GasesKinetic Theory of Gases,

Gas Laws, Ideal Gases, Partial Pressures,

Graham’s Law of Effusion

Characteristics of Gases

• HONClBrIF: diatomic gases• Noble gases are monoatomic.• Many molecular compounds with low molar masses are gases at R.T.

• Gases form homogeneous mixtures regardless of gas types.

• Substances that are liquids or solids at R.T. are called “vapors”. Ex: water vapor

Kinetic-Molecular Theory of Gases

• A gas is composed of particles (molecules or atoms).

• Gas particles are considered to be small, hard spheres (billiard balls).

• Particles have insignificant volume compared to total volume occupied by the gas.

• Particles are far apart from each other.

• Between the particles is empty space.

Kinetic Theory of Gases, Cont.

• No attractive or repulsive forces exist between particles.

• Particles move rapidly in constant random motion (random walk).

• Movement is independent; particles travel in straight paths until a collision.

• Collisions between gas particles are perfectly elastic. (Newton’s Cradle). This is true at constant temperature and low to moderate pressure.

Kinetic Theory of Gases, Part III

• Average kinetic energy (EK) of a gas molecule is proportional to the absolute temperature (K). If K doubles, EK doubles.

• At any given temperature, all gases have the same average EK.

• At a fixed temperature, the average speed of a lighter gas is greater than the average speed of a heavier gas.

Pressure• Gas pressure: the force exerted by a gas per unit surface area of an object.

• P = F/A• Gas pressure is the result of simultaneous collisions of billions of rapidly-moving particles with an object.

• Vacuum: no gas particles, no pressure

Atmospheric Pressure: the collision of air molecules

with objects.

Atmospheric Pressure

Units of Pressure

• SI Unit is the pascal (Pa); kPa is used more frequently.

• Other units include: mmHg, bar, torr, psi, atmosphere (atm)

• 1 atm = 760 mmHg = 760 torr = 101.3 kPa

• A barometer measures air pressure.

Standard Atmospheric Pressure (STP)

•STP is the typical pressure at sea level

•STP is the pressure needed to support a column of mercury

760 mm high.•STP for a gas is 0°C and 1 atm.

The Manometer• A manometer is sometimes used in the laboratory to measure gas pressures near atmospheric pressure.

• Operates on a principle similar to that of a barometer.

• Interesting fact: a blood pressure cuff is a sphygmomanometer.

A Manometer Calculation

• A gas sample is in a flask attached to an open-end manometer in a room with atmospheric pressure of 764.7 torr. The mercury level in the open-arm end of the manometer has a height of 136.4mm, and that is the arm in contact with the gas has a height of 103.8mm. What is the gas pressure in atm?

Solution to Manometer Calculation

• Pressure of gas > atmospheric pressure. (How do we know this?)

• Pgas = Patm + Ph

• The gas pressure equals the atmospheric pressure plus the difference in height between the two manometer arms.

• Pgas = 764.7 torr + (136.4 torr - 103.8 torr) = 797.3 torr = 1.049 atm

The Gas Laws• Four variables are needed to define the condition of a gas:

• Temperature (T), always in K• Pressure (P), in various units (kPa, atm)

• Volume (V), usually L or mL• Number of gas moles (n), in mol• The gas laws describe the relationships between these variables.

Boyle’s Law, 1662

• Pressure-Volume Relationship

• At constant n and T, P 1/V or PV = constant as P, V

• For use in calculations, P1V1 = P2V2

• Example of Boyle’s Law: breathing

Boyle’s Law Graph: Volume vs.

Pressure

A Boyle’s Law Calculation

• A fixed quantity of gas at 23.0C has a pressure of 748 torr and occupies a volume of 10.3 L. Calculate the volume (L) if the pressure increases to 1.88 atm and the temperature remains constant.

• 5.39 L

Another Boyle’s Law Calculation

• A gas with a volume of 4.0 L at 90.0 kPa expands until the pressure drops to 20.0 kPa. What is the new volume if the temperature doesn’t change?

• 18 L

Charles’ Law, 1787

• Temperature-Volume relationship

• At constant n and P:

T V as T, V

• For use in calculations,

V1/T1 = V2/T2

Charles’ Law Graphs:

Volume vs. Temperature

Charles’ Law and Kelvin Scale

• We get a straight-line graph of Charles’ Law data.

• Zero volume equals -273.15C.

• Lord Kelvin creates the absolute scale where 0K = -273.15C.

• WE MUST USE KELVIN SCALE FOR CHARLES’ LAW CALCS!

A Charles’ Law Calculation

• We have a fixed quantity of gas at 23.0C and a pressure of 748 torr. The gas volume is 10.3 L. If the temperature rises to 165.0C, what will the new volume be if the gas pressure remains constant?

• Volume = 15.2 L

Another Charles’ Law Calculation

• A gas with a volume of 300. mL at 150.0C is heated until its volume is 600. mL. What is the new temperature of the gas if the pressure remains constant during the heating process?

• 846 K

Gay-Lussac’s Law, 1802

• Temperature-Pressure relationship

• At constant n and V,

TP as T, P

• Temperature must be in Kelvin• For use in calculations,

P1/T1 = P2/T2

Example: pressure cooker

A Gay-Lussac’s Law Calculation

• The gas in a used aerosol can is at a pressure of 103 kPa at 25.0C. If David throws the can into a fire, what will the gas pressure be when the temperature reaches 928.0C?

• Pressure = 415 kPa

Another Gay-Lussac Calculation

• A sealed cylinder contains nitrogen gas at 1.00 x 103 kPa pressure and a temperature of 20.0C. When the cylinder is left in the sun, the temperature rises by thirty degrees. What is the new gas pressure in kPa?

• 1.10 x 103 kPa

Avogadro’s Law• Mole-Volume relationship• At constant T and P:

n V as n,V

• In other words, doubling the moles of a gas will cause V to double if P and T are held constant.

Molar Volume of a Gas

• Avogadro’s Hypothesis: Equal volumes of gases at the same temperature and pressure contain equal numbers of molecules.

• Molar Volume of a Gas: At STP, 1 mole of any gas = 22.4L

= 6.02 x 1023 particles

Combined Gas Law

• At constant n,

P1V1/T1 = P2V2 /T2

Calculation: The volume of a gas-filled balloon is 30.0 L at 313 K and 153 kPa. What would the volume be at STP?

39.5 L

Another Combined Gas Law Calculation

• A gas at 155 kPa and 25.0C has an initial volume of 1.00 L. If the pressure increases to 605 kPa and the volume decreases to 0.342 L, what is the final temperature of the gas in Kelvin?

• 398 K

Ideal Gas Law•PV = nRT

• P = pressure (various units) V = volume (L)

n = number of gas moles (mol) T = temperature (K only) R = ideal gas constant. R has many units. Most common: 8.314 L• kPa/mol•K and 0.0821 L•atm/mol•k

Ideal Gas Law, Cont.

• A gas that obeys this equation is said to behave ideally.

• This law accounts for the properties of most gases under a wide variety of circumstances.

• This law works best for gases at low pressure and high temperature.

An Ideal Gas Law Calculation

• A deep underground cavern contains 2.24 x 106 L of methane gas at a pressure of 1.50 x 103 kPa and a temperature of 315 K. How many kg of CH4 does the cavern contain?

• 2.06 x 104 kg methane

Another Gas Law Calculation

• A rigid hollow sphere contains 251 mol of He gas at a temperature of 621K and a pressure of 1.89 x 103 kPa. How many liters of gas are in the sphere?

• 686 L

Yet Another Ideal Gas Law Calculation

• If 4.50 g of methane gas (CH4) are in a 2.00-L container at 35.0C, what is the pressure in the container (atm)?

• 3.55 atm

Real Gas Deviation from Ideal Behavior

• Ideal Gas Law is useful, but all real gases fail to obey it to some degree.

• At high pressures, real gases do not behave ideally.

• The deviation from ideal behavior is small at lower pressures (below 10 atm).

Deviations from Ideal Gas Law, Cont.

• As temperature increases, a gas acts more ideally.

• The deviations from ideal behavior increase as temperature decreases, becoming significant near the temperature at which the gas is converted to a liquid.

Why Don’t Real Gases Behave Ideally?…

Volume • Contrary to the Kinetic-Molecular Theory, real gases do have finite volumes.

• At low pressure, gas volume is negligible compared with the container volume. Gas particles move freely.

• At higher pressures, the volume of the gas particles is a larger fraction of the total space available. Free space is more limited. Gas volumes are greater than those predicted by the Ideal Gas Law.

Why Don’t Real Gases Behave Ideally?…

Attractive Forces• Contrary to the K-M Theory, real gases particles are attracted to each other.

• At high pressure, particles experience more attraction because they are closer together.

• As temperature decreases, gas particles move slower and experience more attraction.

• More attraction = more deviation from ideal behavior.

The van der Waals Equation

• Correction of Ideal Gas Law.

•P = (nRT/V-nb) - (n2a/V2) where nb corrects for volume and n2a/V2 corrects for molecular attraction.

• Another form of van der Waals: [P + (n2a/V2)](V-nb) = nRT

A van der Waals Calculation

• Use the van der Waals equation to estimate the pressure (atm) exerted by 1.000 mol of chlorine gas in 22.41L at 0.0C. The needed constants a and b are on page 395 in your book.

• 0.990 atm (For an ideal gas, the pressure would be 1.000 atm.)

Another van der Waals Calculation

• Consider a sample of 1.000 mol of carbon dioxide gas confined to a volume of 3.000 L at 0.0C. Calculate the pressure using the ideal gas equation and the van der Waals equation.

• Ideal gas: 7.473 atm Real gas: 7.182 atm

Gas Densities and Molar Mass

• Ideal Gas equation can be used to measure and calculate gas density.

• Gas densities in g/L.• Rearrange equation: n/V = P/RT• Multiply both sides by M = molar mass (g/mol)

• nM/V = PM/RT• So…density of a gas equals:

d = PM/RT

Gas Density• Gas density depends on m.m., pressure, and temperature.

• A less dense gas will lie above a more dense gas in the absence of mixing. (CO2 from a fire extinguisher lies lower than O2 on a fire, putting out fire.)

• Hotter gas is less dense than a cooler one (hot air rises).

Gas Density Calculations

• Calculate the density of carbon tetrachloride vapor at 714 torr and 125C.

• 4.43 g/L• The molar mass of Saturn’s largest moon is 28.6 g/mol. If its surface temperature is 95K and the pressure is 1.6 atm, calculate the density of its atmosphere.

• 5.9 g/L

Another Gas Density Calculation

•Calculate the molar mass of dry air if it has a density of 1.17 g/L at 21.0°C and 740.0 torr.

•29.0 g/mol

Dalton’s Law of Partial Pressures• What if we have a mixture of gases?

• The total pressure of a mixture of gases equals the sum of the pressures each would exert if it were present alone.

•Ptotal = P1 + P2 + P3 +…

Dalton’s Law of Partial Pressures,

Cont.• If gases are in a mixture, they are all at the same temperature and volume. If they follow the Ideal Gas Law, then…

• Ptotal = (n1 + n2 +n3+…)RT/V or…

• Ptotal = (ntotal )RT/V• The total pressure at constant T and V is determined by the number of gas moles present, regardless of the number of gases in the mixture.

A Partial Pressures

Calculation• Determine the total pressure of a gas mixture that contains: Poxygen = 20.0 atm; Pnitrogen= 46.7 atm; and Phelium = 26.7 atm.

• 93.4 atm

A More Complicated Partial Pressures

Calculation• A gaseous mixture made from 6.00 g of oxygen and 9.00 g of methane is placed in a 15.0-L vessel at 0C. What is the partial pressure (atm) of each gas? What is the total pressure (atm) in the vessel?

• P of oxygen = 0.281 atm; P of methane = 0.841; total P = 1.122 atm

Partial Pressures and Mole Fraction

• The mole fraction (X) is a number without units that shows the ratio of the number of moles of one gas component to the total number of moles in a gas mixture.

•P1 = (n1/nt)Pt = X1 Pt• (The mole fraction is like a percentage without the % sign.) Note that the sum of the mole fractions of a mixture must equal 1.

A Mole Fractions Calculation

• The total pressure on Saturn’s moon is 1220 torr. Calculate the partial pressures of its gases if they have the following mole fractions: nitrogen, 0.82; argon, 0.12; methane, 0.06.

• Nitrogen: 1.0 x 103 torr Argon: 1.5 x 102 torr Methane: 73 torr

Gas Effusion and Diffusion

• Effusion: the escape of gas molecules through a tiny hole. Ex: slow leakage through a balloon’s pores.

• Diffusion: the spread of a gas throughout open space. Ex: diffusion of perfume molecules in a room.

Graham’s Law of Effusion

• The effusion rate of a gas is inversely proportional to the square root of its molar mass.

• To compare the effusion rates of two gases at same T and P:

r1/r2 = √(M2/M1)

• where r1= rate of first gas r2 = rate of second gas M = molar mass of gas one and gas two.

How Does Effusion Work?

• At the same temperature, a lighter gas effuses faster than a heavier gas. Why?

• A gas molecule can only escape through a hole by randomly hitting it. A gas molecule that moves faster gets more chances to hit the hole. At the same temperature, lighter molecules move faster than heavier molecules.

An Effusion Calculation

Which gas effuses faster: hydrogen or chlorine? How much faster?

Hint: use r1/r2 = √(M2/M1) and make r1 = the lighter gas

• About 6 times (5.92) faster.