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H A&S 220c Energy and Environment: Life Under the Pale Sun
29 Oct. 2004
Added notes and typos are in bold font
Exercises on gases, pressure, temperature, mechanical and thermal energy.
1. Suppose 2.5 liters of hydrogen gas are cooled from 250C (room temperature) to
-2000C. This is done with the pressure constant. What will the volume of the gas then be?
Using Pv=nR* T, we relate the two states (1 and 2) by v1/v2 = T1/T2, so we
don’t need to use the constant R* or even know the number of moles of gas, n. But we do
need to use the absolute Kelvin temperature scale. So T1 = 298.15K, T2 = 73.15K, and
therefore v2 = v1 T2/T1 = 2.5 x 73.15/298.15= 0.613 liters. (A liter is 1000 cm3). [note,
the other form of the equation of state of a gas is P = ρ R T where R = R*/M where
ρ is the density (mass/volume) of the gas, and R = R*/(molecular weight of gas) ]
2. Why is moist air less dense than dry air at the same temperature and pressure?
Moist air has water vapor in it. Amadeo Avogodro, in 1811, was one of the first
to think of a gas as being composed of molecules, and proposed that the same number of
molecules are present for any kind of gas for which the temperature and pressure are the
same. This leads to a ‘standard’ reference: 22.4 liters of gas at temperature 00C and
pressure 1 atmosphere (105 Newton m-2) are called 1 mole of gas, comprising
6.02x1023 molecules; The Kinetic Molecular Theory, when it introduces Avogadro’s
number, argues that this is true, regardless of the size or mass of the molecules. The
temperature is a measure of the average kinetic energy of a gas molecule, while the
pressure is proportional to the kinetic energy per molecule multiplied by the total number
of molecules (so, T and P differ by a factor n/v, the number of moles of gas per unit
volume).
If you have two gases, say pure oxygen and pure hydrogen, their molecular
weights are 32 and 2, grams per mole, respectively (this comes from looking up the atomic weights, 16
and 1 respectively, and doubling because there are two atoms per molecule; the numbers mean that a volume of 22.4 liters of oxygen
will weigh 32 grams).
The hydrogen weighs less by a factor of 16, even though it exerts the
same pressure and has the same temperature. This is why helium or hydrogen filled
balloons rise! The balloon can only increase the pressure above atmospheric and hence
would be expected to make the balloon heavier, but in fact it is lighther than air. With
their mass being so much smaller, molecules of hydrogen must move faster than
molecules of air to exert the same pressure. The ratio of speeds is √(ratio of masses), or 4.
For the question above, the molecular weight of water is 18 grams per mole whereas
dry air has a molecular weight arising from 21% oxygen (32 grams/mole) plus 78%
nitrogen (28 grams/mole). As with the balloon, the lighter water molecule makes
moist air lighter than dry air at the same pressure and temperature, and the speed
of the water molecules is higher to keep the pressure and temperature the same. We
discussed the water vapor content of air in class briefly; it ranges from 0 to about
4% depending on temperature and proximity to the ocean or other water.
[ By ‘molecular weight’ we really mean the mass in grams of a certain standard volume
of gas, which is 22.4 liters (or 22.4 cm3). Note, the pressure inside the balloon is actually
greater than atmospheric pressure, but not very much. Your lungs can only exert a
pressure of a few percent of atmospheric pressure.]
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Thus an interesting property of temperature is that in a gas made up of two different
species of molecules, say oxygen and hydrogen, the KE of both kinds of molecule will be
the same, and hence the light molecules will be moving much faster than the heavy
molecules. If you cut a small hole in a container of gas, the light molecules will stream
out first, leaving the heavy ones behind. What
2b. The narwhal dives deeper than 1500m in the ocean to find Greenland halibut. What
is the pressure at that depth, compared with the pressure of the atmosphere on the Earth?
Here we must add an important idea, air or water sitting on the solid Earth exerts
a pressure (force per square meter) equal to the weight of the fluid overhead, per square
meter. Look upward and think about a column of air 1 m square, and as tall as the
atmosphere. Its mass 104 kg. So it weighs Mg or 104 x 9.8 Newtons, or kg m sec-2 (the
unit of force has the units of mass x acceleration). So atmospheric pressure is a little less
than 105 Newtons per square meter, and the pressure of a 1500m tall column of water is
1000 x 9.8 x 1500 or 1.5 x 107 Newtons/m2. That is mass per unit volume times g
times height of the column. or 150 times the atmospheric pressure. The physical stress
and compression of the body of the narwhal must be incredibly great.
3. What is the average distance between molecules in air at room temperature?
There are 6.02x 1023 molecules of gas in one mole…which occupies 22.4 liters at
atmospheric pressure and temperature of 0C. This is a density, call it n, of 6 x 1023/22.4,
or 2.7 x 1022 molecules per liter. A liter is 1000 cm3, for example a cube 10 cm (0.1m) on
a side. If the molecules were equally spaced, then n1/3, or 3.0 x 107 is the average number
of molecules along one edge of the cube. The spacing in cm. is then 10/n1/3, or 3.3 x 10-7
cm, or 3.3 x 10-9 m. Compare this with the size of the molecules. A good reference size
for a small atom like hydrogen is 1 Angstrom (this is written as 1 Å) which is 10-10 m.
Oxygen molecule is about 5 Å in diameter. Therefore the molecules in air are roughly 8
times as far apart as the the width of the individual molecules.
At what pressure would air cease to be described by Kinetic Molecular Theory, which
assumes that the molecules are far apart, compared with their individual diameter?
The equation of state for air, Pv = nR* T or P = ρ RT, tells us that the density ρ of
the gas will increase in proportion to the pressure, if the temperature is constant. So, a
ten-fold increase in pressure will cause a ten-fold increase in density and put air
molecules close to one another. Since forces between molecules will then be stronger,
there is the possibility of liquefying air…causing a change of state. But this doesn’t
actually happen until reaching -1830C…very cold.
4. The typical winds in the atmosphere are 10 m sec-1, while in the fast westerly jetstreams they are typically 100 m sec-1 (which is why it takes longer to fly from NY to
Seattle than the reverse). In the oceans the typical currents are 0.1 m sec-1 while
concentrated currents like the Gulf Stream flow at about 1 m sec-1. Compare the kinetic
energy (per kg. of fluid) for these 4 examples. The density of air at ground is about 1.25
kg m-3, the density of ocean water is about 1030 kg m-3. The atmospheric jet-stream is a
little trickier. The pressure in the atmosphere decreases as one goes upward; at 10 km
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altitude, where the jet-streams live, it is about ¼ of the air pressure at the ground. To
estimate the density there, assume the temperature to be constant.
For winds near the ground, the kinetic energy is ½ MV2 J, or ½ ρ V2 J/m3.
where ρ is the density of air in kg m-3. Note the second expression is not Joules per kg,
but Joules per cubic meter of air. The first part above gives ½ x 1.25 x102 = 62.5 J m-3;
for the second, use Pv = ρRT to give ρjetstream/ρground = Pground/Pjetstream=0.25. So ρjetstream=
.25 x 1.25 = 0.31 kg m-3 and ½ ρv2 = ½ x 0.31 x 1002 = 1.6 x 103J m-3. The ocean
currents are slower but the density of water is 800 times that of air, so KE = ½ x 1030 x
(0.1)2 = 5.15 J m-3 and ½ x 1030 x 12 = 515 J m-3 for the two cases. Despite the low
density of air, it wins by a bit, in terms of its kinetic energy, compared with the ocean.
Now, for kinetic energy per kg. of fluid, that is just ½ V2, so the numbers are
50 J. (typical winds), 5000 J (jet stream), 0.005 J (typical ocean current), 0.5 J (Gulf
Stream). You can see how much it matters to say whether your energy calculation is
looking at a given volume of fluid or a given mass of fluid.
Exercises on thermal energy
1. How much heating is required to raise one liter of water from room temperature to the
boiling point? How much would this cost if electricity is $0.10 per kilowatt hour?
One liter is 103 cm3 or 10-3 m3 [1 meter3 = (100cm)3 = 106 cm3.] The density of
fresh water is about 1 gram per cm3 or 1 kg per liter. The specific heat capacity is about
4000 Joules per kg, per 0C of temperature change. To raise water from 200C to 1000C
which is the boiling point at our standard atmospheric pressure, thus requires (10020)x4000 Joules, or 320,000 Joules of energy (or 320 kiloJoules, KJ).
Our electrical energy is given in kilowatt hours (kwh) which are not the same
units as Joules. To convert, note 1 kwh = 1000 J sec-1 x 3600 sec =3.6x106 J, 3.6 million
Joules (the number of seconds per hour is 3600)….check the units. The number of kwh
to give us the required 320,000 J is thus 3.2x105/3.6x106 = 0.089, a bit less than 1/10 of a
kwh. It would cost 0.89 cents at 10 cents per kwh.
Does this seem right? A burner on your stove, turned up high, is probably using
about 1000 watts of electrical power, and a liter (slightly more than a quart) of water will
come to a boil in 0.089 hours, or 5.3 minutes. This seems about right.
2. An Inuit native in cold surroundings must have energy intake sufficient to maintain
body temperature (core temperature of 98.6 F, or 37C) and also to do work…the
mechanical energy needed for daily life. He or she must also drink enough water to
supply the biochemical needs. Basal metabolism requires, each day, about 1 kcal x kg of
body weight x 24 hrs. During sleep this is reduced to 0.1 kcal x kg body weight per hour
Women: 1200-1450 kcal/day
note: 1 cal = 4.187 Joules; 1 kcal (food calorie) = 4187 J.;
1 quart = 0.9463 liters
1 liquid oz. = 6 tsp = 2 tbsp = 29.57 milliliters
men: 1600-1800 kcal/day
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a minimal amount of water is 1 ml. (1 cubic cm or cc.) of water per kcal of food intake.
For the above kcal ranges this is 34 to 68 fluid ounces (900-2100 ml or 0.9 – 2.1 liters) .
Add digestion, etc. you get roughly 2 liters per day. Very likely more than this in the
harsh cold environment, with metabolism higher.
Calculate the energy required to use ice as a source of the 2 liters per day of water:
Warm the ice from -10C to 0, melt it, then warm the water to 37C body temperature.
For 1 kg of water this is
10 x 2095 (Cp for ice 20,950) + 337,000 (melt ice: latent heat) + 37 x 4180 (Cp for
water, 154,660) = 512,610 J ,
about ½ million J, or ½ candy bar. A daily caloric intake of
2500 kcal = 10.5 x 106 J
….10 candy bars 1 gm sugar = 16 KJ = 3.75 Kcal of chemical energy…
or 16 MJ per kg. Thus 1 kg, 2.2 lbs of sugar contain 16 MJ, or 3750 Kcal
of energy. But remember the human body is an engine, and it runs at less than 20%
efficiency in creating mechanical work…the rest goes into heat and waste chemical
energy.
Compare with hydrocarbon energy below.
http://www.nutrition.org.uk/information/energyandnutrients/energy.html
Fuel Type
Energy Content (kJ/g)
Kerosene 46.3
Paraffin Wax 42.0
Ethanol 26.8
Isopropyl 24.04
Methanol 19.9
Gasoline 48.
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Questions: most of these questions have an ‘essay’ aspect: they should be answered with
a page-long or long paragraph of discussion, and not just a few words.
1. What is temperature?
3. We have had readings describing Greenland and its natives, and the Chuckchi Coast
natives of the Siberian Arctic (the book by H.A. Sverdrup). You may also have picked
up some ideas about the Canadian Inuit or the Alaskan Inupiat. Describe some
significant differences and their relation to geography of the different Arctic regions.
Refer to the attached map of the Arctic.
4. European explorers, settlers, and the native populations of the Arctic have interacted
in both positive and negative ways. Pick two examples of such interaction, and describe
both the good and the bad that came of them. .
5. With energy being converted, stored and moved from place to place in the natural
world, and in the world of humans, there are many ‘equivalences’ we make use of. For
example, 1 million Joules of chemical energy stored in a candy bar can turn into about
200,000 Joules of useful mechanical work (20% efficiency) when eaten. Gasoline
contains about 45 million Joules per kg, and makes mechanical energy at about 20%
efficiency. Make a quick ‘sketch’ of one day in the life of a Greenland native, and one
day in the life of a Seattle stock-broker, comparing their energy use. It is not necessary to
do a complete calculation, but show some examples of their energy inputs and outputs.
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6. In the McNeill chapters on the atmosphere, we see a history of severe local air
pollution, recorded as far back as Roman civilization, and global pollution occurring
really for the first time in the 20th Century. Using some imagination, and your
recollections of the Ehrlich book on Greenland, characterize the air quality experienced
by the Greenland Inuit, during typical activities. Does it seem better or worse than that
experienced by a citizen of London in 1950, or a citizen of Seattle in 2004?