Gases Chapter 5 Unit 3

The Properties of Gases Only four quantities are needed to define the state of a gas The quantity of gas, n (in moles) The temperature of the gas, T (in Kelvins) The volume of the gas, V (in liters) The pressure of the gas, P (in atmospheres, torr, mm of Hg, or kPa)

5.1 Pressure pressure: the weight or force that is produced when something presses or pushes against something else (force/unit area) Barometer: a device to measure atmospheric pressure, invented in 1643 by Evangelista Torricelli The original barometer used the height of a column of mercury to measure gas pressure

Units of Pressure 760 mm Hg = 760 torr = 1.00 atm = 101.325 kPa (the SI unit of measurement) At sea level all of the above define standard pressure, but usually 1.00 atm is used to define standard pressure Most AP questions use atm, but the conversion from mm Hg and torr is on the sheet of equations and constants

Manometer Manometer: 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)

Manometer

Pressure Conversions The pressure of a gas is measured as 49 torr. Represent this pressure in both atmospheres and pascals Answer: 6.4 x 10-2 atm, 6.5 x 103 Pa Rank the following pressures in decreasing order of magnitude: 75 kPa, 300. torr, 0.60 atm, and 350. mm Hg. Answer: 75 kPa < 0.60 atm < 350. mm Hg < 300. torr

5.2 Gas Laws The gas laws are several mathematical laws that relate the properties of gases The laws were derived from experiments that involved careful measurements of the properties so relationships could be discovered AP exams can show these laws graphically

Boyle’s Law The first quantitative experiments on gases were performed by Irish chemist Robert Boyle (1627-1691) He studied the relationship between pressure and volume of a trapped gas The AP “cheat sheet” gives us PV = nRT for gases If nRT is held constant, then PV = k with “k” being a constant

The relationship discovered is inverse Boyle’s law holds precisely only at very low pressures We will assume that gases obey Boyle’s law unless otherwise stated A more useful form of Boyle’s law lets us predict the new volume of a gas when the pressure is changed (at constant T) P1V1 = P2V2

Example Sulfur dioxide, a gas that plays a central role in the formation of acid rain, is found in the exhaust of automobiles and power plants. Consider a 1.53-L sample of gaseous SO2 at a pressure of 5.6 x 103 Pa. If the pressure is changed to 1.5 x 104 Pa at a constant temperature, what will be the new volume of the gas? Answer: 0.57 L

Charles’ Law If a given quantity of gas is held at a constant pressure, then its volume is directly proportional to the absolute temperature. Must use Kelvin for the temperature scale The useful form is : V1 = V2 T1 T2 Jacques Charles was a French physicist

Example A sample of gas at 15.0°C and 1.00 atm has a volume of 2.58 L. What volume will this gas occupy at 38°C and 1.00 atm? Remember to change °C to Kelvin Answer: 2.79 L

Gay Lussac’s Law Gay-Lussac’s law shows the relationship between pressure and temperature (must use Kelvin for temperature) The relationship is direct P = kT or P/T = k Useful form: P1 = P2 T1 T2

Example The gas in a container is at a pressure of 3.00 atm at 25°C. Directions on the container warn the user not to keep it in a place where the temperature exceeds 52°C. What would the gas pressure in the container be at 52°C? Answer: 3.27 atm

Avogadro’s Law Avogadro’s law: for a gas at constant temperature and pressure, the volume is directly proportional to the number of moles of gas. V = kn (“k” is sometimes shown as “a”) “V” is volume, “k” is a proportionality constant, and “n” is the number of moles

Example Suppose we have a 12.2-L sample containing 0.50 mol oxygen gas at a pressure of 1.0 atm and a temperature of 25°C. If all this O2 were converted to ozone (O3) at the same temperature and pressure, what would be the volume of the ozone? Answer: 8.1 L

Combined Gas Law A gas sample frequently undergoes changes in temperature, pressure, and volume all at the same time. The amount of gas remains constant Combined gas law: expresses the relationship between pressure, volume, and temperature of a fixed amount of gas

Example A sample of diborane gas (B2H6), a substance that bursts into flame when exposed to air, has a pressure of 345 torr at a temperature of 15°C and a volume of 3.48 L. If conditions are changed so that the temperature is 36°C and the pressure is 468 torr, what will be the volume of the sample? 3.07 L

5.3 Ideal Gas Law If you consider all the laws together, you get: PV = nRT where P is pressure, V is volume, n is number of moles, R is a proportionality constant called the universal gas constant, and T is temperature Useful only at low P and high T and you can assume these are the conditions unless told otherwise

Universal Gas Constant You should always use liters for volume and Kelvin for temperature The gas constant changes depending on the units used for pressure On the “cheat sheet” you are given the gas constant for atm, torr (also mm Hg) and the one you use when dealing with energy

Example A sample of hydrogen gas (H2) has a volume of 8.56 L at a temperature of 0°C and a pressure of 1.5 atm. Calculate the moles of H2 molecules present in this gas sample. Answer: 0.57 mol H2

More Practice with Gas Laws 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 temperature. Calculate the final pressure. Answer: 4.4 atm

More A sample of methane gas that has a volume of 3.8 L at 5°C is heated to 86°C at constant pressure. Calculate its new volume. Answer: 4.9 L A sample containing 0.35 mol argon gas at a temperature of 13°C and a pressure of 568 torr is heated to 56°C and a pressure of 897 torr. Calculate the change in volume that occurs. Answer: 3 L decrease

5.4 Gas Stoichiometry Solve for the volume of one mol of gas at STP: V = nRT V = (1) (0.08206) (273) = P 1 22. 4 L This is the molar volume of a gas at STP

Examples A sample of nitrogen gas has a volume of 1.75 L at STP. How many mol of N2 are present? Answer: 7.81 x 10-2 mol N2

More Quicklime (CaO) is produced by the thermal decomposition of CaCO3. Calculate the volume of CO2 at STP produced from the decomposition of 152 g CaCO3 by the reaction: CaCO3(s)  CaO(s) + CO2(g) Answer: 34.0 L CO2

One More A sample of methane gas having a volume of 2.80 L at 25°C and 1.65 atm was mixed with a sample of oxygen gas having a volume of 35.0 L at 31°C 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 125°C. Answer: 2.47 L CO2

Molar Mass of a Gas The densities of gases are reported in g/L not g/mL Since n = m/M and substituting m/V as gas density (d) into the ideal gas law equation gives you: Molar mass (M) = dRT P “Molar mass kitty cat”

Gas Density/ Molar Mass The density of a gas was measured at 1.50 atm and 27°C and found to be 1.95 g/L. Calculate the molar mass of the gas. Answer: 32.0 g/mol

5.5 Dalton’s Law of Partial Pressures For a mixture of gases in a container, the total pressure exerted is the sum of the pressures that each gas would exert if it were alone Ptotal = P1 + P2 + P3 + … The symbols P1, P2 and P3 refer the partial pressure of each individual gas

Moles of Gas For a mixture of ideal gases, it is the total number of moles of particles that is important, not the identity of the gases

Total Pressure P1 = n1RT, P2 = n2RT, P3 = n3RT V V V Ptotal = n1RT, + n2RT, + n3RT V V V Ptotal = (ntotal) RT V

Example Mixtures of helium and oxygen can be used in scuba diving tanks to help prevent “the bends”. For a particular dive, 46 L He at 25°C and 1.0 atm and 12 L O2 at 25°C 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 25°C. Answers: PHe = 9.3 atm, PO2 = 2.4 atm, PT = 11.7 atm

Mole Fraction The ratio of the number of moles of a given component in a mixture to the total number of moles in the mixture (no units) X = moles A total moles PA = Ptotal x XA and XA = PA Ptotal

Examples 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. Answer: 0.210 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. Answer: 593 torr

Water Displacement It is common to collect a gas by water displacement. This means some of the pressure is due to water vapor collected as the gas is passing through. You correct for this by looking up the partial pressure due to water vapor by knowing the temperature

Example A sample of KClO3 (potassium chlorate) was heated in a test tube and decomposed by the reaction: 2KClO3(s)  2KCl(s) + 3O2(g)

Continued The O2 produced was collected by water displacement at 22°C at a total pressure of 754 torr. The volume of gas collected was 0.650 L, and vapor pressure of water at 22°C is 21 torr. Calculate the partial pressure of O2 in the gas collected and the mass of KClO3 in the sample that was decomposed. Answers: PO2 = 733 torr, 2.12 g KClO3

5.6 Kinetic Molecular Theory of Gases Assumptions of the 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 directly proportional to its Kelvin temperature

Explanations Boyle’s law: If the volume is decreased that means that the gas particles will hit the wall more often, thus increasing pressure Charles’ law: when a gas is heated, the speed of its particles increase and thus hit the walls more often and with more force, so the volume must increase

More 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.

More 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.

One More Dalton’s law: the P exerted by a mixture of gases is the sum of the partial pressures since gas particles are acting independent of each other and the volumes of the individual particles do not matter.

Maxwell-Boltzmann Distribution Curve The distribution shows how the speeds of molecules are distributed at different temperatures for an ideal gas. The Maxwell-Boltzmann distribution is often represented with the following graph. It is frequently used to show the energy distribution at different temperatures Temperature is NOT on any axis

5.7 Graham’s Law of Diffusion and Effusion Diffusion: the term used to describe the mixing of gases The rate of diffusion is the rate of mixing Matter will move from an area of high concentration to an area of low concentration

Diffusion in Cells

Effusion Effusion: the passage of a gas through a tiny orifice into an evacuated chamber The rate of effusion measures the speed at which the gas is transferred into the chamber rate of effusion of gas 1 = √M2 rate of effusion of gas 2 √M1

Example 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. Answer: 13.2

5.8 Real Gas vs. Ideal Gas Most gases behave ideally until you reach high pressure and low temperature A gas acts close to ideally when: The pressure is low The temperature is high The gas is nonpolar