THE PROPERTIES OF GASES Chapter 4. THE PROPERTIES OF GASES THE NATURE OF GASES 4.1 Observing Gases 4.2 Pressure 4.3 Alternative Units of Pressure THE GAS LAWS 4.4 The Experimental Observations 4.5 Applications of the Ideal Gas Law 4.6 Gas Density 4.7 The Stoichiometry of Reacting Gases 4.8 Mixtures of Gases 2012 General Chemistry I

THE NATURE OF GASES (Sections 4.1-4.3) 4.1 Observing Gases Many of physical properties of gases are very similar, regardless of the identity of the gas. Therefore, they can all be described simultaneously. Samples of gases large enough to study are examples of bulk matter – forms of matter that consist of large numbers of molecules Two major properties of gases: Compressibility – the act of reducing the volume of a sample of a gas Expansivity - the ability of a gas to fill the space available to it rapidly

4.2 Pressure - SI unit of pressure, the pascal (Pa) - Pressure arises from the collisions of gas molecules on the walls of the container. - - SI unit of pressure, the pascal (Pa)

Measurement of Pressure Barometer – A glass tube, sealed at one end, filled with liquid mercury, and inverted into a beaker also containing liquid mercury (Torricelli) where h = the height of a column, d = density of liquid, and g = acceleration of gravity (9.80665 ms-2)

open-tube and (b) closed tube system Manometer This is a U-shaped tube filled with liquid and connected to an experimental system, whose pressure is being monitored. -Two types of Hg manometer: open-tube and (b) closed tube system

4.3 Alternative Units of Pressure - 1 bar = 105 Pa = 100 kPa - 1 atm = 760 Torr = 1.01325×105 Pa (101.325 kPa) - 1 Torr ~ 1 mmHg Weather map mbar

THE GAS LAWS (Sections 4.4-4.6) 4.4 The Experimental Observations Boyle’s law: For a fixed amount of gas at constant temperature, volume is inversely proportional to pressure. This applies to an isothermal system (constant T) with a fixed amount of gas (constant n).

- For isothermal changes between two states (1 and 2),

Charles’s law: For a fixed amount of gas under constant pressure, the volume varies linearly with the temperature. This applies to an isobaric system (constant P) with a fixed amount of gas (constant n).

The Kelvin Scale of Temperature If a Charles’ plot of V versus T (at constant P and n) is extrapolated to V = 0, the intercept on the T axis is ~-273 oC. - Kelvin temperature scale T = 0 K = -273.15 oC, when V → 0. - Celsius temperature scale t (oC) = T (K) - 273.15 0 oC = 273.15 K

Another aspect of gas behavior (Gay-Lussac’s Law) This applies to an isochoric system (constant V) with a fixed amount of gas (constant n).

- This defines molar volume Avogadro’s Principle Under the same conditions of temperature and pressure, a given number of gas molecules occupy the same volume regardless of their chemical identity. - This defines molar volume

The Ideal Gas Law This is formed by combining the laws of Boyle, Charles, Gay-Lussac and Avogadro. The ideal gas law: Gas constant, R = PV/nT. It is sometimes called a “universal constant” and has the value 8.314 J K-1 mol-1 in SI units, although other units are often used (Table 4.2).

Table 4.2. The Gas Constant, R The ideal gas law, PV = nRT, is an equation of state that summarizes the relations describing the response of an ideal gas to changes in pressure, volume, temperature, and amount of molecules; it is an example of a limiting law. (it is strictly valid only in some limit: here, as P 0.)

4.5 Applications of the Ideal Gas Law - For conditions 1 and 2, - Molar volume - Standard ambient temperature and pressure (SATP) 298.15 K and 1 bar, molar volume at SATP = 24.79 L·mol-1 - Standard temperature and pressure (STP) 0 oC and 1 atm (273.15 K and 1.01325 bar) - Molar volume at STP

EXAMPLE 4.4 In an investigation of the properties of the coolant gas used in an air-conditioning system, a sample of volume 500 mL at 28.0 oC was found to exert a pressure of 92.0 kPa. What pressure will the sample exert when it is compressed to 30 mL and cooled to -5.0 oC?

Calculating the pressure of a given sample

Using the combined gas law when one variable is changed

Using the combined gas law when two variables are changed

4.6 Gas Density Molar concentration of a gas is the number moles divided by the volume occupied by the gas. Molar concentration of a gas at STP (where molar volume is 22.4141 L): This value is the same for all gases, assuming ideal behavior. Density, however, does depend on the identity of the gas.

Gas Density Relationships For a given P and T, the greater the molar mass, the greater its density. At constant T, the density increases with P. In this case, P is increased either by adding more material or by compression (reduction of V). Raising T allows a gas to expand at constant P, increases V and therefore reduces its density. Density at STP

4.7 The Stoichiometry of Reacting Gases Molar volumes of gases are generally > 1000 times those of liquids and solids. e.g. Vm (gases) = ~ 25 L mol-1; Vm (liquid water) = 18 mL mol-1 Reactions that produce gases from condensed phases can be explosive. e.g. sodium azide (NaN3) for air bags

EXAMPLE 4.6 The carbon dioxide generated by the personnel in the artificial atmosphere of submarines and spacecraft must be removed form the air and the oxygen recovered. Submarine design teams have investigated the use of potassium superoxide, KO2, as an air purifier because this compound reacts with carbon dioxide and releases oxygen: 4 KO2 (s) + 2 CO2(g) → 2 K2CO3(s) + 3 O2(g) Calculate the mass of KO2 needed to react with 50 L of CO2 at 25 oC and 1.0 atm. Vm = 24.47 Lmol-1; 1 mol CO2 -> 2 mol KO2; MKO2 = 71.10 gmol-1

4.8 Mixtures of Gases - A mixture of gases that do not react with one another behaves like a single pure gas. Partial pressure: The total pressure of a mixture of gases is the sum of the partial pressures of its components. (John Dalton). P = PA + PB + … for the mixture containing A, B, … - Humid gas: P = Pdry air + Pwater vapor (Pwater vapor = 47 Torr at 37 oC) mole fraction: the number of moles of molecules of the gas expressed as a fraction of the total number of moles of molecules in the sample.

EXAMPLE 4.7 Air is a source of reactants for many chemical processes. To determine how much air is needed for these reactions, it is useful to know the partial pressures of the components. A certain sample of dry air of total mass 1.00 g consists almost entirely of 0.76 g of nitrogen and 0.24 g of oxygen. Calculate the partial pressures of these gases when the total pressure is 0.87 atm.

THE PROPERTIES OF GASES Chapter 4. THE PROPERTIES OF GASES MOLECULAR MOTION 4.9 Diffusion and Effusion 4.10 The Kinetic Model of Gases 4.11 The Maxwell Distribution of Speeds REAL GASES 4.12 Deviations from Ideality 4.13 The Liquefaction of Gases 4.14 Equations of State of Real Gases 2012 General Chemistry I

MOLECULAR MOTION (Sections 4.9-4.11) 4.9 Diffusion and Effusion Diffusion: gradual dispersal of one substance through another substance Effusion: escape of a gas through a small hole into a vacuum

Graham’s law: At constant T, the rate of effusion of a gas is inversely proportional to the square root of its molar mass: Strictly,Graham’s law relates to effusion, but it can also be used for diffusion. - For two gases A and B with molar masses MA and MB,

- Rate of effusion and average speed increase as the square root of the temperature: Combined relationship: The average speed of molecules in a gas is directly proportional to the square root of the temperature and inversely proportional to the square root of the molar mass.

4.10 The Kinetic Model of Gases Kinetic molecular theory (KMT) of a gas makes four assumptions: 1. A gas consists of a collection of molecules in continuous random motion. 2. Gas molecules are infinitesimally small points. 3. The molecules move in straight lines until they collide. 4. The molecules do not influence one another except during collisions. - Collision with walls: consider molecules traveling only in one dimensional x with a velocity of vx.

The change in momentum (final – initial) of one molecule: 2mvx All the molecules within a distance vxDt of the wall and traveling toward it will strike the wall during the Interval Dt. If the wall has area A, all the particles in a volume AvxDt will reach the wall if they are traveling toward it.

The number of molecules in the volume AvxDt is that fraction of the total volume V, multiplied by the total number of molecules: The average number of collisions with the wall during the interval Dt is half the number in the volume AvxDt: The total momentum change = number of collisions × individual molecule momentum change

Force = rate of change of momentum = (total momentum change)/Dt for the average value of <vx2> Mean square speed: Pressure on wall:

where vrms is the root mean square speed, or - The temperature is proportional to the mean square speed of the molecules in a gas. - This was the first acceptable physical interpretation of temperature: a measure of molecular motion.

What is the root mean square speed of nitrogen EXAMPLE 4.7 What is the root mean square speed of nitrogen Molecules in air at 20 oC?

4.11 The Maxwell Distribution of Speeds Maxwell derived equation 22, for calculating the fraction of gas molecules having the speed v at any instant, from the kinetic model. v = a particle’s speed DN = the number of molecules with speeds in the range between v + Dv N = total number of molecules; M = molar mass f(v) = Maxwell distribution of speeds For an infinitesimal range, average speed

as M increases, the fraction of molecules with - Molar mass (M) dependence: as M increases, the fraction of molecules with speeds greater than a specific speed decreases. - Temperature dependence: as T increases, the fraction of molecules with speeds greater than a specific speed increases.

REAL GASES (Sections 4.12-4.14) - Deviations from the ideal gas law are significant at high pressures and low temperatures (where significant intermolecular interactions exist). 4.12 Deviations from Ideality - Gases condense to liquids when cooled or compressed (attraction). - Liquids are difficult to compress (repulsion). Deviation from ideal gases

Long range attractions; smaller Z, condensation of gases Compression factor (Z): the ratio of the actual molar volume of the gas to the molar volume of an ideal gas under the same conditions. For an ideal gas, Z = 1 Long range attractions; smaller Z, condensation of gases Short range repulsions; larger Z, low compressibility of liquids and solids, finite molecular volume

- For many gases, attractions dominate at low pressure (Z < 1), while repulsive interactions dominate at high pressure (Z > 1).

4.13 The Liquefaction of Gases Joule-Thomson effect: when attractive forces dominate, a real gas cools as it expands. In this case expansion requires energy, which comes from the kinetic energy of the gas, lowering the temperature. The effect is used in some refrigerators and to effect the condensation of gases such as oxygen, nitrogen, and argon.

i.e. Adiabatic cooling; temperature - Linde refrigerator for the liquefaction of gases i.e. Adiabatic cooling; temperature decrease under isentropic expansion of any gas (w 0)

4.14 Equations of State of Real Gases Virial equation: van der Waals equation: or –nb volume excluded since molecules cannot overlap b volume excluded by 1 mol ~ molar volume in the liquid state pressure reduced due to attractions between pairs of molecules

Virial expansion of the van der Waals equation At low particle densities

Van der Waals Parameters for some Common Gases Table 4.5 Van der Waals Parameters for some Common Gases

Model of gas 1. A large number of gas molecules in ceaseless, random, and straight motion. 2. The average speed and the spread of speeds increase with T and decrease with m. 3. Molecules travel in straight lines until they collide with other molecules or the container wall. 4. Widely separated. Intermolecular forces have only a weak effect on the properties. 5. Repulsions increase the molar volume, whereas attractions decrease the molar volume.

Refrigerant gas (a = 16. 2 L2 atm mol–2, b = 0. 084 L/mol), 1 Refrigerant gas (a = 16.2 L2 atm mol–2, b = 0.084 L/mol), 1.50 mol in 5.00 L at 0 oC; Estimate the pressure. EXAMPLE 4.9