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THE PROPERTIES OF GASES

- A gas uniformly fills any container, is easily compressed and mixes completely with any other gas.
- Only four quantities define the state of a gas:

a. the quantity of the gas, n (in moles)

b. the temperature of the gas, T (in KELVINS)

c. the volume of the gas, V (in liters)

d. the pressure of the gas, P (in atmospheres)

PRESSURE

- A measure of the force that a gas exerts on its container.
- Force is the physical quantity that interferes with inertia.
- Gravity is the force responsible for weight.
- Newton’s 2nd Law: Force = m × a
- The units of force follow: N = kg × m/s2
- Pressure - Force ÷ unit area; N/m2

PRESSURE

- Standard Pressure
- 760.00 mm Hg
- 760.00 torr
- 1.00 atm
- 101.325 kPa ≈ 105 Pa
- The SI unit of pressure is the Pascal; 1 Pa = 1 N/m2

PRESSURE

- Barometer - measures gas pressure (especially atmospheric). 1 mm of Hg = 1 torr
- Manometer—a device for measuring the pressure of a gas in a container. The pressure of the gas is given by h [the difference in mercury levels] in units of torr (equivalent to mm Hg).

PRACTICE ONE

The pressure of a gas is measured as 49 torr. Represent this pressure in both atmospheres and pascals.

PRACTICE TWO

Rank the following pressures in decreasing order of magnitude (largest first, smallest last): 75 kPa, 300. torr, 0.60 atm, and 350. mm Hg.

THE GAS LAWS

- Boyle’s Law: V and P;

inversely proportional.

- Charles’ Law: T and V; directly proportional.
- Gay-Lussac’s Law: P and T; directly proportional.
- Avogadro’ Principle: moles and P or V; directly proportional.

BOYLE’S LAW

THE LAW: the volume of a confined gas is inversely proportional to the pressure exerted on the gas: P1V1= P2V2

- P ∝ 1/V plot = straight line

GOOD HABITS

EVERY TIME you do a gas laws problem:

- Write what you know and what you are trying to find
- Write the formula
- Plug in the numbers with units and solve with the correct number of sig figs.

PRACTICE THREE

Consider a 1.53L sample of gaseous SO2 at a pressure of 5.6 × 1O3 Pa. If the pressure is changed to 1.5 × 104 Pa at a constant temperature, what will be the new volume of the gas ?

PRACTICE FOUR

- Using the results listed below, calculate the Boyle’s law constant for NH3 at the various pressures.

Experiment Pressure (atm) Volume (L)

1 0.1300 172.1

2 0.2500 89.28

3 0.3000 74.35

4 0.5000 44.49

5 0.7500 29.55

6 1.000 22.08

PV vs. P

- What is the y-intercept? How about the 3rd graph on page two?
- Molar Volume of a gas: 22.42L

CHARLES LAW

- THE LAW: If a given quantity of gas is held at a constant pressure, then its volume isdirectly proportional to the absolute temperature.

V1T2 = V2T1

You must use the Kelvin!

K = °C + 273

CHARLES’ LAW

- Where do all the gases cross the x-intercept?
- If the volume is zero, what is the temperature?
- -273.15ºC or 0K

PRACTICE FIVE

A sample of gas at 15ºC and 1 atm has a volume of 2.58 L. What volume will this gas occupy at 38ºC and 1 atm ?

GAY-LUSSAC’S LAW

THE LAW: An increase in temperature increases the frequency of collisions between gas particles. In a given volume, raising the KELVIN temperature also raises the pressure.

P1 T2= P2T1

You must use Kelvin!

AVOGADRO’S LAW

Volume: 22.42L 22.42L22.42L

Mass: 39.95g 32.00g 28.02g

Quantity: 1 mol 1 mol 1 mol

Pressure: 1 atm 1 atm 1 atm

Temperature: 273K 273K 273K

AVOGADROS’S LAW

- The volume of a gas, at a given temperature and pressure, is directly proportional to the quantity of gas.
- Equal volumes of gases under the same conditions of temperature and pressure contain equal numbers of molecules.
- In gas law problems, moles is designated by an “n”.
- One mole of a gas has a volume of 22.42 L (dm3) at STP. It also has 6.02 x 1023 particles of that gas.

PRACTIVE SIX

Suppose we have a 12.2-L sample containing 0.50 mol oxygen gas (O2) at a pressure of 1 atm and a temperature of 25ºC. If all this O2 were converted to ozone (O3) at the same temperature and pressure, what would be the volume of the ozone ?

HINT

PTV

HINT

PVT

- Put the scientists' names in alphabetical order. Boyle’s uses the first 2 variables, Charles’ the second 2 variables and Gay-Lussac’s the remaining combination of variables.

COMBINED GAS LAW

From the Boyle’s, Charles’, and Gay-Lussac’s laws, we can derive the

CombinedGas Law:

P1V1 T2 = P2V2 T1

Mnemonic:Potato and Vegetable on top of the Table for P1V1 = P2V2

T1 T2

STANDARDS

T = 0°C = 273 K

V = 22.4 L (at STP)

P = 1.00 atm= 101.3 kPa

= 760.0 mm Hg = 760.0 torr

Remember only kPa has limited sigfigs.

IDEAL GAS LAW

Ideal Gas Equation:

PV = nRT

“R” is the universal gas constant.

V ∝ (nT)/P

replace ∝ with constant, R

UNIVERSAL GAS CONSTANTS

R = 0.08206 L• atm

mol • K

R = 62.36 L•mmHg

mol • K

R = 62.36 L • torr

mol • K

R = 8.314 L • kPa

mol • K

Why are there four constants?

IDEAL GAS LAW

Remember:

- Always change the temperature to KELVINS and convert volume to LITERS
- Check the units of pressure to make sure they are consistent with the “R” constant given or convert the pressure to the gas constant (“R”) you want to use.

PRACTICE SEVEN

A sample of hydrogen gas (H2) has a volume of 8.56 L at a temperature of 0ºC and a pressure of 1.5 atm. Calculate the moles of H2 molecules present in this gas sample.

PRACTICE EIGHT

Suppose we have a sample of ammonia gas with a volume of 3.5 L at a pressure of 1.68 atm. The gas is compressed to a volume of 1.35 L at a constant temp. Use the ideal gas law to calculate the final pressure.

PRACTICE NINE

A sample of methane gas that has a volume of 3.8 L at 5ºC is heated to 86ºC at constant pressure. Calculate its new volume.

PRACTICE TEN

A sample of diborane gas (B2H6) has a pressure of 345 torr at a temp. of -15ºC and a volume of 3.48 L. If conditions are changed so that the temp. is 36ºC and the pressure is 468 torr, what will be the volume of the sample?

PRACTICE ELEVEN

A sample containing 0.35 mol argon gas at a temp. of 13ºC and a pressure of 568 torr is heated to 56ºC and a pressure of 897 torr. Calculate the change in volume that occurs.

GAS STOICHIOMETRY

VOLUME 1 mol 22.42 L @ STP

1 mole 1 mole

PARTICLES MOLE MASS

6.02 x 1023 molar mass

Use the ideal gas law to convert quantities that are NOT at STP.

HINT

- You must have a balanced equation to do a stoichiometry problem.

PRACTICE TWELVE

Use PV = nRTto solve for the volume of one mole of gas at STP.

PRACTICE THIRTEEN

A sample of nitrogen gas has a volume of 1.75 L at STP. How many moles of N2 are present?

PRACTICE FOURTEEN

Calculate the volume of CO2 at STP made from the decomposition of 152 g CaCO3 by the reaction CaCO3(s) → CaO(s) + CO2(g).

PRACTICE FIFTEEN

A sample of methane gas having a volume of 2.80 L at 25ºC and 1.65 atm was mixed with a sample of oxygen gas having a volume of 35.0 L at 31ºC and 1.25 atm. The mixture was then ignited to form carbon dioxide and water. Calculate the volume of CO2 formed at a pressure of 2.50 atm and a temperature of 125ºC.

DETERMINING DENSITY

This modified version of the ideal gas equation can also be used to solve for the density of a gas.

PV = nRTbcomes D = PM

RT

DETERMINING DENSITY

- D = m = PMMor D = PMM

V RTRT

- The density of gases is g/L NOT g/mL.
- Mnemonic given in notes.

PRACTICE SIXTEEN

- What is the approximate molar mass of air?
- What is the approximate density of air?
- List 3 gases that float in air.
- List 3 gases that sink in air.

PRACTICE SEVENTEEN

The density of a gas was measured at 1.50 atm and 27ºC and found to be 1.95 g/L. Calculate the molar mass of the gas.

DALTON’S LAW OF PARTIAL PRESSURES

THE LAW: The pressure of a mixture of gases is the sum of the pressures of the different components of the mixture:

Ptotal= P1 + P2 + P3 +.....Pn

DALTON’S LAW OF PARTIAL PRESSURES

Also uses the concept of mole fraction, χ

χ A = moles of A

moles A + moles B + moles C + . . .

so now, PA = χ A / Ptotal

The partial pressure of each gas in a mixture of gases in a container depends on the number of moles of that gas. The total pressure is the SUM of the partial pressures and depends on the total moles of gas particles present, no matter what they are.

PRACTICE SEVENTEEN

For a particular dive, 46 L He at 25ºC and 1.0 atm and 12 L O2 at 25ºC and 1.0 atm were pumped into a tank with a volume of 5.0 L. Calculate the partial pressure of each gas and the total pressure in the tank at 25ºC.

PRACTICE EIGHTEEN

The partial pressure of oxygen was observed to be 156 torr in air with a total atmospheric pressure of 743 torr. Calculate the mole fraction of O2 present.

PRACTICE NINETEEN

The mole fraction of nitrogen in the air is 0.7808. Calculate the partial pressure of N2 in air when the atmospheric pressure is 760. torr.

WATER DISPLACEMENT

- It is common to collect a gas by water displacement which means some of the pressure is due to water vapor collected as the gas was passing through the water.
- You must correct for this.
- You look up the partial pressure due to water vapor in a table by knowing the temperature.

PRACTICE TWENTY

A sample of solid potassium chlorate (KClO3) was heated in a test tube (see the figure above) and decomposed by the following reaction: 2 KClO3(s) → 2 KCl(s) + 3 O2(g) The oxygen produced was collected by displacement of water at 22ºC at a total pressure of 754torr. The volume of the gas collected was 0.650 L, and the vapor pressure of water at 22ºC is 21torr. Calculate the partial pressure of O2 in the gas collected and the mass of KClO3 in the sample that was decomposed.

KINETIC MOLECULAR THEORY

Assumptions of the KMT Model:

- All particles are in constant, random motion.
- All collisions between particles are perfectly elastic.
- The volume of the particles in a gas is negligible.
- The average kinetic energy of the molecules is its Kelvin temperature.

KINETIC MOLECULAR THEORY

- This neglects any intermolecular forces as well.
- Gases expand to fill their container, solids/liquids do not.
- Gases are compressible; solids/liquids are not appreciably compressible.

KINETIC MOLECULAR THEORY

Boyle’s Law: If the volume is decreased, the gas particles will hit the wall more often, thus increasing pressure.

KINETIC MOLECULAR THEORY

Charles’ Law: When a gas is heated, the speed of its particles increase and thus hit the walls more often and with more force. The only way to keep the P constant is to increase the volume of the container.

KINETIC MOLECULAR THEORY

Gay-Lussac’s Law: When the temperature of a gas increases, the speeds of its particles increase, the particles are hitting the wall with greater force and greater frequency. Since the volume remains the same this would result in increased gas pressure.

KINETIC MOLECULAR THEORY

Avogadro’s Law:An increase in the number of particles at the same temperature would cause the pressure to increase if the volume were held constant. The only way to keep constant P is to vary the V.

DISTRIBUTION OF MOLECULAR SPEED

- Although the molecules in a sample of gas have an average KE (and therefore an average speed), the individual molecules move at various speeds and they stop and change direction according to the law of density measurements and isolation → they exhibit a distribution of speeds.
- Some move fast, others relatively slowly.
- Collisions change individual molecular speeds but the distribution of speeds remains the same.

DISTRIBUTION OF MOLECULAR SPEED

Maxwell’s equation:

- Urms means root mean square velocity which is the measure of the average velocity of particles in a gas.
- Use the “energy R” or molar gas constant, 8.314 J/K• mol for this equation since kinetic energy is involved.

DISTRIBUTION OF MOLECULAR SPEED

- By taking the root of the square of the average velocities, you can acquire the average speed of gaseous particles.
- The root-mean-square velocity takes into account both molecular mass and temperature, two factors that directly affect the KE of a material.
- What happens if we change to a gas that has a higher MM?
- What happens if we lower the temperature?

PRACTICE TWENTY-ONE

Calculate the root mean square velocity for the atoms in a sample of helium gas at 25ºC.

DISTRIBUTION OF MOLECULAR SPEED

- If we could monitor the path of a single molecule it would be very erratic.
- Mean free path—the average distance a particle travels between collisions. It’s on the order of a tenth of a micrometer - very small.
- Examine the effect of temperature on the numbers of molecules with a given velocity as it relates to temperature. They heat up, they speed up.

DISTRIBUTION OF MOLECULAR SPEED

- Drop a vertical line from the peak of each of the three bell shaped curves — that point on the x-axis represents the AVERAGE velocity of the sample at that temperature.
- Note how the bells are “smashed” as the temperature increases.

the Maxwell-Boltzmann Distribution which describes particle speeds of gases.

GRAHAM’S LAW OF EFFUSIONAND DIFFUSION

- Effusion is closely related to diffusion.
- Diffusion is the term used to describe the mixing of gases. The rate of diffusion is the rate of the mixing.
- Effusion (pictured at left)is the term used to describe the passage of a gas through a tiny orifice into an evacuated chamber as shown on the right.
- The rate of effusion measures the speed at which the gas is transferred into the chamber.

GRAHAM’S LAW OF EFFUSIONAND DIFFUSION

Graham's Law of Effusion: The rates of effusion of two gases are inversely proportional to the square roots of their molar masses at the same temperature and pressure.

If two bodies of different masses have the same kinetic energy, the lighter body moves faster.

GRAHAM’S LAW OF EFFUSIONAND DIFFUSION

- REMEMBER rate is a change in a quantity over time, NOT just the time!
- If they give you time, divide the time into 1 to get the rate.

PRACTICE TWENTY-TWO

Calculate the ratio of the effusion rates of hydrogen gas (H2) and uranium hexafluoride (UF6), a gas used in the enrichment process to produce fuel for nuclear reactors.

PRACTICE TWENTY-THREE

A pure sample of methane is found to effuse through a porous barrier in 1.50 minutes. Under the same conditions, an equal number of molecules of an unknown gas effuses through the barrier in 4.73 minutes. What is the molar mass of the unknown gas?

REAL vs. IDEAL GASES

- Most gases behave ideally until you reach high pressure and low temperature. (Remember, either of these can cause a gas to liquefy)
- Under very high pressure, real gases have trouble compressing completely. The ideal gas law fails. Ideal gases have no volume, but real gases do.

van der Waals EQUATION

- corrects for negligible volume of molecules and accounts for inelastic collisions leading to intermolecular forces.
- Pressure is increased (IMFs lower real pressure, you’re correcting for this)
- Volume is decreased (corrects the container to a smaller “free” volume).

a and b are van der Waals constants.

INTERPRETATION

- When PV / nRT= 1.0, the gas is ideal.
- All of these are at 200K.
- Note the pressures where the curves cross the dashed line [ideality].

INTERPRETATION

- This graph is just for nitrogen gas.
- Note that although non-ideal behavior is evident at each temperature, the deviations are smaller at the higher temperature.

THE AP EXAM

- Don’t underestimate the power of understanding these graphs.
- AP loves to ask questions comparing the behavior of ideal and real gases - not an entire free-response gas problem on the real exam.
- Gas Laws are tested extensively in the multiple choice since it is easy to write questions involving them! You will most likely see PV = nRTas one part of a problem in the free response, just not a whole problem!

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