Our
Atmosphere
Half the Earth is
always cloudy since
rising air in "lows"
bring clouds while
falling air in "highs"
doesn't. And what
goes up must come
down!
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The atmosphere, its physics & chemistry, are topic enough for several
courses! Our text gives it terribly short shrift. Fortunately, your loquacious
professor has already written at length to other classes on the atmosphere,
but before he sends you there, remember that to return here,
you must use your Back button in your browser (because all the links in
the previous course's pages connect only its own pages).
With that caveat understood, click on
this to read about gravity, pressure,
oxygen, and ancient Life. There: wasn't that fascinating? :) OK, where's
Pavlov's Bell (that imaginary chime I ring to alert you to material on which
you'll be tested rather than all that disparaged understanding I keep
trying to throw at you? All right...it's the composition: 77% N2,
21% O2, 1% other stuff (mostly argon, of all things, and a
soon to be critical 0.03% CO2).
Let's see, what's the Web equivalent of Pavlov's Bell? Aha, the most annoying
trick in HTML...ahem...the testable item from that other page was the
of the atmosphere. Oooo...that is annoying!
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Gas Laws
Charles must have
been fascinated by
hot air balloons.
They illustrate his
law about volume
increasing with
temperature at
fixed pressure.
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OK, quick, what's Gay-Lussac's Law? Don't know? Don't care? Don't blame you.
Still the notion that pressure varies directly with temperature in a gas whose
volume (and number of moles) is fixed, is a useful tool regardless of whoever
came up with it. Personally, I'm convinced that text authors believe that you'll
memorize all those individual Gas Laws in preference to the one PV=nRT equation
which encapsulates them all because the individual laws, like Gay-Lussac's
say, seem simpler; after all, it just says that P/T is fixed under some circumstances
(fixed n and V). So it appears you only have to memorize a trivial formula. But
there are four times as many of them! PV=nRT may be inobvious until you see
that it's just Boyle's, Charles', Gay-Lussac's, and Avogadro's all tied up with
a pretty bow. (Maybe pink.)
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So let me send you to a lecture of mine which sort of derives PV=nRT but
has some more things to say about "why" en route. Again, your
Back button will return you here from this link to
Ideal Gases.
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Using
the Tool |
So PV=nRT is compact and efficient. Is it useful?
Surely we wouldn't have wasted your time with it if it wasn't.
Aside from the profundity of its derivation from the Kinetic Theory of Gases,
a blindingly successful application of mathematical physics from the 19th century,
the Ideal Gas equation is de rigeur for helping us chemists convert
between moles, n, and volumes, V, or even pressures, P, both of which are
pretty easy to measure in gases. Pressure gets measured with manometers
that transform it into readings of differential heights of mercury columns.
Volume gets measured as (invisible) gases displace (visible) fluids, often
water.
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Unfortunately, for that volume measurement, all gases mix perfectly in
all other gases. In particular, the water vapor above the liquid water that
traps the sample gas, mixes in with it. Fortunately, since the gases are
(for all practical purposes) ideal, both the sample and the water vapor
establish their own pressure independent of the other's presence!
And we know the (saturated) vapor pressure of water (well, we
personally don't, but thermodynamicists who study the vaporization process
certainly do, and they've given us tables to go by). For example, at 100°C,
water will have a vapor pressure of exactly 760 torr = 1 standard atmosphere.
That's why it boils at that temperature. At 25°C, it's vapor pressure
is only 23.76 torr, a small correction to the sample's pressure.
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This correction works since, being ignorant of one another, the two gases's
pressures simply add to the observed pressure. So if we observe the (damp)
sample gas pressure to be 760 torr (at 25°C), then 24 torr of that isn't
the sample; it's the water. So the sample is only 736 torr or a fraction of
736/760 = 0.968 of the moles (or the volume) in that gas phase.
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Dalton's Law
of Partial
Pressures |
Indeed, regardless of how many gases occupy a given volume, the total
pressure of the mixture is just the sum of the partial pressures of each
gas. In fact that ratio we just took above of one partial pressure compared
to the total pressure will, for ideal gases, be Xi, the mole fraction
of that component, i. This is because V and T are the same for all the
gases in the mixture (duhh), so P is directly proportional to n. So it's easy
to obtain the mole % (100×the mole fraction), and the weight % isn't that
much harder.
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Clearly the weight of the mixture is the sum of the weights of the gas
components. So in any mole of mixture, each gas contributes
Xi×MWi (where MW is its molecular weight). The
sum of all those terms is the total weight, and the ratio of each term to the
total (×100%) is its weight percent.
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Other examples of the utility of PV=nRT are given in this
link as is a discussion
of the relative velocities of different molecular weight gases. That
too comes from simple Kinetic Theory!
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| Non-Ideal Gases |
What happens when gas molecule do interact? At the distances
they usually see one another, they attract one another weakly. That means
that a molecule about to strike the wall does so just a little more gently
than it should because it feels tiny tugs at its skirttails from its buddies
nearby in the gas. So the Ideal Gas pressure is higher than the measured one
by the amount of this tugging; that is reflected by replacing P in the ideal
gas equation by ( P + n²a/V² ) where n is still the number
of moles, and V the volume, but a is a measure of the attraction
between gas molecules.
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Similarly, the molecules are not without a volume of their own. We have
only to try to condense them to a liquid to discover that, cheek-by-jowl,
they occupy space. While it is nice to know that's because their electrons
don't permit other atom's electrons to squeeze into their atoms, the upshot
is that this molar volume, b, is strictly off-limits for any other
molecule to occupy...so V overestimates the free volume in which the
gas molecules can move. We correct V to become available V by replacing
it with ( V - nb ) in the Ideal Gas Equation.
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With both corrections, the Ideal Gas Equation becomes Van der Waals'
equation:
( P + n²a/V² )×( V - nb ) = nRT
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