Atmospheric Pressure

Fig 10.1 pg. 401

The Characteristics of Gases

• There are three phases for all substances: solid, liquid, and gases.
• Gases are highly compressible and occupy the full volume of their containers.
• When a gas is subjected to pressure, its volume decreases.
• Gases always form homogeneous mixtures with other gases.
• Gases only occupy about 0.1 % of the volume of their containers.

Pressure

Atmospheric Pressure and the Barometer

• Pressure is the force acting on an object per unit area:

• Gravity exerts a force on the earth's atmosphere.
• A column of air 1 m² in cross section exerts a force of 10⁵ N.
• The pressure of a 1 m² column of air is 100 kPa.
• SI Units: 1 N = 1 kgm/s²; 1 Pa = 1 N/m².

Torricelli Barometer

Fig. 10.2 pg. 402

Pressure

Atmospheric Pressure and the Barometer

• Atmospheric pressure is measured with a barometer.
• If a tube is inserted into a container of mercury open to the atmosphere, the mercury will rise 760 mm up the tube.
• Standard atmospheric pressure is the pressure required to support 760 mm of Hg in a column.
• Units: 1 atm = 760 mm Hg = 760 torr = 1.01325 x 10⁵ Pa = 101.325 kPa

Open-End Manometer

Fig. 10.3 pg. 403

Pressure

Pressures of Enclosed Gases and Manometers

• The pressure of gases not open to the atmosphere are measured in manometers.
• A manometer consists of a bulb of gas attached to a U-tube containing Hg.
• If the U-tube is closed, then the pressure of the gas is the difference in height of the liquid (usually Hg).
• If the U-tube is open to the atmosphere, a correction term needs to be added:
  1. If P_gas < P_atm then P_gas + P_h2 = P_atm.
  2. If P_gas > P_atm then P_gas = P_atm + P_h2.

Boyle's Law

Fig. 10.6 pg. 405

Gas Laws

The Pressure-Volume Relationship: Boyle's Law

• Mercury is added to U-tube with both ends open to atmosphere. since both ends now experience same pressure mercury level is equal in both columns.
• Now left-hand column is sealed at the top, giving rise to picture shown in (a). Gas trapped in left-hand column above mercury is at atmospheric pressure.
• Next additional mercury is added to right-hand column until gas volume in left-hand column drops to half original value. Since volume has dropped in halp pressure must be doubled (PV = const) to about 2 atm.
• Gas pressure in left-hand column (approx. 2 atm) must equal atmospheric pressure pushing down on right-hand column plus pressure from excess mercury height in right-hand column.
• Since atmosphere pushes down on right-hand column with about 1 atm of pressure, excess mercury must exert remainder of pressure, ie. about 1 atm. Therefore 760 mm of excess mercury column height corresponds to (exerts) 1 atm of pressure.
• 1 mm of excess mercury height known as 1 Torr (in honor of Torricelli) of pressure. "Standard" atmospheric pressure chosen by convention to be 760 Torr or mm Hg of pressure.

Volume and Pressure

Fig. 10.7 pg. 405

Gas Laws

The Pressure-Volume Relationship: Boyle's Law

• Boyle's Law: the volume of a fixed quantity of gas is inversely proportional to its pressure.
• Mathematically:

• A plot of V versus P is a hyperbola.
• Similarly, a plot of V versus 1/P must be a straight line passing through the origin.
• Working of the lungs:
  • As we breathe in, the diaphragm moves down and the ribs expand. Therefore, the volume of the lungs increases.
  • By Boyle's law, when the volume of the lungs increases, the pressure decreases. Therefore, the pressure inside the lungs is less than atmospheric pressure.
  • Therefore, air is forced into the lungs.
  • As we breathe out, the diaphragm moves up and the ribs contract. Therefore, the volume of the lungs decreases.
  • By Boyle's law, the pressure decreases and air is forced out.

P-V Relationships

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P-V Relationships

• As the pressure on a gas increases at constant temperature, the volume decreases.
• Boyle's Law: pressure and volume are inversely proportional. I.e. the product of pressure and volume for a given amount of gas at a constant temperature is a constant.
• Thus, the plot of Pressure versus Volume is hyperbolic.
• The plot of Pressure versus 1/V is linear.

Volume and Temperature

Fig. 10.9 pg. 406

Gas Laws

The Temperature-Volume Relationship: Charles's Law

• We know that hot air balloons expand when they are heated.
• Charles's Law: the volume of a fixed quantity of gas at constant pressure increases as the temperature increases.
• Mathematically:

• A plot of V versus T is a straight line.
• When T is measured in ^oC, the intercept on the temperature axis is
• -273.15^oC.
• We define absolute zero, 0 K = -273.15^oC.
• Note: the value of the constant reflects assumptions about the amount of gas and pressure.

Law of Combining Volumes

Fig. 10.10 pg. 407

Gas Laws

The Quantity-Volume Relationship: Avogadro's Law

• Gay-Lussac's Law of combining volumes: at a given temperature and pressure, the volumes of gases that react are ratios of small whole numbers.
• Avogadro's Hypothesis: equal volumes of gas at the same temperature and pressure will contain the same number of molecules.
• Avogadro's Law: the volume of gas at a given temperature and pressure is directly proportional to the number of moles of gas.
• Mathematically:

• We can show that 22.4 L of any gas at 0^oC contains 6.02 x 10²³ gas molecules.

Electrolysis of Water

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Electrolysis of Water

• When an electric current is passed through water, the water decomposes into hydrogen and oxygen.
• Hydrogen is generated at the negative electrode.
• Oxygen is generated at the positive electrode.
• The volumes of hydrogen and oxygen give the proportions of H and O in water.
• The volume of hydrogen is twice the volume of oxygen.
• Therefore, in water molecules there are two H atoms and one O atom per molecule of water.

Gas Stoichiometry Tables

Text slide.

Gas Stoichiometry Tables

Nitrogen and hydrogen gases react to form ammonia gas according to the reaction shown below. At a certain temperature and pressure, 1.2 L of N₂ reacts with 3.6 L of H₂. If all N₂ and H₂ consumed, what volume of NH₃ produced at same temperature and pressure?

..........................N₂(g) + 3 H₂(g) Æ 2 NH₃(g)

volume	1.2	3.6	.
coefficient	1	3	2
vol equiv's	.	.	.

• Build stoichiometry table around balanced reaction.
• Notice grams and MM unnecessary because volume given.
• Volume can take the place of moles because volume proportional to moles with gases.
• Extra columns for "other" info if necessary (not in this problem).

Ideal Gas Law

Text slide.

Ideal Gas Law

• Ideal gas law relates P, V, T in all cases without having to memorize different gas laws for Boyle's, Charles', or Avogadro's Law problems.
• PV = nRT, where:
  • P = pressure in atmospheres only
  • V = volume in liters only
  • n = gas quantity in moles only
  • R = gas constant, 0.0821 L atm/mol_K, and
  • T = temperature in degrees Kelvin only.
• In some problems all but one variable given. In these first convert all values to proper units, then solve for unknown with PV = nRT.
• In other problems (ie. problem 10.19 pg. 399) you given only a couple variables which change values under two different sets of conditions. Write ideal gas law, scratch out variables which same in both sets of conditions, and write gas law as proportionality, ie. V µ T, or V = CT, where C is a constant. Now solve for C using first set of values, and use this value of C to solve unknown variable under new conditions.

Density and Molar Mass

Text slide.

Density and Molar Mass

• Because given volume (ie. 1 liter) of gas contains a set number of gas molecules (or atoms) the heavier one gas molecule is the heavier the number of molecules contained in one liter of gas is. Mass of one liter of gas is its density in g/L. Density is proportional to molar mass.
• To do calculations of this type choose one liter volume of gas and replace the word "density" with the word "mass" throughout the problem. Use PV = nRT and plug in all known values, using correct units, and remembering that V = 1 liter.
• Example: Calculate the density of SF₆ vapor at 455 Torr and 32 ^oC.
• Example: Calculate molar mass of a gas with density 6.345 g/L at 22 ^oC and 743 Torr.

Collection of Gas Over Water

Fig. 10.16 pg. 419

Gas Mixtures and Partial Pressures

• Since gas molecules are so far apart, we can assume they behave independently.
• Dalton's Law: in a gas mixture, the total pressure is given by the sum of partial pressures of each component:

• Each gas obeys the ideal gas equation.
• Let n_i be the number of moles of gas i exerting a partial pressure P_i, then

where is the mole fraction (n_i/n_t).

Collecting Gases over Water

• It is common to synthesize gases and collect them by displacing a volume of water.
• To calculate the amount of gas produced, we need to correct for the partial pressure of the water:

Partial Pressures

Text slide.

Partial Pressures

• Pressures of different gases in a mixture are additive. Total pressure of gas mixture is sum of pressures of individual component gases; these component gas pressures called partial pressures.
• To work partial pressure problems first work out number of moles of each component or mole fraction of each component gas.
• Ratio of partial pressure of a gas to total pressure is equal to mol fraction of gas (moles of individual gas/total moles) in mixture.
• Example: Mixture contains 3.15 g each of CH₄, C₂H₄, and C₄H₁₀ in 2.00-L flask at 64 ^oC. Calculate partial pressure of each gas and total pressure.

Distribution of Molecular Speeds

Fig. 10.18 pg. 421

Kinetic-Molecular Theory

• Gas molecules have an average kinetic energy.
• Each molecule has a different energy.
• There is a spread of individual energies of gas molecules in any sample of gas.
• As the temperature increases, the average kinetic energy of the gas molecules increases.
• 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:

Distribution of Molecular Speeds

Fig. 10.19 pg. 423

Molecular Effusion and Diffusion

• Average kinetic energy of a gas is related to its mass:

• Consider two gases at the same temperature: the lighter gas has a higher rms than the heavier gas.
• Mathematically:

• The lower the molar mass, M, the higher the rms for that gas at a constant temperature.

Kinetic Energy in a Gas

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Kinetic Energy in a Gas

• At any temperature, the molecules in a gas are moving.
• Some molecules move faster than others.
• As the temperature increases, the average kinetic energy of the gas molecules increases.
• Notice from the graphs that there is a spread of speeds for individual gas molecules.
• As the temperature increases, the average speeds of the molecules increase.
• Consider two gases with two different molecular masses (He and Ne).
  • At the same temperature, the two gases have the same average kinetic energy.
  • However, since Ne is more massive than He, the Ne molecules move more slowly.
  • That is, there is a greater spread to high speeds for He than for Ne.
  • As temperature increases, the average speeds of He and Ne increases.

Pressure

Fig. 10.17 pg. 421

• Kinetic-Molecular Theory
• Theory developed to explain gas behavior.
• Theory of moving molecules.
• Assumptions:
  1. Gases consist of a large number of molecules in constant random motion.
  2. The volume of individual molecules is negligible compared to the volume of the container.
  3. Intermolecular forces (forces between gas molecules) are negligible.
  4. Energy can be transferred between molecules, but total kinetic energy is constant at constant temperature.
  5. The average kinetic energy of molecules is proportional to temperature.
• Kinetic molecular theory gives us an understanding of pressure and temperature on the molecular level.
• The pressure of a gas results from the number of collisions per unit time on the walls of container.
• The magnitude of pressure is given by how often and how hard the molecules strike.

Effusion of Gas

Fig. 10.20 pg. 424

Molecular Effusion and Diffusion

Graham's Law of Effusion

• Effusion is the escape of a gas through a tiny hole (a balloon will deflate over time due to effusion).
• The rate of effusion can be quantified.
• Consider two gases with molar masses M₁ and M₂, the relative rate of effusion is given by

• Only those molecules that hit the small hole will escape through it.
• Therefore, the higher the rms, the more likelihood of a gas molecule hitting the hole.
• We can show

Diffusion of Gas

Fig. 10.22 pg. 426

Molecular Effusion and Diffusion

Diffusion and Mean Free Path

• Diffusion of a gas is the spread of the gas through space.
• Diffusion is faster for light gas molecules.
• Diffusion is significantly slower than rms (consider someone opening a perfume bottle: it takes while to detect the odor but rms at 25^oC is about 1150 mi/hr).
• Diffusion is slowed by gas molecules colliding with each other.
• Average distance of a gas molecule between collisions is called mean free path.
• At sea level, mean free path is about 6 x 10^-6 cm.

Graham's Law Problems

Text slide.

Graham's Law

• v₂/v₁ = t₁/t₂ = (MM₁/MM₂)^1/2 or MM₂/MM₁ = (v₁/v₂)² = (t₂/t₁)²
• To work problems you need to convert a molar mass ratio to a velocity ratio or vice versa depending on what the problem gives you. A time ratio (how long does it take...) is just the reciprocal of a velocity ratio.
• Best way to calculate ratios is to work with all ratios greater than one (numerators greater than denominators) and remember that heavier gas molecules move slower than light gas molecules (take more time to escape through openings, etc.).
• To go from speed or time ratio to molar mass ratio square the speed ratio, and to go from molar mass ratio to speed (time) ratio take the square root. To remember this remember T µ 1/2 mv²; v is squared in this formula and velocity ratio is squared to calculate mass ratio also.
• Example: What is MM of a gas that takes 105 s to escape through an opening that takes 31 s for equal amount of O₂ to escape through?

Real Gases

Fig. 10.23 pg. 427

Real Gases: Deviations from Ideal Behavior

• From the ideal gas equation,

• For 1 mol of gas, PV/RT = 1 for all pressures.
• In a real gas, PV/RT varies from 1 significantly.
• The higher the pressure, the more the deviation from ideal behavior.

Real Gases

Fig. 10.24 pg. 428

Real Gases: Deviations from Ideal Behavior

• From the ideal gas equation,

• For 1 mol of gas, PV/RT = 1 for all temperatures.
• As temperature increases, the gases behave more ideally.
• The assumptions in kinetic molecular theory show where ideal gas behavior breaks down:
  1. the molecules of a gas have finite volume;
  2. molecules of a gas do attract each other.