CHM 101 GENERAL CHEMISTRY
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FALL
QUARTER 2008 |
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Section
2 |
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Lecture
Notes – |
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(last revised: |
5.1
Pressure
- Recall the common properties of gases:
o
A gas will
uniformly fill any container.
o
A gas is easily
compressed.
o
A gas mixes
completely with any other gas.
o
A gas exerts pressure on its surroundings.
- The Pressure of the Atmosphere: Figure 5.1 pictures a dramatic demonstration
of the pressure exerted by the earth’s atmosphere. Panel a) shows a metal can,
open to the atmosphere and full of steam. When the can is sealed and
allowed to cool, the steam inside condenses and the pressure inside the
can falls to a small fraction of atmospheric pressure. Panel b) shows the
consequence of this imbalance in pressure: the can collapses. The pressure
of the atmosphere crushes it.
Panel a) 
Panel b) 
- Measuring the Pressure of the Atmosphere: Atmospheric pressure can be measured with a barometer, a device pictured in
Figure 5.2. A long glass tube is filled with mercury, a liquid metal with
a high density (13.6 g/cm3), is inverted in a dish of mercury.
Some of the mercury flows from the tube into the dish, creating a vacuum
at the top of the tube, but most of the mercury stays up in the tube,
seemingly defying gravity.
In fact, the mercury in
the tube exerts pressure on the mercury in the dish that matches the pressure
exerted by the atmosphere on the mercury in the dish. The height of the mercury
column is proportional to the actual pressure of the atmosphere, and changes in
this height can be used to detect and measure changes in atmospheric pressure.
- Atmospheric Pressure at Various Elevations: Here are some average barometric readings:
o
Sea Level: 760
mm Hg
o
D-103 (~6,300
feet): 610 mm Hg
- Units of Pressure: The most common units of pressure we will use
are
o
millimeters of mercury (abbreviated: mm Hg).
o
torr (another name for millimeters of mercury, honoring
Evangelista Torricelli, inventor of the barometer).
o
standard atmosphere (abbreviated: atm).
However, in the SI
system, pressure is measured in units of
o
the pascal (abbreviated: Pa).
(In physics, pressure is
defined as force per unit area, and since the SI system measures force in
newtons (N) and area in meters-squared (m2), 1 Pa = 1 N/m2.)
We will avoid using the
pascal in CHM 101.
- Pressure Conversion: Here are some equivalence statements to use
for converting among different pressure units:
|
1 mm Hg |
= |
1
torr |
|
1 atm |
= |
760
mm Hg |
|
1 atm |
= |
760
torr |
|
1 atm |
= |
101,325
Pa |
|
1 atm |
= |
29.92
in Hg |
|
1 atm |
= |
14.7
lb/ft2 |
- Sample Exercise 5.1 (p. 181): The pressure of a sample of gas is measured
as 49 torr. Convert this to atmospheres and to pascals.
o
The first step
is to derive an appropriate unit factor from the equivalence statements in the
above table:
o
Then:
o
For the second
conversion, we can use the unit factor:
o
This gives us:
- Measuring Pressure in a Closed Container: The traditional laboratory instrument for
measuring the pressure of a confined gas is the manometer, pictured in
Figure 5.3ab. Here the height of a column of mercury in a tube measures
the difference in pressure between the container (a part of the manometer)
and the atmosphere.
- Sample Exercise: The barometric pressure is determined to be
750. mm Hg (or 750. torr). a) Suppose the height
of mercury in a manometer like that shown in Figure 5.3a is 100. mm. What is the pressure of the gas sample within the
manometer bulb? b) Suppose the height of mercury in a manometer like that
shown in Figure 5.3b is 100. mm. What is the
pressure of the gas sample within the manometer bulb? c) Suppose the
manometer were of the closed tube type and the space above the column of
mercury were evacuated (i. e., in vacuum). Again the height of the column
of mercury is 100 mm. What is the pressure of the gas sample within the
manometer bulb?
o
In case a),
the pressure in the bulb is less than atmospheric pressure, so we subtract:
Pgas = 750. torr
— 100. torr = 650. torr
o
In case b),
the pressure in the bulb is greater than atmospheric pressure, so we add:
Pgas = 750. torr
+ 100. torr = 850. torr
o
In case c),
the pressure in the bulb is measured against vacuum (i. e., 0. torr)
Pgas = 100. torr
5.2
Gas Laws: Boyle, Charles, & Avogadro
- Boyle’s Law: The first quantitative experiments on gases were performed by
Robert Boyle (1627-91). He used an apparatus that allowed him to vary the
pressure on a sample of gas and accurately measure both the pressure and
the volume. (He also held the temperature constant.) His apparatus is
pictured in Figure 5.4.
As you can see, the
apparatus is a manometer. Some of his actual data are listed in Table 5.1.
(Since he was from
The data show clearly
that the product of pressure and volume holds constant over the entire range of
pressures. They can be represented by Boyle’s Law:
The constant, k, holds for a given sample of gas at a
particular, fixed temperature.
- Charles’ Law: The French physicist, Jacques Charles
(1746-1823) discovered the gas law that bears his name. He discovered a
linear relationship between the volume of a gas sample and its temperature
when pressure is held constant. (His adventures as a balloonist may have
led him toward this discovery.) Mathematically his discovery can be
expressed as:
The
constant, b, holds for a given
sample of gas at a particular, fixed pressure. Figure 5.8 shows plots for several different gases
showing the linear relationship between V and T for
several different gases. Interestingly, the constant, c, depends only on the temperature scale; all the
plots extrapolate to the same temperature at zero volume. More interestingly,
if T is measured using the Celsius scale (developed during the time of
Charles), the constant, c, has the value 273.2 C°.
This means that if T is
expressed in kelvins, then Charles’ Law simplifies to:
as plotted in Figure 5.9:
- Avogadro’s Law: Avogadro’s Law states that equal volumes of
gases at the same conditions of temperature and pressure contain equal
numbers of gas particles. Mathematically this can be written:
In this equation, a is
a constant for fixed values of temperature and pressure, and n is number of gas particles.
5.3
The Ideal Gas Law
- Summarizing the
previous section, we have the three gas laws:
|
Boyle’s Law: |
|
(at
constant T and n) |
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Charles’ Law: |
|
(at constant P and n) |
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Avogadro’s Law: |
|
(at constant T and P) |
- We can further
summarize the three laws as: volume is inversely proportional to the pressure
and directly proportional to the temperature and to the number of moles of
gas:
A slight rearrangement
gives us the familiar form of the Ideal Gas Law:
The constant, R, is called the Universal Gas Constant. If P is expressed in atmospheres, V in liters, n in moles, and T in
kelvins, then R has the
following value:
- Sample Exercise 5.6 (p. 187): A sample of hydrogen gas (H2)
has a volume of 8.56 L at a temperature of 0 °C and
a pressure of 1.5 atm. Calculate the number of moles of H2 gas
present in the sample.
o
We solve the
ideal gas law for n, the number of moles of gas:
o
Now we plug in
our data and our value for the gas constant:
- Sample Exercise 5.7 (p. 188): We have just determined that a
particular sample of ammonia gas has a volume of 7.0 mL at a pressure of
1.68 atm. Keeping the temperature (and the mass
of ammonia) constant, we compress the sample to a volume of 2.7 mL. What
is its pressure now?
o
Since n and T remain
constant while the pressure and volume both change, we can write:
and
o
This gives us
the following (which you should recognize as Boyle’s law):
o
We have data
for everything but the final pressure, P2, so we rearrange and plug in:
o
We have
applied the ideal gas law at a pressure where its predictions begin to deviate
from actual gas behavior, so the measured new volume might be a bit different
from this calculated value.
- Standard Temperature and Pressure: Many of the measurements we chemists take
must be made under carefully controlled conditions of temperature and
pressure. A particularly easy combination to control is 0.0 °C and
1.000 atm. This combination has come to be defined as standard temperature and pressure (STP). (That ease of control
of pressure at 1.0 atm applies to laboratories at altitudes close to sea
level; in D-103 our normal atmospheric pressure is approximately 0.8 atm,
and we must adjust our experiments accordingly.)
·
The Volume of 1.000 mol of gas at STP: We can compute the volume of a mole of (an ideal)
gas at STP by applying the Ideal Gas Law:
·
This volume,
22.42 L, of an ideal gas at 0.0 °C and 1.000 atm (STP) can be called the molar volume.
- Validity of the Ideal Gas Law: The Ideal Gas Law is amazingly good, but it
is not perfect. For example: it is totally invalid for a given substance
under conditions of high enough pressures and low enough pressures to
cause the substance to condense to a liquid or to a solid.
·
Limiting Law: It is best to consider the Ideal Gas Law as a limiting law; it
predicts behavior that a real gas approaches at low pressure and high
temperature.
5.6
The Kinetic Molecular Theory of Gases: The kinetic
molecular theory (KMT) of gases is a theory
that attempts to explain the simple
behavior of gases that is described
by the ideal gas law. (Recall that laws describe while theories explain.)
We start by making a model of a single gas particle; then we model the behavior
of a gas as if it were composed of a large collection of these single particle
models.
- Postulates of the KMT:
o
1) The
particles are small enough compared to the distances between them that we can
neglect their size. Therefore we can model an individual particle as a mass
that has no volume.
o
2) The
particles are in constant motion and frequently collide with the walls of their
container. These collisions are the cause of the pressure exerted by the gas.
o
3) The
particles exert no forces upon each other, neither attractive
nor repulsive.
o
4) The average
kinetic energy of a collection of gas particles is assumed to be directly
proportional to the Kelvin temperature of the gas.
- The Postulate of Zero Volume: The zero volume postulate, 1), obviously
could never apply to a liquid. Take one mole of liquid nitrogen for
example, as pictured in Figure 5.14a.
The entire mole of
nitrogen fits in a 50 mL beaker, and the molecules are all crowded together and
taking up most of the liquid volume. However, the same mole of nitrogen gas,
when warmed up to 273 K and held at 1 atm pressure (i. e., at STP), occupies
22.4 L, and even nearest neighbor molecules are rather far apart. Compared to
the gas volume of 22.4 L, the volume occupied by gas molecules is essentially
zero, as seen in Figure 5.14b.
- No Intermolecular Forces: The zero force between molecules postulate
(3) seems reasonable in view of the large separations between molecules.
- Motion of the Particles: The second postulate says that the particles
are in constant motion. Thus they all have velocities. But do they all
have the same velocity? The answer is, no. There is a distribution of
velocities, as shown in Figure 5.20, a plot of the velocities of oxygen
molecules at at a temperature of 273 K. The plot shows a range from 0 to
about three times the velocity characteristic of the peak of the
distribution:
- KMT and the Ideal Gas Law: Let us see how this model accounts for the
behavior of an ideal gas as summarized by the ideal gas law (PV = nRT).
o
Pressure and Volume (Boyle’s Law): If we hold temperature and number
of moles constant, the ideal gas law reduces to Boyle’s law, where pressure is
inversely proportional to volume:
The KMT tells us if we
reduce the volume, we increase the frequency of gas particle collisions with the
vessel walls and thus increase the pressure. This is illustrated in Figure
5.15.
o
Pressure and Temperature: If we hold the number of moles and the volume
constant, we can rearrange the ideal gas law to the following, where pressure
is proportional to temperature:
The KMT tells us if we
increase the temperature, we increase the velocity of the gas particles and
thus the frequency of their collisions with the vessel walls. This increases
the pressure as illustrated in Figure 5.16:
o
Volume and Temperature (Charles’ Law): If we hold the number of moles and
the pressure constant, the ideal gas law rearranges to Charles’ law, where
volume is proportional to temperature:
The KMT tells us if we increase
the temperature, we increase the velocity of the gas particles so if the
pressure remains constant, the volume must expand, as illustrated in Figure
5.17:
o
Volume and Number of Moles (Avogadro’s Law): If we hold the pressure and
temperature constant, the ideal gas law can be rearranged to Avogadro’s law,
where the volume is proportional to the number of moles:
The KMT tells us if we
increase the number of moles while holding the pressure constant, the volume
must expand, as illustrated in Figure 5.18:
o
Mixtures of Gases (
