Unit 7: Gas Behavior

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Gas Pressure and Kinetic Molecular Theory

Assumptions of Kinetic Molecular Theory

1. Gases are composed of atoms or molecules that can be considered to be widely spaced points without volume

2. Gas molecules move randomly in all directions, with a velocity that is related to their temperature

3. Collisions between gas molecules or with objects are perfectly elastic; no energy is lost due to interactions between particles or with their container

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★Pressure is the force exerted on an area (like the walls of a container or a liquid surface) by gas molecules as they move in random directions and collide with it

★The atmosphere exerts enough downward force to push a column of mercury exactly 760mm upward

3 Gas Pressure and Kinetic Molecular Theory

How “gas laws” follow from Kinetic Molecular Theory

1. As molecules move faster at higher temperatures, they have more kinetic energy so they collide with the walls of their container with greater force. That force is pressure.

2. As gas molecules exert more force on the walls of their container, the container expands outward from that force and increases its volume until the inward pressure on the walls is equal.

4 Gas Pressure and Kinetic Molecular Theory

Boyle’s LawPressure and volume are inversely related: Boyle’s Experiment

★ A sample of air is sealed in one side of a U-shaped tube by pouring mercury into the other side

★ As mercury is poured to a height that rises above the air on the other side, the air compresses

★ The volume of the air sample and the height of the mercury above it are measured to find the relationship between pressure against the gas and volume of the gas

Robert William Boyle (1627-1691)

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Boyle’s LawPressure and volume are inversely related: Boyle’s Experiment

Constant amount of gas at constant

temperature

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Boyle’s LawPredicted by Kinetic Molecular Theory

★ Consider pressure: it’s the force exerted by randomly moving molecules colliding with the walls of their container

★ As the container gets smaller (volume decreases) there are more molecules colliding per area, so the pressure increases because there’s more force exerted on the same area

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Boyle’s LawExample: Using Boyle’s Law

To predict pressure-volume relationships at constant temperatureYou drink a 250mL bottle of water on a plane at cruising altitude where the cabin pressure is 0.840 atm, and you close the bottle. What will its volume be when you land at seal level, where the pressure is 1.00 atm?

At cruising altitude At sea level

The bottle will actually crush to a volume of 210mL when you land. Try it!

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Charles’s LawTemperature and volume are directly related

★ Unpublished qualitative observation that was later attributed to him by Joseph Gay-Lussac

Jacque Charles (1746-1823)

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Charles’s LawTemperature and volume are directly related: Gay-Lussac’s Experiment

1) Gay-Lussac filled a balloon with known volumes of gas by submerging a constant-volume glass bell with a stopcock in water and letting atmospheric pressure on the water force the air out through the top of the bell

2) Placed the balloon in a water bath, heated the bath, and measured the volume change as the balloon expanded and displaced more water

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Gay-Lussac’s LawTemperature and pressure are directly related: Gay-Lussac’s Experiment

Joseph Luis Gay-Lussac (1778-1850)

★Studies on the relationship between temperature and pressure were based on behavior of “air thermometers”

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Gay-Lussac’s LawTemperature and pressure are directly related

1) Gay-Lussac filled a balloon with known volumes of gas by submerging a constant-volume glass bell with a stopcock in water and letting atmospheric pressure on the water force the air out through the top of the bell

2) Placed the balloon in a cage in a water bath, heated the bath, and measured the pressure change by observing how high it pushed a column of mercury in a tube

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Charles’s and Gay-Lussac’s LawsPredicted by Kinetic Molecular Theory

★ Volume should be related to temperature at constant pressure: As molecules move faster at higher temperatures, their force on the walls of the container should make the container expand to keep the forces from the inside and outside equal

★ Pressure should be related to temperature at constant volume: If the walls of the container don’t expand to spread the force of gas particle collisions out as they get more violent, then the force per area will increase

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Absolute TemperatureAn important implication of Charles’s and Gay-Lussac’s Laws:

There’s no pressure if there’s no thermal energy

y=302.92x- 275.73

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0 0.2 0.4 0.6 0.8 1 1.2 1.4

Tempe

rature,D

egreesC

Pressure,Atm

Figure4:Temperaturevs.PressureforAir

★Extrapolating to zero pressure gives a temperature where gases lose the property of “pressure” because they have no thermal energy so they don’t exert force on the container. ★Absolute zero is -273.150C, which is the basis of

the Kelvin scale of temperature

Real student laboratory data was used to extrapolate to absolute zero. The data produced a result within a few degrees of the accepted value for absolute zero.

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The Combined Gas LawCombining Boyle’s, Charles’s, and Gay-Lussac’s Laws

★ Pressure is inversely proportional to volume:

★ Volume is directly proportional to temperature:

★ Pressure is directly proportional to temperature:

★ Therefore:

So for a constant amount of gas:

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The Combined Gas LawExample: Using the Combined Gas Law

To predict pressure, volume, or temperature when they all change

A weather balloon is filled with 1000L of helium at sea level, where the pressure is 760mm Hg (1.00 atm) and the temperature is 220C (295K), and it rises to the edge of the troposphere where the temperature is -600C (213K) and the pressure is 150mm Hg (0.197 atm). What is its volume at the edge of the troposphere?

At release At high altitude

You would need a balloon with more than

3670L volume to not burst

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Avogadro’s LawVolume is directly proportional to the number of gas particles

★Didn’t actually perform his own experiments - just interpreted the results of others brilliantly

★Published a comment on the implications of other scientists’ work in 1810

★Avogadro’s hypothesis implied that the same number of particles of any gas will have the same volume

★We now know that the molar volume of any gas is very close to 22.4L at 00C (273.15K) and 1 atm

Amedeo Avogadro (1776-1856)

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Avogadro’s LawSince volume is proportional to moles of gas,

Stoichiometric coefficients can also be used to express volume ratios at a given temperature and pressure

★There are 5L of O2 and 1L of propane in the reactants ★There are 3L of CO2 and 4L steam in the products

C3H8(g) + 5O2(g) → 3CO2(g) + 4H2O(g)Consider:

If there are 6L of reactant gases and the pressure is held constant, then

Notice that there is more gas volume in the products, meaning the volume must have expanded! We’ll discuss the implications of that in a later unit.

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Avogadro’s Law

Combined gas law

Avogadro’s Law (Molar volume is constant for all gases at a

given temperature and pressure)

Now rename the constant “R” and get rid of the division

PV=nRTIdeal gas law

R= 0.08206 L•atmmol•K

Universal Gas Constant

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Implications lead to the “ideal gas” law

Dalton’s Law

★ If volume is proportional to number of gas particles the same way for all gases and pressure is proportional to volume, then a mixture of gases can be treated as one gas and the total pressure of the mixture can be treated as the sum of the “partial pressures” of each gas:

Pt = X1P1 + X2P2 + … + XnPn

Total Pressure “Partial pressure” of a given gas in the

mixture

Percentage of total number of

molecules or “mole fraction” of the given gas

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All gases follow the same ideal gas law: Leads to Dalton’s Law of Partial Pressures

Gas-Phase Chemistry★ Gases can be “collected over water” in a eudiometer to measure how

much gas is produced by a chemical reaction

1) The cylinder (eudiometer) is filled with water and inverted in the water bath

2) Gases from the reaction are collected by bubbling into the eudiometer, where they displace water

3) After the reaction, the eudiometer is submerged so the water level is even with the level of the water bath. That means that the pressure of the gas inside is the same as atmospheric pressure.

4) The partial pressure of water in the gas sample (which only varies with temperature) is subtracted so only the number of moles of the gas from the reaction and not the water vapor is quantified in the calculations

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Using Dalton’s Law and the Ideal Gas Law

Gas-Phase ChemistryExample: collecting gases over water

The following reaction was performed in the laboratory, and product gases were collected over water in a eudiometer:

2Na(s) + 2H2O(l) → 2NaOH(aq) + H2(g)An unknown amount of sodium produced 435mL of gases at 743.00 mm Hg atmospheric pressure and 20.00C (at which the partial pressure of water vapor is 17.54 mm Hg). What mass of sodium was in the test tube?

PV=nRT

Calculate the pressure of ONLY H2 in the eudiometer by subtracting out the water vapor and converting to atmospheres

Solve for “n” which is the number of moles of hydrogen gas produced

Set up the equation and substitute in known values for variables

Use the stoichiometry of the chemical equation to find how many moles of sodium were reacted, and convert to grams

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Real Gas BehaviorKinetic Molecular Theory has some incorrect assumptions

1. Gas particles are matter, and matter has volume. Assuming that the widely spaced gas particles are points with no volume is not entirely accurate.

2. Gas particles do, in fact, still move in random directions with a kinetic energy that is related to their temperature. This assumption remains accurate.

3. The collisions between particles or with their container are NOT perfectly elastic because gas particles are attracted to each other electrostatically by “intermolecular forces” that will be explained in detail later in your study of chemistry.

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Real Gas BehaviorThe Van der Waals Equation accounts for “excluded volume” of

molecules and intermolecular attraction

★ The term “n/V” is the number density of gas particles (moles per liter), and it is squared to account for interaction occurring between two particles. The constant “a” is unique to each gas and expresses a degree of interaction between its molecules.

★ The constant “b” is the volume taken by each mole of molecules themselves, unique to each gas, and is multiplied by “n” or the number of moles of molecules to correct for their volume.

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Real Gas BehaviorWhen does the ideal gas approximation fail?

1. At very high pressures, the gas molecules are so close together that the volume of the particles themselves (excluded volume) becomes a significant fraction of the volume that they occupy so the “volume” is effectively lower. They no longer behave like “widely separated points with no volume.”

2. At very low temperatures, the gas particles have so little kinetic energy that they no longer overcome the attractive forces between them, so they are attracted to each other rather than the walls of the container. Pressure becomes lower than would be predicted by the ideal gas approximation.

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Kinetic Energy and Temperature: Diffusion and Effusion

Temperature is a measure of kinetic energy of particles

★In physics, kinetic energy is related to the mass and velocity of an object:

★The “average” velocity of gas particles is expressed as their “root mean square” velocity, which is related to temperature by “Maxwell-Boltzmann statistics” that are the physical origin of the Universal Gas constant

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Kinetic Energy and Temperature: Diffusion and Effusion

★ Diffusion: Gas molecules moving randomly to eventually fill their volume uniformly • Example: someone’s fragrance fills a room quickly if

they’re wearing a lot of it ★ Effusion: Gas molecules moving through an aperture from a

volume at high pressure to a volume at low pressure to fill it uniformly • Example: gas escapes from a balloon into the outside

environment ★ Rates of diffusion and effusion are derivable from the

definitions of kinetic energy and root mean square velocity of gas particles:

Graham’s Law of Diffusion and Effusion

Thomas Graham (1805-1869)

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Kinetic Energy and Temperature: Diffusion and Effusion

Example: Using Graham’s LawIf a helium-filled balloon effuses to emptiness in 8 hours, then how long would it take for a nitrogen-filled balloon to reach the same point?

Set up the equation and substitute in known values for variables. Rate is the inverse of time, so Graham’s Law can be arranged to use time instead of rate.

Substitute known values for variables. “Gas 2” is helium (He) and “gas 1” is nitrogen (N2) so its effusion time can be solved as “time1”

time1 = 21.2 hrs According to Graham's Law; if a helium balloon effuses to emptiness in 8 hours, then a nitrogen balloon will take 21.2 hours to reach the same level of emptiness, given the atomic or molecular weights of both gases. 28