Chm 1316 HFC II Gas Class Notes

The added bonus to Charles' Law is that a plot of V vs. T for an ideal gas (or a real gas a really low pressures) shows its origin at T = - 273.16°C which we conveniently relabel as 0.00 K. And the very linearity of that plot means that an ideal gas makes a first principles thermometer!

The bonus to Avogadro's Law is that we get to measure number of molecules (in moles) by measuring volume! And since we can weigh known volumes of gas, we can weigh molecules (well, OK, only gaseous molecules, but it's a start).

So if you insist on SI units, P is in Pascals (N/m² = Nm/m³ = J/m³ = kg m^-1 s^-2) and V is in m³. That's instructive since it makes PV in J! We'll soon see that PV is work; it's nice to know it has the units for it. So a little algebra shows R's units to be J mol^-1 K^-1. And experimental measurements with ideal gases give its value as R = 8.3145 J mol^-1 K^-1.

If you're sane, on the other hand, you'll probably be measuring P in atmospheres and V in Liters which renders R = 0.082058 atm L mol^-1 K^-1. And, using that value, you can show that at Standard Temperature and Pressure (0°C and 1 atm), the molar volume of an (ideal) gas is 22.414 L or 22,414 cm³. For water, that's some 1244 times its volume as a liquid (18.016 cm³ at the slightly higher 25°C)!

Contemporary practice is to standardize pressure as the bar defined as 100 kPa. It's sort of convenient since 1 atm = 101.325 kPa. So 1 bar = 0.98692 atm, and you can convert at leisure.
 

Note that since pressure is a force per unit area measurement, the cross-section and even the shape of the mercury tube is irrelevant! Bigger ones certainly weigh more, but the force is spread out over the larger area. Bulgy ones weigh more too, but the extra force is distributed over the non-vertical glass (which might break if it's not really sturdy!); so their pressures are still only a function of the height of the Hg.
 

We'd be in lots worse trouble if we tried to substitute water (density 1.0 g/cm³) for Hg (13.6 g/cm³) because (a) we'd have a barometer 0.760×13.6=10.3 m high (we'd need a 3-story building) and (b) the vapor pressure of water at room temperature is about 24 torr, giving a water barometer over 3% error! Nevertheless, trees have to give it a go as they siphon water up their trunks to survive. But wait! Some trees exceed 10.3 m in height (Pacific Coast Redwoods have reached 112 m!) and still water gets pulled up to their leaves. So since straws can't work over (0.97)×10.3 m, trees have to be more clever. We'll find out how later.

1. Gas molecules are so few and far between that it's safe to imagine that their important interactions are with the vessel walls.
2. Collisions with the walls impart momentum changes which are the forces of pressure. [Newton's Law: F = d(mv)/dt]
3. Higher molecular velocities increase momenta and hence the force of pressure but fixing T fixes a unique velocity distribution.
4. The root mean squared velocity, v_RMS, is the key since ½mv²=3(½)kT.

Actually, it's a consequence of statistical mechanics that you won't see until Physical Chemistry, but it is consistent with the instinctive notion that higher temperatures ought to correlate with higher energies. And, in particular, for a gas, the interesting energies are the translational ones (which permit the molecules to run into the walls) as opposed, say, to the rotational ones, although the latter also scale with T.

If you want to see how a Newtonian (classical mechanical) analysis gets from ½mv²=3(½)kT to PV=nRT, jump with this link (using your BACK key to return). It's nice to know that one part of the connection is that R=kN_Av.

But we fixate on a different effect of this effusion of gas out of a small hole. Suppose two different gases were inside. Since they must share the same T (another piece of intuition...if they didn't initially, the colder one would warm up, the hotter cool down, until they had the same T), that equipartition theorem insists they share the same average kinetic energy. So if one is 4x the molecular weight of the other, it must be going only ½ as fast on average so that ½mv² is the same for both!

So we can use relative leak rates (what a weird concept) to compare gases of unknown molecular weight to those of known molecular weights in effusion experiments. But gaseous diffusion (spreading) depends on velocities in the same way; heavier ones diffuse slower than lighter ones. And that's not even taking into consideration gases so heavy that they "settle out."

Why is it more dense?

Density r = M/V and M = n(MW) so r = (MW)n/V = (MW)P/RT by the ideal gas law, and gas density is directly proportional to MW, molecular weight! So since equal volumes (of different gases) contain the same number of moles (for fixed P,T), the weights of those volumes are in direct proportion to their molecular weights. And we have another method for "weighing" molecules.

You can witness them first-hand locally at the Plano Annual Balloon Festival in Bob Woodruff Park.
 

The sun warms the equator more than it does the poles. That's because cos(0°) = 1 at the equator indicating the sun's rays are fully perpendicular to the ground, but cos(90°) = 0 at the poles with the sun's rays only tangential. However, Nature abhors a non-uniformity like that, and She does whatever it takes to transport heat from the equator to the poles. Most of it goes oceanic; that is, warm waters circulate up eastern coasts (in our hemisphere anyway), deposit their heat in northern latitudes, and circulate back down again on western coasts. (Means you want to swim on an eastern seaboard.)

Since it was damp (near 100% humidity in the tropics), the temperature soon falls enough to condense the water vapor into clouds which coalesce into raindrops and give us spectacular tropical thunderstorms. But for our purposes, the important feature is what the condensation does for that rising parcel: it took energy from the sun to evaporate that water in the first place, and the water vapor gives that energy back when it condenses! So while the rain in forming, the parcel can't cool off, and it's density difference becomes impressive. That means it rises faster, and the storm cloud grows higher.

Oops, the margin gave it away. We have to sneak a little atmospheric chemistry in here (early) to answer the question. And it has to do with the sun's germicidal ultraviolet (shorter wavelength) rays. They'd kill off life right quick (as we say here in Texas) if they reached the surface (where life lives). And they used to do so! Life hid in the ocean (about a fathom down, 6' landlubber, is sufficient shielding). And as Life did its photosynthetic thing, it cast off O₂(g) which bubbled out of the sea. After it had oxidized all the surface metals, it started to accumulate in the atmosphere. And set up its own UV shield as follows:

And that latter absorption converts light, hv, into heat that keeps the stratosphere (where these reactions occur) warm. These reactions make the stratosphere! They keep hv_UV off our doorsteps until we interfere.

Net result: ·Cl kills off ozone catalytically, reducing the "ozone shield" and increasing UV at Earth's surface. Estimates are that for every 1% increase in UV, there's a 2% increase in skin cancers, but that's only the sensationalism. What's truly disturbing is the way UV will interfere with the "Green Revolution" (efficient food production) upon which nations have come to depend.

It's no surprise that the nations of the world are banning these chlorofluorocarbons. Those polar ozone "holes" are working their way equatorward.

Polarward.

The complete circuit (after all, the parcel has to head south along the ground to be uplifted again in the tropics) is called a Hadley Cell.

If you want to know (much) more about atmospheric circulation, there's an entire course on it in progress at Columbia University; the topic we've touched on here is on Monday of Week 5 of their Climatology course.

Some of it is trivial acid-base chemistry. The partial pressure of CO₂ is currently P_CO2 = 3.3×10^-4 atm. At that concentration, atmospheric CO₂(g) is in equilibrium with CO₂(aq) which is the anhydride of H₂CO₃, a weak acid. The 1st acid dissociation constant, 4.3×10^-7 and [CO₂(aq)] conspire to render atmospheric water at a pH of 5.6, as one text likes to put it, "the acidity of flat beer."

Weak as it is, that carbonic acid is sufficient to dissolve mountains (over geologic time, mind you), releasing, among other ions, Ca²⁺ from limestone, the most easily attacked material:

which flows to the ocean, reacting to its pH 8 by redepositting CaCO₃ not only on the ocean floor but also in the shells of much marine life.

The ocean is a vast resevoir of hydrogen carbonate ion, but while it's a very large sink, it isn't a very quick sink. So mankind's extensive use of fossil fuels for combustion is steadily increasing the CO₂(g) in the atmosphere faster than the ocean can get rid of it. That's good news to plants since photosynthesis proceeds even better with a more abundant supply of CO₂(g). And, if fact, life as we know it demands a supply of a gas with CO₂'s light absorption properties!
 
 

You're right; that's not the Earth's average temperature. So something else is happening to retain extra heat. Well, the Earth is slowly leaking heat from its interior (from radioactive decay), but not nearly enough to account for the actual average Earth temperature (a few degrees above freezing). Instead, it is due to our warm and fuzzy IR blanket, mostly powered by CO₂. Those molecules intercept IR light on its way off Earth, and then they reradiate it again in the IR. Sounds like a wash, but that reradiation is down as often as it is up! So the Earth's surface gets to revisit about half the warming infrared each molecule nabs. And the more CO₂, the warmer we get.

We'll see that first at the poles. Last year the polar ice pack was only 30% of its recent historical thickness, and only a small fraction of that change can be attributed to El Niño. The rest is quite likely to be our first unequivocal measurement of Global Warming. Now the trick is to wean ourselves of fossil fuel combustion before we trick the Earth into a new (and unpleasant simply because we're used to this one) climate with the concomitant disruptions in the Earth's capacity for support (or is that tolerance) of the human race.
 
 

We pay for it when the SO₂(g) gets oxidized to SO₃(g), the thirsty anhydride of H₂SO₄. In one sense, we're lucky SO₃(g) is so hydroscopic (water-loving) since it pulls water vapor out of the air to coalesce around SO₃(g) molecules as the start of a sulfuric acid raindrop! True, that doesn't sound lucky, but it means that SO₃(g) doesn't last long in the atmosphere. Unlucky, of course, is any plant, animal, lake, or stream that rain should drop on. While the pH of lakes ought to be 5.6 (if the lakebed itself has no acid-base character), but the lakes downwind of coal-burning factories (there's sulfur in cheap coal) have acid concentrations over 10 times higher. In those lakes, the fish have long since died. The Norwegians and Swedes are not at all happy at the coal-burning British, for example.

Fortunately, it's a problem with an economical fix. The sulfur oxides can easily be captured by scrubbing the gases on their way out of the plant, and since H₂SO₄ is such a terrifically useful chemical (and the one in highest production in the world), this "pollutant" can be sold!

But this fix only works for a centralized polluter. Decentralized pollution is the job of the internal combustion engine. No one is going to sell the nitrogen oxides, NO_X (shorthand for NO, NO₂, etc.), which are an almost inevitable product of using air to burn gasoline in our vehicles. Their oxidation from N₂ to NO to NO₂ to NO₃ and the reactions:

demonstrate that nitric is also an acid rain acid.

Here the fix is the same as that for Global Warming: change the combustion of fossil fuels. The most promising possibility is the fuel cell which oxidizes combustibles in a controlled manner in a battery cell. No high temperature is involved to encourage NO_X formation. And if the combustibles aren't hydrocarbons, we could lose the excess CO₂ as well. The best fuel bet is H₂ which we could make by electrolysis of seawater (using some cleaner energy source such as solar or fusion). Instead of compressing the dihydrogen into dangerous cylinders of explosive gas (remember the Hindenberg?), it may soon be possible to absorb H₂ into metal sponges which will release it slowly and safely to the fuel cell.
 
 

Bear in mind that O: and ·OH aren't ions; they're neutral radicals. And although in very low concentration ( [·OH] ~ 10⁵ molecules/cc ), they're of critical importance in smog formation and many of other atmospheric chemistry phenomena.

. . . more later