KINETIC MOLECULAR THEORY OF GASES
1. Gases are composed of many tiny particles separated by large distances and whose
volume is negligible compared to the volume of their container.
2. Attractive forces between particles are negligible due to their high velocities and
large separation from other particles.
3. Gas particles travel at high velocities, randomly colliding with each other and with
the walls of their container creating pressure.
4. Their velocity and Kinetic Energy are directly proportional to their absolute
temperature.
5. Gas particles have constant Kinetic Energy at a constant temperature and thus their
collisions with the walls of their container and other gas particles are perfectly elastic.
PROPERTIES OF GASES
1. Transparency: Gases are mostly empty space and offer little interference to light.
2. Gases are readily compressed by applying pressure.
3. Gases can expand without limit due to their weak attractive forces and high KE.
4. Gases diffuse (mix with one another) due to their random motion and high velocity.
5. Gases expand when heated. As temperature and thus KE increase particles travel
faster and exert greater pressure on the walls of their container.
STANDARD MOLAR VOLUME OF GASES
It is observed that 1 mole of any gas (6.02  1023 particles) occupies 22.41 L at
0 C, i.e., at 273.15 K and at standard atmospheric pressure (1.0 atm). These standard
conditions for gases are referred to as STP. Memorize them.
Ideal gases (theoretical gases) obey this rule exactly.
Almost all real gases follow this quite closely except at very high pressures and at very
low temperatures where some deviation occurs.
ABSOLUTE TEMPERATURE
For gas calculations we use the Absolute temperature scale. On this scale, the
coldest possible temperature, where all molecular motion stops, is assigned a value of 0
K. To interconvert between the Kelvin and Celsius temperature scales, use these
formulas.
C + 273 = K
and
K - 273 = C
Absolute zero (0 K) = -273.15 C. The Kelvin temperature is always 273 degrees higher
than the same temperature on the Celsius scale.
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GAS LAWS
For gases we consider 4 variables, P, V, T, and number of moles (n).
1. Charles Law: V  T at constant P and n.
Example: Marshmallows swell when roasted over a campfire.
2. Boyle’s Law: V  1/P at constant V and n.
Example: Scuba divers must exhale as they ascend to the surface.
3. Guy Lussac’s Law: P  T at constant V and n.
Example: Popcorn pops when roasted.
4. Avogardro’s Law: V  n at constant P and T.
Example: Blowing air into a paper bag causes it to fill.
5. For P-n we have: P  n at constant V and T.
Example: The pressure in a tire decreases when in has a leak.
6. For T-n we have: T  1/n at constant P and V.
Example: The only way to add more gas to an air cylinder without a pressure
increase is by cooling it.
7. When only “n” is constant, P, V, and T are variable. The formula below applies.
P1  V1
P V
= 2 2
T1
T2
where 1 and 2 represent two different sets of conditions.
8. Ideal Gas Equation: When all 4 variables change, i.e., P, V, T, and n, we can use:
PV = nRT
where R is the Molar Gas Constant
R has a variety of values depending upon which units are used to calculate it.
To solve for values of R, simply rearrange the ideal gas equation to isolate R and
substitute in standard values of P and V for 1 mole of gas at 273 K.
VALUES OF THE GAS CONSTANT “R”
R = 0.08206 atm Lmol-1K-1 = 8314 PaLmol-1K-1 = 62.364 torrLmol-1K-1
=
8.314 Pam3mol-1K-1
=
8.314 Jmol-1K-1
= 8.314 Pam3mol-1K-1
Be sure to choose the R-value which is consistent with the units of P and V chosen.
OTHER FORMS OF THE IDEAL GAS EQUATION
m
 m
then PV = nRT becomes P  V =    R  T and rearranging
 M
M
 m
 m
P  M =    R  T and since  =   we can write P  M =   R  T
 V
 V
Since n =
gives
2
Be very careful about units when using these derived formulas. If mass units of grams
are used then  will be g/mL or g/L or g/m3 depending upon the units of volume you
choose.
PRESSURE
A column of air extending from sea level to the top of the atmosphere (~ 80 km)
exerts the same pressure (force per unit area) as a column of Hg 760 mm high.
1 atmosphere of pressure = 1atm = 760 mm Hg = 760 torr = 76 cm Hg
= 29.92 in. Hg = 14.70 lb/in2 = 101325 N/m2 = 101325 Pa = 101.325 kPa
Dalton’s Law of Pressure: The Total pressure exerted by a mixture of gases equals the
sum of the partial pressures of each gas in the mixture.
PT = P1 + P2 + P3 + etc.
VAPOR PRESSURE OF WATER
When water is placed in a container, some of its molecules in the liquid phase
evaporate into the vapor space above the liquid and join other vapors and gases and add to
the pressure of gases in the container. We say that the partial vapor pressure of water
adds to the partial vapor pressures of other gases, e.g., N2 and O2 in air. The vapor
pressure of water at atmospheric pressure has been measured at various temperatures and
this data is readily obtained from chemical and physical tables. The vapor pressure of
water ranges from 4.5 mm Hg at its freezing point (0 C) to 760 mm Hg at its boiling
point (100 C). Note that at the boiling point of a liquid, its vapor pressure equals
atmospheric pressure. This is the definition of boiling point, i.e., the temperature at
which the vapor pressure of a liquid is the same as atmospheric pressure. At this
temperature vapor bubbles form in the liquid and the liquid boils.
Any gas samples collected over water will be “wet”, i.e., contain some water
vapor. The vapor pressure of the “dry” gas can be calculated by subtracting the vapor
pressure (partial pressure) of water at the given temperature from the total pressure of the
“wet” gas mixture.
Pgas = PTotal - PH2O
PROBLEMS
1.
2.
3.
4.
Convert 5.0 psi to kPa, to cm Hg, and to atm.
Convert 0.8 atm. to torr, to mm Hg, and to in. Hg.
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Convert 40.0 in Hg to Pa, to N/m , and to atm.
The highest mountain in the world (Mt. Everest, in Nepal) rises approximately 8.9 km
above sea level. Atmospheric pressure at this elevation is about 0.33 atm.
Atop Mt. Everest, water will boil at 72 C. Explain why.
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5. Aviators flying high altitude jets must wear breathing apparatus. Explain why.
6. With reference to the KMT, explain why real gases deviate from ideal gas behavior at
very high pressures or very low temperatures.
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GAS LAW PROBLEMS
1.
20 mL of a gas at 30 C is cooled to 0 C. Calculate its final volume. Ans: 19 mL
2.
What volume will a gas occupy at 1.0 atm if it occupies 10 mL at 2.5 atm? Ans: 25 mL
3.
What is the final temperature of a gas (in C) at 15 atm if it is initially at 100 C and 2 atm?
Assume the volume of the gas is held constant. Ans: 2525 C
4.
0.5 mol of a gas occupies 20 L. Calculate the new volume when 1.5 more moles of the gas
are added at the same temperature and pressure. Ans: 80 L
5.
A tire at 30 psi pressure contains 2.0 moles of air. Calculate the number of moles of air
which are released when the valve is opened and the pressure drops to 20 psi. Assume the
temperature and volume are constant. Ans: 0.7 mol
6.
In a laboratory experiment, a glass flask contains 10.0 mol of gas at 125 C under high
pressure. It is desired to add 5 more moles of the gas to the flask without causing a change
in pressure. How can this be accomplished? What is the final temperature of the flask?
Ans: -8 C
7.
25.0 mL of a gas is collected in the laboratory at 22.0 C and 745 mm Hg.
Calculate its volume at standard conditions. Ans: 22.7 mL
8.
Calculate the density of chlorine, sulfur hexafluoride, and hydrogen gases at STP.
Ans: 3.17 gpL, 6.52 gpL, 0.089 gpL
9.
a) Calculate the density of hydrogen gas at 700 mm Hg and 30 C. Ans: 0.074 gpL
b) Calculate the density of hydrogen gas collected over water at 700 mm Hg and 30 C.
Ans: 0.071 gpL
10. a) Calculate the # mol. carbon dioxide in 10.5 g this gas. Ans: 0.239 mol.
b) Calculate the # L at STP that 10.5 g of the gas would occupy. Ans: 5.35 L
c) Calculate the # L at 95 kPa and 25 C that 10.5 g of the gas would occupy. Ans: 6.22 L
d) Calculate the # L at 95 kPa and 25 C that 10.5 g of the gas would occupy if collected
over water. Ans: 6.44 L
e) Calculate the # L of carbon dioxide collected at 95 kPa and 25 C over water after
complete neutralization of 50.0 g sodium carbonate with excess hydrochloric acid.
Ans: 10.57 L
11. a) Calculate the # mol. of gas in 20 L at STP. Ans: 0.893 mol.
b) Calculate the # mol. of gas in 20 L at 730 mm Hg and 40 C. Ans: 0.75 mol.
c) Calculate the molar mass of a gas if 32.9 g occupy 20 L at 730 mm Hg. Ans; 44.0 g
12. Calculate the molar mass of ethyl ether given that 1.65 g of it occupied 691 mL at 40 C and
84.0 kPa. Ans: 73.9 g/mol
13. What volume of hydrogen gas, collected at 30 C and 0.921 atm will be formed by reacting
50.0 g aluminum with excess sulfuric acid? Ans: 75.0 L
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