Download Wind and the Gas Laws PV = nRT PV = nRT

Wind and the Gas Laws
For any gas or gas mixture, we can relate the pressure of the gas (P), the volume it occupies (V), with the
temperature of the gas (T) and the number of gas particles (n) according to the ideal gas law. R is a
proportionality constant.
PV = nRT
In this section we'll discuss how the gas law relates to wind and energy transfer in the atmosphere.
Outline
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Ideal Gas Law
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Ideal and Non-ideal Gases
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Wind
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Homework
PV = nRT
Pressure, Volume, Temperature, Moles
We know that temperature is proportional to the average kinetic energy of a sample of gas. The
proportionality constant is (2/3)R and R is the gas constant with a value of 0.08206 L atm K-1 mol-1 or
8.3145 J K-1 mol-1.
(KE)ave = (2/3)RT
As the temperature increases, the average kinetic energy increases as does the velocity of the gas particles
hitting the walls of the container. The force exerted by the particles per unit of area on the container is the
pressure, so as the temperature increases the pressure must also increase. Pressure is proportional to
temperature, if the number of particles and the volume of the container are constant.
What would happen to the pressure if the number of particles in the container increases and the
temperature remains the same? The pressure comes from the collisions of the particles with the container.
If the average kinetic energy of the particles (temperature) remains the same, the average force per
particle will be the same. With more particles there will be more collisions and so a greater pressure. The
number of particles is proportional to pressure, if the volume of the container and the temperature
remain constant.
What happens to pressure if the container expands? As long as the temperature is constant, the average
force of each particle striking the surface will be the same. Because the area of the container has
increased, there will be fewer of these collisions per unit area and the pressure will decrease. Volume is
inversely proportional to pressure, if the number of particles and the temperature are constant.
There are two ways for the pressure to remain the same as the volume increases. If the temperature
remains constant and so the average force of the particle on the surface, adding additional particles could
compensate for the increased container surface area and keep the pressure the same. In other words, if
temperature and pressure are constant, the number of particles is proportional to the volume.
Chemistry 102
Prof. Shapley
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Another way to keep the pressure constant as the volume increases is to raise the average force that each
particle exerts on the surface. This happens when the temperature is increased. So if the number of
particles and the pressure are constant, temperature is proportional to the volume. This is easy to see
with a balloon filled with air. A balloon at the Earth's surface has a pressure of 1 atm. Heating the air in
the ballon causes it to get bigger while cooling it causes it to get smaller.
Partial Pressure
According to the ideal gas law, the nature of the gas particles doesn't matter. A gas mixture will have the
same total pressure as a pure gas as long as the number of particles is the same in both.
For gas mixtures, we can assign a partial pressure to each component that is its fraction of the total
pressure and its fraction of the total number of gas particles. Consider air. About 78% of the gas particles
in a sample of dry air are N2 molecules and nearly 21% are O2 molecules. The total pressure at sea level is
1 atm, so the partial pressure of the nitrogen molecules is 0.78 atm and the partial pressure of the oxygen
molecules is 0.21 atm. The partial pressures of all of the other gases add up to a little more than 0.01 atm.
Atmospheric pressure decreases with altitude. The partial pressure of N2 in the atmosphere at any point
will be 0.78 x total pressure.
Gas Molar Volume at Sea Level
Using the ideal gas law, we can calculate the volume that is occupied by 1 mole of a pure gas or 1 mole of
the mixed gas, air. Rearrange the gas law to solve for volume:
V = nRT/P
The atmospheric pressure is 1.0 atm, n is 1.0 mol, and R is 0.08206 L atm K-1 mol-1. Let's assume that the
temperature is 25 deg C or 293.15 K. Substitute these values:
V = (1.0 mol)(0.08206 L atm K-1 mol-1)(298.15 K)/(1.0 atm) = 24.47 L = 24 L (to 2 sig. fig.)
Gas Velocity and Diffusion Rates
Kinetic molecular theory can derive a quantity related to the average velocity of of a gas molecule in a
sample, the root mean square velocity. You can see the derivation in the appendix to Zumdahl's textbook
or read about it on an online source. The calculations are beyond the scope of this course.
This velocity quantity is equal to the square root of 3RT/M where M is the mass of the particle.
Chemistry 102
Prof. Shapley
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The relative rate of two gases leaking out of a hole in a container (effusion) as well as the rate of two
gases moving from one part of a container to another (diffusion) depends on the ratio of their root mean
square velocities.
Can apply this to isotope separation for nuclear reactors? Remember that uranium fuel for commercial
reactors must be enriched to 3-5% U-235. Its natural abundance is only about 0.7% with the remainder
U-238. The uranium is converted to a volatile form, UF6. Let's calculate the rate at which the lighter
235UF6 would pass through a small hole from one gas centrifuge to the next relative to the heavier gas
238UF6.
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mass of 235UF6 = (6)(18.9984 g) + (235.0439 g) = 349.0343 g
•
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mass of 238UF6 = (6)(18.9984 g) + (238.0508) = 352.0412 g
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rate of effusion of 235UF6/238UF6 = 352.0412/349.0343 = 1.0086
Now you can see why row-after-row of gas centrifuges are necessary for isotope separation!
Chemistry 102
Prof. Shapley
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Real Gases
Assumptions of the Ideal Gas Law
The ideal gas law is based on a series of assumptions on gas particles.
1.
All gas particles are in constant motion and collisions between the gas molecules and the walls of
the container cause the pressure of the gas.
2.
The particles are so small that their volume is negligible compared with the volume occupied by
the gas.
3.
The particles don't interact. There are no attractive or repulsive forces between them.
4.
The average kinetic energy of the gas particles is proportional to temperature.
The first assumption is true at any temperature above absolute zero.
The fourth assumption is true for small gas molecules. When there are multiple atoms in a molecule,
some of the internal energy goes into molecular vibrations rather than in translational motion. In very
large molecules, this can be a significant part of the total energy so the KEave would be a little lower than
predicted by the ideal gas law. (We'll discuss molecular vibrations in more detail next time.)
Atomic/Molecular Volume
For most gases at atmospheric pressure, the volume of the gas particle really is insignificant. Look at
helium, for example, with an atomic radius of 31 x 10-12 m.
helium atom volume = 4/3πr3 = (4/3)(3.1415)(29791 x 10-36 m3) = 1.2 x 10-31 m3
volume of 1.0 mol He atoms = 1.2 x 10-31 m3 x 6.02 x 1023
= 7.5 x 10-8 m3 = 7.5 x 10-8 m3 x 103 L/m3 = 7.5 x 10-5 L
We know that 1.0 mol of gas should occupy 24.47 L of volume. The atoms of helium occupy 7.5 x
10-5/24.47 or 0.00031% of the volume of the gas. For helium at 1 atmosphere pressure, this is an excellent
assumption.
For very large gas molecules and at very high pressures, the volume of the the particles can become
significant. There is a correction to the ideal gas law than can be applied under these conditions that
accounts for the actual volume available.
PV' = nRT
Atomic/Molecular Interactions
All atoms and molecules in the gas phase have both repulsive and attractive interactions with other atoms
and molecules.
Chemistry 102
Prof. Shapley
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Most of the volume of any atom is due to its negatively charged electron cloud. The electrostatic
interaction of any two areas of the same charge is repulsive so when two atoms encounter each other in
the gas phase their electron clouds tend to push them apart. The repulsive interaction is especially strong
when the gas is under high pressure because the number of collisions between gas particles increases with
pressure.
How does this interaction affect pressure? Remember that
pressure is due to the force of the gas particles with a unit area
of the container. Gas particles will hit the wall of the container
with a little more force if they have been pushed by the
electron cloud of another particle.
The position of an electron cloud around a gas particle is not
constant. It can fluctuate and so leave the particle with partially
positive parts and partially negative parts. The negative
electron cloud of another gas-phase atom would be
electrostatically attracted to the positive part of the first
particle. Some gas molecules have permanent dipoles, areas of
excess positive charge and excess negative charge, due to polar
bonds.
The attractive forces between gas phase particles are called
van der Waals forces. To the extent that the gas particles stick
together, even very briefly, the number of free particles is
reduced along with the gas pressure.
There is a correction to the ideal gas law than can be
applied to account for the small pressure changes that
result from the inter-particle forces.
P'V' = nRT
Wind
Wind in the Stratosphere
Wind is movement of the gases in the air from one location to another. One reason for movement of gases
is the rotation of the Earth. As the planet rotates, it drags along the gases close to the surface. The upper
levels of the atmosphere are not pulled along as well as the gases in the troposphere. So gases in the
stratosphere and above move relative to a position on the Earth's surface in the direction opposite to the
rotation.
Surface Pressure Differences
Surface pressure is the weight of the column of gases above the point on the surface. The pressure at sea
level is about 1 atmosphere (1013 mPa) but we know that the pressure changes with weather conditions.
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Weather maps, such as the one below, show regions of higher pressure and regions of lower pressure.
Wind arises when air from an area of high pressure flows into an area of low pressure.
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Prof. Shapley
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Conversion of Solar Energy to Wind
Differences in the absorption of solar energy causes the pressure differences at the surface that create
wind. Some of the electromagnetic energy from the sun is absorbed by the surface of the planet. This is
then released as heat.
Different parts of the surface absorb different amounts of radiation and so release different amounts of
heat. There is more solar radiation at the equator than at the poles, so heat released depends on distance
from the equator. Water tends to reflects more radiation than land does. Dark surfaces, such as vegetation
or paved surfaces, absorb more heat than light surfaces such as deserts.
From the gas law, we know that heating a gas will result in an increase in its volume. Gases above a hot
region of the Earth's surface will expand. This reduces the weight of the gases in the air column above this
spot and so reduces the surface pressure. Colder, denser air will tend to move into this low pressure area.
Capturing Wind Energy
The kinetic energy of the wind can be
captured and converted into electricity by
wind turbines.
The Twin Groves Wind Farm is located
outside of Bloomington, Illinois on two
prominent moraines in eastern McLean
County. It consists of 120 turbines with a
capacity of 198 MW, enough to power
approximately 54,000 Illinois homes energy
each year.
There is a proposal to build another wind
farm just outside Urbana.
Chemistry 102
Prof. Shapley
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There are many changes in the form of energy, finally resulting in electrical power.
1.
Mass destroyed in the fusion of hydrogen atoms in the core of the sun is converted to...
2.
electromagnetic radiation that travels to the Earth, is absorbed by the surface, and is converted
to...
3.
heat energy that is released and lowers air density at the surface and is converted to...
4.
the kinetic energy of the wind that turns the blades of a wind turbine and is converted to...
5.
electrical energy.
Chemistry 102
Prof. Shapley
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