Chapter 10: Gases

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Chapter 10 Gases
http//mc2.cchem.berkeley.edu/Java/molecules/inde
x.html
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Global Warming and Greenhouse Gases

• Carbon dioxide
• Methane
• Nitrous oxide
• HFCs
• PFCs
• SF6

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Chapter Outline

• Characteristics of gases
• Pressure
• The Gas Laws
• The Ideal Gas Equation
• Gas Densities in gas law calculations
• Gas Mixtures and Partial Pressures
• Kinetic Molecular Theory
• Molecular Effusion and Diffusion
• Deviations from Ideal Behavior

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Gases

• In the gas phase, molecules are relatively far
  apart. This makes gases very compressible.
• A gas expands to fill the volume of its
  container.
• Gases are usually measured
• by volume, so a relationship
• between volume and number
• of moles is needed.

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Pressure

• Gases exert a pressure on the walls of their
  container.

• Pressure is defined as force per unit area

SI unit 1 Nt/m2 1 Pascal (Pa)
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Atmospheric Pressure

• The atmospheric pressure can be measured using a
  barometer.

Standard atmospheric pressure supports a column
of mercury about 760 mm high. 760 mm Hg 1 atm
1.013?105 Pa
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Pressure Measurements

• A manometer is a
• U-shaped tube usually
• containing mercury.
• An open-end manometer
• measures pressure
• relative to atmospheric
• pressure.

Pgas Ph Patm
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Pressure Measurements

• If the barometer reads 753.3 mm Hg, what is the
  atmospheric pressure in atm and kPa?

0.9912 atm
753.3 mm Hg
100.4 kPa
0.9912 atm

• What is the pressure in a vessel when the
  mercury height is 106 mm?

Pgas Patm Ph 753.3 mmHg 106 mmHg
859.3 mmHg
1.131 atm
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The Gas Laws

• The gas laws are experimental relationships among
  pressure (P), volume (V), temperature (T), and
  moles (n).
• Boyles law (V,P)
• Charles law (V,T)
• Avogadros law (V,n)
• Ideal gas law (V,P,T,n)
• All gases behave similarly. The gas laws assume
  ideal behavior.

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Boyles Law
V1 P1 V2P2
P1 1.0 atm V1 4.0 L
P2 1.0 atm V2 1.0 L
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Boyles Law

• In 1662, Robert Boyle discovered that volume is
  inversely proportional to pressure.

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Charles Law

• Activity How could we discover the relationship
  between volume and temperature?
• Hint 1 think about a hot air balloon

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Charles Law

• Charles discovered that volume is directly
  proportional to temperature.

The volume of a gas extrapolates to zero at
-273?C. This must be the lowest temperature
possible.
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Charles Law

• Lord Kelvin proposed an absolute temperature
  scale defined by
• T(K) T(C) 273.15
• Expressed in absolute temperature, Charles law
  is

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Conceptual Question

• For which of the following changes is it not
  clear whether the volume of a particular sample
  of an ideal gas will increase or decrease?
• A) increase the temperature and increase the
  pressure
• B)increase the temperature and decrease the
  pressure
• C) increase the temperature and keep the pressure
  constant
• D) keep temperature constant and decrease the
  pressure
• E) decrease the temperature and increase the
  pressure

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Avogadros Law

• In 1811, Avogadro proposed that Equal volumes of
  gases at the same temperature and pressure
  contain equal numbers of molecules.
• It follows that the volume of a gas at constant
  temperature and pressure is proportional to
  number of moles.

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Avogadros Law
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Summary

• V ? 1 (constant T,n)
• P
• V ? T (constant P,n)
• V ? n (constant T,P)

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The Ideal Gas Equation

• Combining the gas laws gives

where R is called the Gas constant
The Ideal gas law is usually written as
PV nRT
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Standard Temperature and Pressure

• In order to compare two gases, we choose a
  standard temperature and pressure
• STP 0C and 1 atm
• What is the volume of a mole of gas at STP?

1 mol
273.15 K

1 atm
22.41 L
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Ideal Gas Calculations

• How many moles of N2 are in a 750 mL vessel at
  26C and 625 mm Hg?

0.750 L
V 750 mL
P 625 mm Hg
0.822 atm
T 26
273.15
299 K
0.750 L
0.822 atm

0.0251 mol
299 K
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Processes Involving Ideal Gases

• Lets describe a process by
• State 1 ? State 2
• or P1,V1,n1,T1 ? P2,V2,n2,T2
• State 1 P1V1 n1RT1 State 2 P2V2
  n2RT2
• Solving for R

If the number of moles stays constant (n1 n2)
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Processes Involving Ideal Gases

• 100 mL of He at 30 atm and 20C is expanded to
  1.0 atm at 20C. What is the final volume?

P1V1 P2V2
Since T1 T2,
30 atm
100 mL
3.0x103 mL
1.0 atm
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Processes Involving Ideal Gases

• A helium balloon has a certain volume at STP. At
  what temperature will the balloon have exactly
  half the volume at the same pressure?

Since P1 P2,
273.15 K
V1
-136.58?C
136.58 K
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Gas Density

• Different gases have the same volume at STP, but
  they have different masses.
• Define density ? d molar mass ? M
• Then mass nM dV

From the ideal gas law
Combining these equation
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Gas Density

• What are the densities of N2 and He at STP?

? 1 atm
N2

1.250 g/L
? 273.15 K
? 1 atm
He

0.1786 g/L
? 273.15 K
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Volumes of Gases in Chemical Reactions

• Gases are often products in chemical reactions
• The ideal-gas equation relates P, V, and T to
  number of moles of gas.
• The n can then be used in stoichiometric
  calculations
• Example The Chemistry of Air bags Å

If an air bag has a volume of 36 L and is to be
filled with nitrogen gas at a pressure of 1.15
atm at a temperature of 26.0 C, how many grams
of NaN3 must be decomposed?
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Gas Mixtures

• Consider a gas mixture with n1 moles of Gas 1, n2
  moles of Gas 2, etc.
• Partial pressure The pressure a gas would
  exert if it were the only gas present.

• Daltons Law The total
• pressure of a mixture is the
• sum of partial pressures

Pt P1 P2 P3 ...
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Gas Mixtures

• If the gases in a mixture behave ideally

...
Since Pt P1 P2 P3 ...
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Gas Mixtures

• What are the partial pressures of 2.00 g H2 and
  8.00 g N2 in a 10.0 L vessel at 273 K?

2.00 g H2
0.992 mol H2
8.00 g N2
0.286 mol N2
273 K
0.992 mol
2.22 atm
10.0 L
0.286 mol
0.641 atm
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Mole Fractions

• The ratio of partial pressure to total pressure
  can be expressed as

Mole fraction of Gas 1
X1
The mole fractions in a mixture must sum to 1
1
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Mole Fractions

• On a 25C day with 100 humidity, the mole
  fraction of H2O vapor is 0.031. What is the
  partial pressure of H2O?

24 mmHg
760 mmHg
0.031
What is the mole fraction of H2O if the relative
humidity is 60?
14 mmHg
24 mmHg
? 0.60
14 mmHg
0.018
760 mmHg
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Measuring Gases

• To measure the amount of gas produced in a
  reaction, it is often collected over water.
• Reaction of magnesium with HCl
• Mg(s) 2 HCl(aq) MgCl2(aq) H2(g)

Patm
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Measuring Gases

• The reaction of a sample containing Mg with
  excess HCl at 22?C and 753.2 mmHg yielded 1207 mL
  of gas. What was the mass of Mg?

22C PH2O
19.83 mmHg
PH2 Patm - PH2O 753.2 mmHg - 19.83
mmHg 733.4 mmHg
0.965 atm
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Measuring Gases

• 22C 273.15 295 K

1207 mL
1.207 L
0.965 atm
1.207 L

295 K
0.0481 mol H2
0.0481 mol Mg
1.17 g Mg
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Kinetic Molecular Theory

• Kinetic theory explains properties of gases based
  on a molecular view.
• The assumptions are
• The molecules are in continuous, random motion.
• A molecule has negligible volume.
• The forces between molecules are negligible.
• The average kinetic energy depends on the
  temperature.

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Kinetic Molecular Theory

• Kinetic molecular theory gives us an
  understanding of pressure and temperature on the
  molecular level.
• Pressure of a gas results from the number of
  collisions per unit time on the walls of
  container.

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Kinetic Molecular Theory

• Gas molecules have an average kinetic energy.
• As the temperature increases, the average
  kinetic energy of the gas molecules increases.

Distribution of of molecular speeds for nitrogen
gas at 0 C and 100 C
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Kinetic Molecular Theory

• As kinetic energy increases, the velocity of the
  gas molecules increases.
• Root mean square speed, u, is the speed of a gas
  molecule having average kinetic energy.
• Average kinetic energy, ?, is related to root
  mean square speed

e 1/2 mu2 m º mass M/NA
u º rms speed Å
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Average Molecular Speed
M º molar mass
Å
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Example N2 and He

• Calculate the average molecular speed for
  nitrogen and helium at 20?C.
• N2 M 28.02 g/mol

? 293 K
3

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Example N2 and He

• Calculate the average molecular speed for
  nitrogen and helium at 20?C.
• He M 4.003 g/mol

? 293 K
3

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Effusion

• The rate of effusion (in mol/s) is proportional
  to the average speed u.
• Effusion is the escape of gas through a small
  opening into a vacuum.

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Grahams Law

• If Gas 1 with molar mass M1 effuses at rate r1
  and Gas 2 with molar mass M2 effuses at rate r2

Grahams law

Å

• If a 5050 mixture of H2 and He undergoes
  effusion for a certain period, what is the
• composition of collected gas?

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Diffusion

• Diffusion is the spread of gas molecules
  throughout a volume. Å
• Diffusion is much slower than the average
  molecular speed because of collisions between
  molecules.

The mean free path is the average distance
between collisions.
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Real Gases

• No gas obeys the ideal gas equation exactly.
  Behavior becomes less ideal at high pressures...

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Real Gases

• ... and at low temperatures.

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The van der Waals Equation

• A more accurate description of gases was proposed
  by van der Waals

b is the molar volume of gas molecules.
a is a measure of the attractive force
between molecules.
a and b are different for each gas.
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The van der Waals Equation

• Calculate the pressure of 1.000 mole of CO2 in a
  3.000 L vessel at 0.0C using ideal gas and van
  der Waals equations.
• Ideal gas equation

1.000 mol ?
273.2 K

3.000 L
7.473 atm
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The van der Waals Equation

• CO2 a 3.59 L2atm/mol2 b 0.0427 L/mol

273.2 K
1.000 mol ?

3.000 L
- 1.000 mol
(1.000 mol)2
-
7.182 atm
(3.000 L)2