KINETICKINETIC THEORYTHEORY OFOF GASESGASES && IDEALIDEAL GASGAS LAWLAW Macroscopic definition of gas state: no shape, no volume Microscopic definition: molecules are separated by large distances, fly freely & interact through elastic collisions only. Is a mixture of gases always homogeneous, always heterogeneous or either? Why are gases compressible, while liquids & solids are incompressible? How does density of gases compare to that of liquids and solids? Objectives: - To introduce the basics of the kinetic theory of gases: - To relate pressure, as macroscopic, measurable parameter of gas state, to molecular motion - To relate temperature to the kinetic energy of moving gas particles - To establish the relationship between the three macroscopic parameters of gas state: temperature, pressure & volume T [K] P [atm] V [L] through the state equation of ideal, or perfect gas, & to introduce the definition of ideal gas. Why important? 1. The gas state is the only state for which quantitative description & prediction of properties is possible in simple theoretical terms. The behavior of liquids & solids is more complicated & hard to explain quantitatively in terms of the properties of constituent particles (atoms, molecules or ions). 2. It is important in chemical calculations. Gas laws permit us to perform stoichiometric calculations of reactions involving gases, at arbitrary conditions (T & P). KINETICKINETIC THEORYTHEORY OFOF MATTERMATTER ININ APPLICATIONAPPLICATION TOTO GASGAS

1. Gases consist of large numbers of molecules that occupy a volume at least 1000 times larger than they would occupy in solid or liquid state. Molecules of gases are far apart. Most of volume occupied by a gas is empty.

2. Molecules of a gas are in constant motion, traveling rapidly along straight lines in random directions & with a random distribution of their speeds (this is why the theory is called "kinetic" - from greek "moving") 3. The only way gas particles interact with each other & with the walls of a container is through elastic collisions. In elastic collisions the particles exchange their kinetic energy, but the sum of kinetic energy of two colliding particles is preserved & not converted into other types of energy. There are no forces of attraction or repulsion between the gas particles, except the elastic repulsion during the collision. 4. The average kinetic energy of gas molecules is proportional to absolute temperature. . EEk == constconst TT Kinetic energy of a single molecule is 2 Ek= ½ mv where m & v are mass & speed of the molecule. In an individual gas (not a mixture of several gases!), the kinetic energy depends on speed only (since all masses are the same). Molecules in any gas travel with various, randomly distributed speeds. This distribution is expressed by a distribution curve, with a peak corresponding to the most probable speed. The higher T of gas is, the higher is the most probable speed: the distribution curve shifts to the higher speeds. Different gases at the same temperature have the same average kinetic energy of their molecules. However their average speeds will be different, since different are their masses: the higher the mass is, the lower is the speed, since kinetic energy is the same. PRESSUREPRESSURE Gas exerts pressure on the wall of the container. PRESSURE (in general, not only for gas) IS FORCE ACTING ON THE UNIT AREA p = F/s With the same effort (force) we can make a hole in a sheet of paper with a sharp needle, though not with a thick stick. This is because the pressure is inverse proportional to the area of contact. For the same reason a sharp knife cuts well, though a blunt one does not.

UnitsUnits ofof PressurePressure pascal, Pa; millimeter of mercury, mm Hg or torr; atmosphere, atm; bar.

Conversion factors : 1 atm = 1.01325 bar = 101.325 kPa = 760 torr = 760 mm Hg to be used in precise calculations.

Approx.: 1 atm  1 bar  100 kPa  760 mm Hg The pressure units are related to atmospheric pressure determined by the weight of the air column above any place on the Earth surface. 1 atm is the pressure of air at the sea level. It goes down at higher altitudes.

Barometer (particular case of a manometer) – instrument to measure gas pressure GAS PRESSURE is due to multiple impacts by the moving molecules against the wall of the container. Gas pressure increases when: - the molecules hit the wall more frequently. This happen when there are more of them in a given volume. If the number of molecules is fixed (constant mass of gas), the frequency of encounters of molecules with the wall, & also the pressure, will increase when the volume of gas is reduced (gas compressed).

- the energy of each impact increases. This energy depends on temperature: the higher the temperature, the stronger each impact, & higher the net pressure. Hence, we expect that: PRESSURE BY A GIVEN AMOUNT OF GAS, IN A SEALED CONTAINER, INCREASES WHEN TEMPERATURE INCREASES, & INCREASES WHEN VOLUME IS REDUCED. These are the expectations from the kinetic theory of gas. Experiments show this is true qualitatively for all gases, & there are simple quantitative relationships between the three variables:

T, V , P for any gas at reasonable conditions. These relationships are called GAS LAWS. There are 4 of them: Boyle's Law: V - P relationship The pressure of a given mass of gas is inverse proportional to its pressure, or PV = const, at any given temperature.

P1V1 = P2V2 at T = const Hence, if we know P & V at some temperature, & then compress the gas to, say, half of its initial volume, the pressure will increase two-fold. This relationship is graphically presented as a

HYPERBOLA This hyperbola is gasgas isothermisotherm. EXAMPLE: A mass of hydrogen occupied a volume of 1L at a pressure of 4 atm. Then the gas was allowed to double its volume by effusion to another container of the same volume. What will be the final pressure in the system? p1=4 atm, V1=1L, V2=2L, p2 ? p1V1 = p2V2 or p2 = p1V1/V2 p2 = 4atm 1L / 2L = 2 atm

Charles' Law: V - T relationship The volume of a given mass of gas is directly proportional to the temperature at any constant pressure.

VV1/T/T1 == VV2/T/T2 at p = const or V/T = const at p = const., This is the equation for GAS ISOBAR This is how the absolute scale of temperature,T(K) has been established, & why this scale is also called gas scale: Charles measured temperature in °C, and found that for various gases, or for one & the same gas but in different amounts, there is one and the same temperature, (-273°C) at which the extrapolation of the straight lines gives zero volume. absolute zero: 0 K = -273°C EXAMPLE: 0.500L of a gas was heated from 250 to 500K at constant pressure. What volume it will ocupy at this new temperature? V1 = 0.5 L, T1 = 250 K,T2 = 500 K, V2 -? V1/T1 = V2/T2 or V2 = (V1/T1) T2 V2 = (0.5 L/250 K ) 500 K = 1.0 L Gay-Lussac's Law: P - T relationship. Pressure of a given volume of gas is directly proportional to absolute temperature

p1/T1 = p2/T2 at V = const ISOSTER All 3 relationships can be combined: For a given mass of gas, pV/T = const. or p1V1/T1 = p2V2/T2 Combined Gas Law AVOGADRO LAW Equal volumes of gases, at the same to & pressure, contain equal number of molecules. 1 mole of any substance contains the same number of molecules - Avogadro number of molecules (by definition of mole). Combining these two statements, One mole of any gas, at the o same t & p, occupies the same volume. Experimentally found: 1 MOLE OF ANY GAS AT STANDARD TEMPERATURE & PRESSURE (0°C, 1 atm) OCCUPIES 22.4 L This permits to extend stoichiometric calculations over volumes of gases: Example: CaCO3(s)+2HCl(aq)  CaCl2(aq)+H2O+CO2

If 200. g CaCO3 were decomposed, what volume of CO2, in L, was released at STP? CO2 (L) = . . 200gCaCO3(1molCaCO3/100gCaCO3) (1molCO2/1molCaCO3) 22.4 L/mol = 44.8 L CO2 In gas reactions, volume ratios are the same as mole ratios. EXAMPLE: The pressure of a gas in a 2.0 L container is 3.0 atm at 273 K. What will be the pressure of that gas if the volume available is increased up to 6.0 L, and the temperature is raised up to 546 K? p1=3.0atm,V1 =2.0L, T1 =273K, V2=6.0 L, T2 =546K, p2-? p1V1/T1 = p2V2/T2 or p2 = (p1V1/T1) (T2/V2) p2=(3.0atm 2.0 L/273 K)(546K/6L)= 2 atm COMBINEDCOMBINED GASGAS LAWLAW 1 mole of any gas at STP (T= 273 K & P = 1 atm) occupies the volume of 22.4 L. Hence, for one mole of gas: pV/T=1atm x 22.4L/273K= 0.0821 L atm/K This is true for 1 mole of any gas at any set of conditions.

UNIVERSAL GAS CONSTANT R=0.0821L.atm/K.mol or pV/T = R, or pV = RT (for 1 mol of gas) If we have n moles of gas, pVpV == nRnRTT Mass of gas can be presented as molar mass M multiplied by the number of moles, n, i.e. m = nM, or n = m/M, and pVpV == ((m/m/MM))RRTT This is the basis to determine MOLECULAR MASS of gases: M = m RT/pV Since m/V = D, M = DRT/p M1/M2 = D1/D2