. Below that temperature, vapour leaves only at the surface. Read the 100 °C row. The fit returns 101.328 kPa, and one atmosphere is 101.325 kPa. That is how you know the fit is sound. It was built from vapour pressures. It hands back the boiling point on its own.
That makes the table a prediction machine. The table runs temperature in, pressure out. Turn it round, pressure in, boiling point out, and it answers for pressures this module never showed. The check is an Indian pressure cooker. The Indian standard IS 2347:2006 sets the nominal cooking pressure at up to 1.0 kgf/cm² gauge. Gauge means on top of the atmosphere. That is 98.07 kPa on top, so 199.39 kPa in all. Run the curve backwards at 199.39 kPa and it says water boils at 120.1 °C. HyperPhysics, which we did not use to build the curve, quotes near 121 °C for the same cooker. Here is the caution. NIST states the range its fit was made for, and 120 °C is about 20 K beyond it. So this is an extrapolation. An extrapolation is a prediction pushed outside the data it was built from. Landing within a degree of an independent figure is a check. It is not a proof.
Run it turned round at a lower pressure too. At 70.12 kPa water boils at 90.0 °C. That pressure is roughly what the standard atmosphere gives at 3,000 m. The standard atmosphere is an agreed table of how air pressure falls with height. Two cautions go with that number. The 3,000 m figure comes from a secondary compiler's table, not from the primary standard. And no measured pressure for any particular Indian hill station was reachable, so this page names none.
Quick check — a second website also quotes 120.1 °C for the cooker. It turns out to have computed that from the same NIST coefficients. Is that a second check?
The slope of the vapour-pressure curve carries a price. It is the energy it costs to turn liquid water into vapour. Module 14 called it the heat of boiling. Between 20 °C and 40 °C the slope of NIST's fit gives 43.88 kJ per mole. Module 12 used 44.00 kJ per mole for the same step, from CODATA. That figure is the gap between CODATA's formation enthalpies for liquid and gaseous water. A formation enthalpy is the heat taken in when one mole of a substance is made from its elements. The gap is one calorimetric measurement of the heat of vaporisation, restated as a difference. Before claiming the two agree, we checked that they are different experiments. They are. The fit was made from measured vapour pressures, and a slope is not a heat. The CODATA figure comes from calorimetry. Calorimetry measures heat by watching a temperature change. So the agreement is evidence, and not an input coming back out. One more thing has to be said. The slope depends on which two temperatures you take it between. Between 20 °C and 30 °C it gives 44.09, and between 0 °C and 25 °C it gives 44.57. The 43.88 above is the interval that agrees least well, which is why it is the one shown.
The awkward part comes next. The number is not constant. Between 80 °C and 100 °C the same slope gives 41.65 kJ per mole. A steam table gives 40.66 kJ per mole at 100 °C. That table is a flagged source, because it does not say what data it was built from. But the direction is plain. It costs less to boil water at 100 °C than to evaporate it at 20 °C. A hotter liquid has already paid part of the price. That is also why module 14's boiling table used a smaller figure for water than module 12 did.
The mirror is the same table, read the other way. The mirror is colder than the bathroom air. Air touching the glass is cooled. NOAA defines the dew point as the temperature to which air must be cooled to reach saturation. The air at the glass drops below its dew point. It now holds more vapour than the saturation pressure allows. Water condenses onto the glass. That is the fog. An hour later the mirror is clear, and nobody wiped it. Commit to why.
The mirror fogs during a shower. An hour later it is clear, and nobody wiped it. Commit — why did it clear?
Commit before you look. First answers are counted anonymously, never named.
The film is a liquid surface, and section 3's rule applies to it. While the shower runs, the room's air is at or near saturation. As many molecules return to the film as leave it. Once the steam has gone, the air is below saturation again. More molecules leave the film than return. The film evaporates, and the mirror clears. That mechanism is our own inference from NOAA's definitions. NOAA does not describe mirrors.
Bread rises by section 1's rule. Yeast makes a gas inside a stretchy dough. The gas hits the walls of every bubble. The bubbles grow, and the dough grows with them. Idli batter rises the same way, with different help. A 1965 paper in Applied Microbiology showed that a bacterium, Leuconostoc mesenteroides, makes the batter rise. The text we could reach does not spell out which gas it makes. So this page says only that the bacteria make a gas, and puts no formula on it.
Popcorn comes last. At 180 °C, the temperature the paper measured, a steam table puts water's own vapour pressure at about 10 bar. That table is the flagged source named above. So water's own curve puts a number near 10 bar inside a kernel at bursting. That is consistent with the flagged figure. It does not confirm it, because the steam table is itself flagged.
Mukherjee, Albury, Pederson, Van Veen & Steinkraus, Applied Microbiology 13(2):227–231, 1965, DOI 10.1128/am.13.2.227-231.1965. Pressure cooker: IS 2347:2006, Bureau of Indian Standards, clause 3.1. Independent boiling-point figure: HyperPhysics, Georgia State University. Altitude pressures: EngineeringToolbox International Standard Atmosphere, a secondary compiler. Steam table: MSU ME 201 (Somerton), flagged.
— Evaporation happens at every temperature. Boiling happens at the one temperature where the vapour pressure catches the air pushing down.
Bench 1 left a loose end. Load the room-air preset and let the meter settle. P × V divided by N × T reads 1.002 times Boltzmann's constant. Now crowd the box. Push the particles up to 400 and shrink the side to 14 nm. Let it settle again. The reading creeps up, to about 1.012 times Boltzmann's. Thin the box right out and it falls back to 1.000. The excess is small. It is also steady, and it grows the more you crowd. Something in the box is not behaving as the ideal gas law says. Commit to a reason before the reveal.
Bench 1's measured constant creeps above Boltzmann's constant when you crowd the box. Commit — what is the most likely reason?
Commit before you look. First answers are counted anonymously, never named.
There are two corrections to the ideal gas, and they pull opposite ways. Real molecules take up room. The space left to fly in is smaller than the box. Hits come sooner, so the pressure is pushed up. Real molecules also attract one another. A molecule on its way to the wall is tugged back by the ones behind it. It lands a little softer, and the pressure is pulled down. The simulation has the first correction and not the second. Its particles have size. They have no attraction at all. So in the bench, only the upward push can show. We checked that by shrinking the simulation's particles, and the excess vanished with them.
Chemists measure the gap with one number. It is called the compressibility factor, written Z. Z = PV ⁄ nRT. For an ideal gas Z is exactly 1. Above 1, the gas is harder to squash than the ideal law says. Below 1, it is easier. The two corrections have a famous equation of their own, the van der Waals equation. It carries two numbers for each gas. The number a stands for the attraction. The number b stands for the room the molecules themselves take up. Both numbers are tabulated, not measured here. The values this bench uses come from a secondary compilation. No primary source for them could be reached, so they are weak. Bench 3 draws Z from those two numbers and nothing else.
Set the bench to 273 K and read the curves left to right. Below about 100 bar most gases dip below Z = 1. Attraction is winning there. Push the pressure higher and every curve climbs above 1. Now the molecules' own volume is winning. At 273 K nitrogen bottoms out at Z = 0.938 near 110 bar. Methane goes lower, to 0.716 near 132 bar. Now click argon. Argon dips deeper than nitrogen, to 0.866 near 164 bar. Yet its a is a little smaller than nitrogen's. So a alone cannot be the whole story of the dip. The depth is set by a against b, and the next paragraph names the temperature that carries it.
One gas breaks the pattern, and it is helium. Helium never dips at all at room temperature. Its Z is above 1 from the first bar. The reason is a temperature that belongs to each gas. Chemists call it the Boyle temperature. Above it, the attraction never gets a turn, and only the volume correction shows. It is worked out from the same a and b. For helium it is 17.5 K. At 273 K helium is more than fifteen times above its own crossover. Nitrogen's Boyle temperature is 425.8 K. At 273 K nitrogen is below its own, so it dips. Argon's is 509.1 K, because its b is small. So at 273 K argon sits further below its own crossover than nitrogen does. That is why it dips deeper. Now a caution, and it is the one section 3's check taught. Put the pressure marker at 100 bar. At 273 K nitrogen reads 0.939. Warm the bench past 426 K and the same marker reads 1.013. That is not a test of nitrogen. The crossover and the curve come from the same two numbers, so the bench had to agree with itself. It shows that the bench is doing its arithmetic right, and nothing more. A real test needs a measured Z for nitrogen, and the panel says we could not reach one.
The bench also has a line it will not cross. Every gas has a critical temperature. Above it, no pressure can turn the gas into a liquid. Below it, there is a range of pressures where the van der Waals equation has three answers for the volume. The substance would be a liquid, not a gas. The bench leaves those curves out. It says which gas it has left out, and at what temperature, rather than drawing something false. Carbon dioxide's critical temperature is 304.18 K. Drop the bench below that and carbon dioxide disappears. That is a deliberate refusal. It is not a missing feature.
Quick check — set the bench to 500 K and put the pressure marker at 100 bar. Nitrogen reads 1.023. Methane still reads 0.983. What does that tell you about methane's own crossover temperature?
— The ideal gas law leaves out two real things, and a two-number model shows roughly where each one starts to matter.
These check themselves. “New numbers” deals a fresh set each time, so there is nothing to memorise. Each stem says how exact to be. Use R = 8.314 J/(K mol), or 0.083145 L bar/(K mol) when the answer is in litres and bar.
Two of the exercises use the curve that made section 3's table. It is NIST's Antoine fit for water. For temperatures from 344 K to 373 K it reads
log10(P ⁄ bar) = 5.08354 − 1663.125 ⁄ (T ⁄ K − 45.622)
1. Module 03's mole, then the gas law: Turn the mass into moles first, then find the pressure. (To ±0.02 bar.)
2. The rms speed: Put the molar mass into kilograms per mole before you take the root. (To ±2 m/s.)
3. Module 13's expansion, revisited: The gas spreads into the larger volume at a fixed temperature. Find ΔS, the entropy change. Section 1 told you why the temperature does not move. (To ±0.05 J/K.)
4. The vapour pressure: Put the temperature into kelvin, then into the fit printed above, and give the vapour pressure in kilopascals. (To ±0.05 kPa.)
Section 3 turned the prediction machine round. It took a pressure and gave a boiling point. Now run it the table's own way, temperature in, pressure out, at a temperature the table never listed. This is what a thermometer in a kettle can tell you about the air above a mountain camp. Use the fit printed above Tier 1.
Water in an open kettle at a mountain camp boils at 81.3 °C. There are three parts.
Part 1. Put the boiling point into kelvin. Then say whether it lies inside the range NIST states for these coefficients.
Part 2. Find the air pressure at the camp, in kilopascals.
Part 3. Compare your pressure with one atmosphere, 101.325 kPa. Then say which is the stronger claim, this one or section 3's pressure-cooker prediction, and why.
Part 1. Add 273.15 to the Celsius reading, so 81.3 plus 273.15 gives 354.45 K. That temperature sits inside the 344 K to 373 K range NIST states for these coefficients. So this is an interpolation, not an extrapolation. An interpolation reads the fit between its own data points, where it was made.
Part 2. Start with the bracket, and subtract 45.622 from 354.45 to get 308.828. Next divide 1663.125 by 308.828, which gives 5.3853. Now take that away from the constant, so 5.08354 minus 5.3853 gives −0.3018. The pressure is 10 raised to the power −0.3018, which comes to 0.499 bar. That figure is water's vapour pressure at 81.3 °C, and it is not yet the air pressure. Water boils when its vapour pressure matches the pressure pushing down on it. So the air at the camp must be pushing down with that same 0.499 bar. One bar is 100 kPa, so the air pressure at the camp is about 49.9 kPa. Call it 50 kPa.
Part 3. One atmosphere is 101.325 kPa, so the camp's air pushes down with about half of that. The water needs only half the vapour pressure to boil, and the table says it reaches that at about 81 °C. This claim is the stronger of the two, and for one reason only. Its temperature lies inside the range the fit was made for. The pressure cooker's 120.1 °C lay about 20 K outside that range, and needed an independent figure to check it. Here no check is needed, because the fit is being read where its own data are.
Module 13 said a gas spreads into an empty space because there are more ways to be spread than to be bunched. This module says the particles were flying at hundreds of metres a second and bouncing off the walls. A friend says these are two different theories, and one of them must be wrong. Answer the friend. Say what each view can do that the other cannot. Then say what both views predict for the thermometer when the tap opens, and why they agree.
They are not two theories. They are two views of one world, and each can do something the other cannot. Module 13's counting says which way things go. It gives the entropy change, and it does that without ever watching a particle. It cannot say how fast the gas spreads, and it cannot say what pressure the wall feels. This module's marbles give the pressure, and they give the speed of spreading. They cannot, on their own, say why the spread never runs backwards. That answer needs the counting.
Both views say the thermometer does not move. Module 13 said it because the gas pushed against nothing, so no work was done and no energy left. This module says it because a bounce changes a direction and not a speed. Those are the same statement, made in two languages. The energy the counting keeps track of is the energy the marbles carry. That is why the two views agree, and it is why neither replaces the other.
You now have the box of marbles nobody can slow down. Wet clothes dry on a cold day for one reason. The fastest few water molecules at the surface always have enough energy to leave. Boiling needs the vapour pressure to catch the air pushing down. Leaving does not. The mirror fogs and clears by the same table, read in the other direction. Popcorn bursts because water's own vapour pressure climbs past what the kernel can hold. Perfume reaches you because its molecules leave the bottle and stagger across the room through a crowd. Steel never reaches you because its atoms are held far harder. The share fast enough to leave is next to nothing. Bread rises because a gas hits the walls of every bubble. Five of the hook's questions are settled. One thread is left hanging. Section 2 drew a cut-off across the tail of the speeds. The next module gives that cut-off a name. It is the activation energy, and it is why reactions need a push.
A gas is mostly empty space. Pressure is just particles hitting the wall.