Borrowed from: L. Scheffler Lincoln High School Gas Laws Borrowed from: L. Scheffler Lincoln High School Scheffler 1

Properties of Gases Variable volume and shape Expand to occupy volume available Volume, Pressure, Temperature, and the number of moles present are interrelated Can be easily compressed Exert pressure on whatever surrounds them Easily diffuse into one another Scheffler 2

Kinetic Molecular Theory Matter consists of particles (atoms or molecules) that are in continuous, random, rapid motion The Volume occupied by the particles has a negligibly small effect on their behavior Collisions between particles are elastic (no Energy is lost) Attractive forces between particles have a negligible effect on their behavior Gases have no fixed volume or shape, but take the volume and shape of the container The average kinetic energy of the particles is proportional to their Kelvin temperature Scheffler 3

Maxwell-Boltzman Distribution Molecules are in constant motion Not all particles have the same energy The average kinetic energy is related to the temperature An increase in temperature spreads out the distribution and the mean speed is shifted upward Scheffler 4

Velocity of a Gas The distribution of speeds of three different gases at the same temperature The distribution of speeds for nitrogen gas molecules at three different temperatures Scheffler 5

Mercury Barometer Used to define and measure atmospheric pressure On the average at sea level the column of mercury rises to a height of about 760 mm. This quantity is equal to 1 atmosphere It is also known as standard atmospheric pressure Scheffler 6

Barometer The mercury barometer was the basis for defining pressure, but it is difficult to use or to transport Furthermore Mercury is very toxic and seldom used anymore Most barometers are now aneroid barometers or electronic pressure sensors Scheffler 7

Pressure Units & Conversions The above represent some of the more common units for measuring pressure. The standard SI unit is the Pascal or kilopascal. (kPa) The US Weather Bureaus commonly report atmospheric pressures in inches of mercury. Pounds per square inch or PSI is widely used in the United States. Most other countries use only the metric system. Two other older units for pressure are still used millimeters of mercury (mm Hg) atmospheres (atm) Scheffler 8

Standard Temperature and Pressure (STP) The volume of a gas varies with temperature and pressure. Therefore it is helpful to have a convenient reference point at which to compare gases. For this purpose, standard temperature and pressure are defined as: Temperature = 0oC 273 K Pressure = 1 atmosphere = 760 torr = 101.3 kPa This point is often called STP Scheffler 9

Dalton’s Law of Partial Pressures The total pressure of a mixture of gases is equal to the sum of the pressures of the individual gases (partial pressures). PT = P1 + P2 + P3 + P4 + . . . . where PT = total pressure P1 = partial pressure of gas 1 P2 = partial pressure of gas 2 P3 = partial pressure of gas 3 P4 = partial pressure of gas 4 Scheffler 10

Dalton’s Law of Partial Pressures Determine the partial pressure of each gas in a vessel that holds 2.50 mole of O2, 1.00 mole of N2 and 0.50 mole of CO2. The total pressure in the vessel is 96.0 kPa. 2.50 mol + 1.00 mol + 0.50 mol = 4.00 mol 2.50/4.00 = .625 x 96.0 kPa = 60.0 kPa O2 1.00/4.00 = .25 x 96.0 kPa = 24.0 kPa N2 0.50/4.00 = .125 x 96.0 kPa = 12.0 kPa CO2 1.00 96.0 kPa Total Scheffler 11

Dalton’s Law of Partial Pressures Applies to a mixture of gases Very useful correction when collecting gases over water since they inevitably contain some water vapor. Scheffler 12

Sample Problem 7 Henrietta Minkelspurg generates Hydrogen gas and collected it over water. If the volume of the gas is 250 cm3 and the barometric pressure is 765.0 torr at 25oC, what is the pressure of the “dry” hydrogen gas? (PH2O = 23.8 torr at 25oC) Scheffler 13

Boyle’s Law According to Boyle’s Law the pressure and volume of a gas are inversely proportional at constant temperature. PV = constant. P1V1 = P2V2 Scheffler 14

Boyle’s Law A graph of pressure and volume gives an inverse function A graph of pressure and the reciprocal of volume gives a straight line Scheffler 15

Sample Problem 1: If the pressure of helium gas in a balloon has a volume of 4.00 dm3 at 210 kPa, what will the pressure be at 2.50 dm3? P1 V1 (conditions 1) = P2 V2 (new conditions) (210 kPa) (4.00 dm3) = P2(2.50 dm3) P2 = (210 kPa) (4.00 dm3) (2.50 dm3) = 340 kPa Scheffler 16

Charles’ Law According to Charles’ Law the volume of a gas is proportional to the Kelvin temperature as long as the pressure is constant V = kT Note: The temperature for gas laws must always be expressed in Kelvin where Kelvin = oC +273.15 (or 273 to 3 significant digits) How do you remember? (Use Zero, get a Zero) V1 = T1 V2 T2 Scheffler 17

Pressure, Temperature, Volume Pressure, temperature and volume of gases also have a relationship to each other. That relationships is summarized with: PTV Scheffler 18

Charles’ Law A graph of temperature and volume yields a straight line. Where this line crosses the x axis (x intercept) is defined as absolute zero Scheffler 19

Sample Problem 2 A gas sample at 40 oC occupies a volume of 2.32 dm3. If the temperature is increased to 75 oC, what will be the final volume? V1 = V2 T1 T2 Convert temperatures to Kelvin. 40oC = 313K 75oC = 348K 2.32 dm3 = V2 313 K 348K (313K)( V2) = (2.32 dm3) (348K) V2 = 2.58 dm3 Scheffler 20

Gay-Lussac’s Law P1 = P2 T2 T1 Gay-Lussac’s Law defines the relationship between pressure and temperature of a gas. The pressure and temperature of a gas are directly proportional P1 = P2 T2 T1 Scheffler 21

Sample Problem 3: The pressure of a gas in a tank is 3.20 atm at 22 oC. If the temperature rises to 60oC, what will be the pressure in the tank? P1 = P2 T1 T2 Convert temperatures to Kelvin. 22oC = 295K 60oC = 333K 3.20 atm = P2 295 K 333K (295K)( P2) = (3.20 atm)(333K) P2 = 3.6 atm Scheffler 22

The Combined Gas Law 1. If the amount of the gas is constant, then Boyle’s Charles’ and Gay-Lussac’s Laws can be combined into one relationship 2. P1 V1 = P2 V2 T1 T2 Scheffler 23

Sample Problem 4: A gas at 110 kPa and 30 oC fills a container at 2.0 dm3. If the temperature rises to 80oC and the pressure increases to 440 kPa, what is the new volume? P1V1 = P2V2 T1 T2 Convert temperatures to Kelvin. 30oC = 303K 80oC = 353K V2 = V1 P1 T2 P2 T1 = (2.0 dm3) (110 kPa ) (353K) (440 kPa ) (303 K) V2 = 0.58 dm3 Scheffler 24

Advogadro’s Law Equal volumes of a gas under the same temperature and pressure contain the same number of particles. If the temperature and pressure are constant the volume of a gas is proportional to the number of moles of gas present V = constant * n where n is the number of moles of gas V/n = constant V1/n1 = constant = V2 /n2 V1/n1 = V2 /n2 Scheffler 25

Universal (Ideal) Gas Equation Based on the previous laws there are four factors that define the quantity of gas: Volume, Pressure, Kelvin Temperature, and the number of moles of gas present (n). Putting these all together: PV nT = Constant = R The proportionality constant R is known as the universal (Ideal) gas constant Scheffler 26

Universal (Ideal) Gas Equation The Universal (Ideal) gas equation is usually written as PV = nRT Where P = pressure V = volume T = Kelvin Temperature n = number of moles The numerical value of R depends on the pressure unit (and perhaps the energy unit) Some common values of R include: R = 62.36 dm3 torr mol-1 K-1 = 0.0821 dm3 atm mol-1 K-1 = 8.314 dm3 kPa mol-1 K-1 Scheffler 27

Sample Problem 5 Example: What volume will 25.0 g O2 occupy at 20oC and a pressure of 0.880 atmospheres? : (25.0 g) n = ----------------- = 0.781 mol (32.0 g mol-1) Data Formula Calculation Answer V =? P = 0.880 atm; T = (20 + 273)K = 293K R = 0.08205 dm3 atm mol-1 K-1 PV = nRT so V = nRT/P V = (0.781 mol)(0.08205 dm3 atm mol-1 K-1)(293K) 0.880 atm V = 21.3 dm3 Scheffler 28

d is the density of the gas in g/L Universal Gas Equation –Alternate Forms Density (d) Calculations PV=nRT d = mass/Volume n= moles …which can be equal to: mass (g) Molar Mass (g/mol) Substituting: = PM RT m V m is the mass of the gas in g d = M is the molar mass of the gas Molar Mass (M ) of a Gaseous Substance dRT P d is the density of the gas in g/L M = Scheffler 29

Sample Problem 6 dRT m M = d = = = 2.21 V P x 0.0821 x 300.15 K M = A 2.10 dm3 vessel contains 4.65 g of a gas at 1.00 atmospheres and 27.0oC. What is the molar mass of the gas? dRT P M = d = m V 4.65 g 2.10 dm3 = = 2.21 g dm3 2.21 g dm3 1 atm x 0.0821 x 300.15 K dm3•atm mol•K M = M = 54.6 g/mol Scheffler 30

“Stupid” Gas Laws… How did all these gas laws “come to be”???? All are based on the Ideal Gas Law for 2 sets of conditions. Remember: PV = nRT we can rearrange things to solve for a constant (R) R=PV/nT n=# moles. (Not going to change when conditions change) Therefore, if moles don’t change: R=PV/T If we look at a gas that changes from one condition P1V1 = nRT1 to a new condition P2V2 = nRT2, we can set up a relationship (moles don’t change and R is constant) P1V1/T1 = R = V2P2/T2 P1V1/T1 = V2P2/T2 (Combined Gas Law) Scheffler 31

“Stupid” Gas Laws (cont.) P1V1 = P2V2 If Temperature is constant P1V1 = P2V2 T1 T2 (Boyle’s Law) T1 T2 P1V1 = P2V2 If Volume is constant P1V1 = P2V2 T1 T2 (Gay-Lassac’s Law) T1 T2 P1V1 = P2V2 If Pressure is constant P1V1 = P2V2 T1 T2 (Charles’ Law) T1 T2 P1V1 = P2V2 If Pressure & T are constant P1V1 = P2V2 n1T1 n2T2 n1T1 n2T2 Scheffler 32

Diffusion Gas diffusion is the gradual mixing of molecules of one gas with molecules of another by virtue of their kinetic properties. NH4Cl NH3 17.0 g/mol HCl 36.5 g/mol Scheffler 33

DIFFUSION AND EFFUSION Diffusion is the gradual mixing of molecules of different gases. Effusion is the movement of molecules through a small hole into an empty container. Scheffler 34

Graham’s Law (For Information only: do not need to know how to do this) Graham’s law governs effusion and diffusion of gas molecules. KE=1/2 mv2 (don’t need to know this) The rate of effusion is inversely proportional to its molar mass. Thomas Graham, 1805-1869. Professor in Glasgow and London. Scheffler 35

Kinetic Molecular Theory Matter consists of particles (atoms or molecules) that are in continuous, random, rapid motion The Volume occupied by the particles has a negligibly small effect on their behavior Collisions between particles are elastic (no Energy is lost) Attractive forces between particles have a negligible effect on their behavior Gases have no fixed volume or shape, but take the volume and shape of the container The average kinetic energy of the particles is proportional to their Kelvin temperature Scheffler 36

Ideal Gases v Real Gases Ideal gases are gases that obey the Kinetic Molecular Theory perfectly. The gas laws apply to ideal gases, but in reality there is no perfectly ideal gas. Under normal conditions of temperature and pressure many real gases approximate ideal gases. Under more extreme conditions more polar gases show deviations from ideal behavior. Scheffler 37

Real Gases For ideal gases the product of pressure and volume is constant. Real gases deviate somewhat as shown by the graph pressure vs. the ratio of observed volume to ideal volume below. These deviations occur because Real gases do not actually have zero volume Polar gas particles do attract if compressed Scheffler 38

In an Ideal Gas --- The particles (atoms or molecules) in continuous, random, rapid motion. The particles collide with no loss of momentum The volume occupied by the particles is essentially zero when compared to the volume of the container The particles are neither attracted to each other nor repelled The average kinetic energy of the particles is proportional to their Kelvin temperature At normal temperatures and pressures gases closely approximate ideal behavior Scheffler 39

van der Waals Equation (For Information only: do not need to know how to do this) The van der Waals equation shown below includes corrections added to the universal gas law to account for these deviations from ideal behavior (P + n2a/V2)(V - nb) = nRT where a => attractive forces between molecules b => residual volume or molecules The van der Waals constants for some elements are shown below Substance a (dm6atm mol-2) b (dm3 mol-1) He 0.0341 0.02370 CH4 2.25 0.0428 H2O 5.46 0.0305 CO2 3.59 0.0437 Scheffler 40