Gaseous States Notes

Page # 1 GASEOUS STATES GASEOUS STATES 1. GASEOUS STATE : When the molecular forces of attraction between the particles of a matter are minimum, the particles exist in a state known as gaseous state or a state of matter in which molecules are far away from each other and free to move in available space is called gaseous state. 1.1 Properties of Gases (a) They do not have definite shape and volume. (b) They can occupy whole space open to then. (c) Gases have unlimited dispensability and high compressibility. (d) They have very low densities because of negligible intermolecular forces. (e) Gases exerts pressure on the wall of the container with perfectly elastic collisions. (f) They diffuse rapidly through each other to form homogeneous mixture against the electric, magnetic and gravitational field. 1.2 Parameters of Gases The characteristics of gases are described in terms of four measurable parameters and it is also called as measurable properties of gases which are (a) Mass (b) Volume (c) Pressure and (d) Temperature (a) Mass (m)- The mass of a gas is denoted by 'm' which is related to the no of moles 'n'. Therefore, n (no of moles) = so, m b m mass in grams b M Molar mass g g (v) P (Pressure) = b g AbAreag F Force (c) Pressure (P)(i) It is force acting per unit area. A confined gas exerts uniform pressure on the walls of its container in all the direction. (ii) It is denoted by 'P' and specified in pascal (Pa). (iii) Other units of pressure are atm, cm hg, mmHg, N/m2, bar, torr. (iv) 1 atm = 76 cm Hg = 760 mm Hg = 1.013 × 105 N/m2 = 1.013 × 105 Pa = 1.013 bar = 760 torr. Mass  Accelaration Area (vi) Pressure exerted by a gas is due to kinetic energy (K.E. = ½ mv2) of the gases molecules. (vii) K.E. of the gas molecules increases, as the temperature is increased so, pressure of a gas is directly proportional to temperature. P  T (d) Temperature (T)(i) The temperature of a gas is denoted by 'T' and it is measured in the unit of kelvin (K). (ii) Other units of temperature are, ºC, ºF, ºR. (iii) K = ºC + 273.15 x0 C (iv) = 5 ey f  32j 0 9 2.GAS LAWS : The certain laws whi ch relate the four parameters are called gas laws. 2.1 Boyle's Law (a) It states that at constant temperature, the volume of a given mass of a gas is inversely proportional to the pressure. (b) Mathematically - 1 (at constant temperature) V or PV = K or P1V1 = P2V2 (c) Graphical representations P  (i) P vs V (ii) P vs = n × M (b) Volume V(i) Gases occupy whole space available to then. The volume occupy by a gas is simply the volume of container in which it is filled. (ii) The volume of a gas is denoted by 'V' and it is measured in units of litre or cubic metre (m3) or cm3 or dm3. (iii) 1 litre = 1 dm3 = 1000 cm3 = 1000 ml = 1 V (iii) log P vs, log V (iv) PV vs P log P  PV log V P 2.2 Charle's law (a) This law states that at constant pressure, the volume of a given mass of a gas is directly proportional to its absolute temperature. (Absolute temperature = ºC + 273.15) Page # 2 GASEOUS STATES (b) Mathematically V  T (a t constant pressure) V = volume of gas T = Absolute temperature V = KT or V =K T Hence, if the volume of a gas of mass is V1 at temperature T1 changes to V2 at T2, pressure remaining constant, V1 V2 then T = T 1 2 or = constant log V – log T = constant (c) For each degree change of temperature the volume of sample of a gas changes by the fraction of 1 so Vt = V0 273.15 of its volume at 0 ºC LM 273.15  t OP N 273.15 Q 2.3 Gay-Lussac's Law or Amonton's Law (a) It states that at constant volume, the pressure of a given mass of a gas is directly proportional to its absolute temperature. (b) Mathematically - P  T (at constant volume) where P = pressure of gas T = Absolute temperature P = KT P or = K T Hence , if the pressure of a gas is P1 at temperature T1 changes to P2 at T2, volume remaining constant. P1 P2 then = T1 T2 = constant log P – log T = constant. (c) Pt This equation is called Charles-gay-lussac equation. where Vt = volume of gas at temperature t ºC V 0 = volume of gas at 0 ºC temperature t = temperature in ºC. FG H Po 1  = where Pt Po t t 273.15 IJ K = Pressure of gas at t ºC = Pressure of gas at 0 ºC = Temperature in ºC. (d) Graphical representation (i) P vs T (ii) P vs 1 T or T vs 1 P (d) Graphical representations - (i) V vs T P P AV T 1/T (iii) log P vs log T (iv) A 1 (ii) V vs or T 1 T vs V log P V (iii) log V vs log T, log V log T V vs T T T Vs T log T — P/T T 1/T (iv) P 2.4 Ideal gas equation (a) It correlate all the four parameters of a gas. (b) It is the combination of Boyle's and charle's law. (c) PV = nRT m PV = RT M The equation is called as ideal gas equation. Where n = no of moles of the gas m = mass of the gas M = Mol. wt. of the gas R = Molar gas constant. Page # 3 GASEOUS STATES (d) For 1 mole of the gas n = 1 PV = RT So P1V1 P2 V2 = T1 T2 = R or P1V1 P2 V2 = T1 T2 Where P1, V1, T1 are the initial pressure, volume and temperature and P2, V2 T2 are final. The above equation is called as ideal gas equation. (e) The unit of R is the unit of work or energy per degree per mole as R R R = Pr essure  volume PV = mole  Temperature nT force  volume force  length Area = = mole  Temperaute mole  temperature = b work energy g mole  temperature (f) Numerical values of R in different units (i) R = 0.0821 litre atm. deg.–1 mole–1 (ii) R = 62.4 litres mm. deg.–1 mole–1 (iii) R = 8.314 × 107 ergs deg.–1 mole–1 (iv) R = 82.05 C.C.atm. deg.–1 mole– (v) R = 2 cals. deg.–1 mole– (vi) R = 8.314 J K–1 mole–1 3. DALTON’S LAW OF PARTIAL PRESSURE : (a) According to this law, when two or more than two chemically inert gases are kept in a closed container, the total pressure exerted by the gaseous mixture is equal to the sum of the partial pressures of individual gases i.e.- P = P1 + P2 + P3 + ..... + Pn (b) Let n1 & n2 be the no. of moles of two inert gases A and B which is filled in a container of volume 'V' at temperature T. So the total pressure of container 'P' may be calculated as PV = (n1 + n2) RT ....(i) Partial pressure of individual gas calculates at PAV = n1 RT ....(ii) PBV = n2 RT ....(iii) On the addition of eq. (ii) & (iii) we get (PA + PB) V = (n1 + n2) RT .... (iv) On the comparison of eq. (i) & (iv) P = PA + PB Dividing by equation (ii) by (i), we get n1 PA = = xA n1  n 2 P P A = xA × P where xA = mole fraction of 'A' Similarly dividing (iii) by (i), we get P B = xB × P so Partial pressure of a component = mole fraction × total pressure 3.1 Applications of Dalton's Law of Partial pressure (a) mole fraction of a gas in a mixture of gas partial pressure of gas Total pressure of the mixture of gas (b) % of a gas in a mixture = = partial pressure of gas × 100 Total pressure (c) Pressure of dry gas which is collected over the water isPTotal = Pmoist air = Pdry gas + Pwater vapour (Note : Pwater so Pdry is called aqueous tension) = PTotal – PWater vapour vapour gas (Note: Aqueous tension is directly proportional to absolute temperature) 3.2 Limitations of Dalton's law of partial pressure (a) It is applicable only inert gases like N2 and O2, N2 and Cl 2 etc. (b) It is not applicable for chemically reactive gases like H2 and Cl 2, CO and Cl 2 etc. 4. AVAGADRO’S LAW : (a) According to this law under the same condition of temperature and pressure, equal volumes of all gas contains equal no. of molecules. V  n (At constant temperature & pressure) Where V = volume n = no of molecules (b) Molar Volume or gram molecular volume 22.4 litres or 22400 ml of every gas at NTP is the volume occupied by its one gram mole and it is called molar volume or gram molecular volume. (c) The mole Concept - Mole is defined as the total amount of substance that contains as many basic units as there are atoms in 12 g of the isotopes of carbon -12. Thus a mole will be defined as the Avogadro no of particles which is equal to 6.023 × 1023. (d) Loschmidt number - It the no of molecules present in the volume of a gas at S.T.P. Its value is 2.617 × 1019 per c.c. 5. GRAHAM’S LAW OF DIFFUSION OR EFFUSION Diffusion: It is the ability of a gas to mix spontaneously and to form a homogenous mixture is known as diffusion. Page # 4 GASEOUS STATES Effusion: It is a process in which a gas is allowed to escape under pressure through a fine orifice from closed container. (d) All collisions between two molecules or between a molecule and a wall are perfectly elastic. LAW (a) This law was proposed by Thomas Graham. (b) Accordi ng to th is l aw, at c onstant temperature and pressure, the rate of diffusion or effusion of a gas is inversely proportional to the square root of its density. Thus and directly proportional to its pressure. (e) All the molecules obey Newton's laws of motion. Rate of diffusion r1 r2 = p1 P2 (r)  P d d2 d1 It pressure is constant r1 r2 = d2 d1 where r1 & r2 are rates of diffusion of two gases and d1 & d2 are densities.  2 × vapour density = Molecular mass  r1 r2 = (f) Gas molecules neither attract nor repel each other. (g) Kinetic energy of gas molecules depend upon the absolute temperature so. Kinetic energy  absolute temperature (h) The force of gravity has no effect on the speed of gas molecules. 6.2 Calculation of Kinetic energy According to postulates of kinetic gas equation- 1 mnv2 3 PV = where P V n = pressure of the gas = volume of the gas = no of molecules present in the given amount of gas = Root mean square speed. v M2 M1 Z where M1 & M2 are the molecular masses of two gases. B Volume of gas diffused (c) Rate of diffusion = Time taken for diffusion 5.1 Applications of graham's law of diffusion (a) To the Detection of marsh gas in mines. (b) Separation of isotopes (c) Determination of density and molecular mass of gases. 6. KINETIC THEORY OF GASES : (a) It was a commonly given for ideal gases. (b) This theory was proposed by Bernoulli and next to developed by Clausius, Maxwell, Kroning and Boltzman. 6.1 Assumptions of Kinetic Theory of gases (a) All the gases are made up of molecules moving randomly in all the directions. (b) The volume of the individual molecule is negligible as compared to the total volume of the gas. (c) Pressure exerted by gas is due to collision of gas molecules with the wall of the container so. Pressure  No of collisions per unit time per unit area by the molecules on the wall of the container l cm Before collision Ux After –Ux collision V Uz A Uy Ux X Y for one mole of the gas PV = RT and n = N 1 mNv2 = RT 3 2 1 or . mNv2 = RT 3 2 1 ( mNv2 = K.E. per mole) 2 2  K.E. = RT 3 3RT  K.E. per mole / Avg. K.E. per mole. 2 K.E. =  Boltzmann constant K = RN = 1.38 × 10–16 ergs K–1 molecules–1 K.E. = 3KT  K.E. per molecule / Avg. K.E. per 2 molecule Page # 5 GASEOUS STATES 6.3 Velocities related to gaseous state (A) Root mean square velocity (RMS velocity) Square root of the mean of the squares of the speed of all the molecules present in the given sample of the gas is called root mean square speed. u = 3PV = M 3RT = M 3P d 8RT m (c) Most Probable SpeedSpeed possessed by maximum number of molecules of a gas at a given temperature. 2RT M MPS = .816 × RMS ; RMS = 1.224 MPS MPS : A.V. speed : RMS = 1 : 1 . 128 : 1.224 7. DIFFERENCE BETWEEN REAL GAS & IDEAL GAS: S.N. Ideal gas Real gas 1. Obeys gas law Obeys only at under all conditions low P and highT. of P and T. 2. 3. 4. 5. Obeys ideal gas equation Intermolecular interaction between gaseous molecules are negligible. condition mostly these two assumptions (a) The force of attraction between gaseous molecules are negligible (b) The volume occupied by the gaseous of gas at high pressure. Both the assumptions Average speed = .9213 × RMS speed RMS speed = 1.085 × Average speed Most probable speed  = Kinetic theory of gases do not hold good at all molecules is negligible compared to total volume (b) Average Speed It is the arithmetic mean of the various speeds of the molecules Average speed ( v ) = 7.1 Explanation of deviation Does not obeys ideal gas equation. Intermolecular interaction between gaseous molecules not negligible. Volume of a particle is negligible as compared to total volume of the gas. Volume of a particle is not negligible as compared to total volume of the gas. Exists only at high temperature and low pressure Exists only at low temperature and high pressure. do not hold good hence deviating from ideal gas. 8. SPECIFIC HEAT OF GASES : (a) Specific Heat - It is the amount of heat required to raise the temperature of one gm of the substance through one degree centigrade. It is usually expressed in "Calories". (b) Calorie - A calorie is defined as the amount of heat required to raise the temperature of one gm of water through 1ºC (more accurately from 15.5 ºC to 16.5 ºC.). The heat so supplied is used up in increasing the internal energy of the molecule, i.e. in raising the temperature or in raising the average K.E. of the molecules. (c) Specific Heat of a gas at constant volume (Cv) The amount of heat required to raise the temperature of one gm of the gas by 1 ºC when the volume is kept constant and pressure is allowed to increase. (d) Specific Heat of a gas at constant Pressure (Cp) The amount of heat required to raise the temperature of one gm of the gas by 1 ºC when the pressure is kept constant and volume is allowed to increase. C V = cV × M and CP = cP × M Where CV and CP are molar specific heat or heat capacity per mole at constant volume and at constant pressure. (e) Ratio for Molar Heats CP (i) For monoatomic gases C = g V 5.0 = = 1.66 3.0 Page # 6 GASEOUS STATES  Molar heat capacity at constant volume (CV) = K.E. = 3RT 2 at 1 ºC  K.E = 3/2 R 3 × 2 = 3. 2 Therefore  CV = 3 calories. 10. CRITICAL CONSTANTS : (a) Critical temperature (T C ) - The temperature above which gas connect be liquified whatever pressure is applied is called critical temperature. K.E. = & CP = Increase in K.E. = 3 5 R + R = R 2 2 CP = FG 3 RIJ H2 K + work done (Since R = 2) 5 × 2 = 5 calories. 2 CP (ii) For diatomic gases C V = 5x 3x In case of diatomic gases x in many cases is 2. CP CV = 7 = 1.40 5 CP (iii) For triatomic gases C V = = 53 33 8 = 1.33 6 9. VANDER WAAL’S EQUATION-A MODIFICATION IN GAS EQUATION : According to Vander Waal Maxwell has made two wrong assumptions in his kinetic theory of gases which lead to wrong results. (a) The forces of attractions between gaseous molecules are negligible under all conditions. (b) The volume occupied by the gaseous molecules is negligible as compared to total volume of gas. (c) Volume Correction - The observed volume is greater than ideal volume & correction term 'nb' has to be subtracted from observed volume in order to get ideal volume. Videal = V – nb Where 'n' is no. of moles of real gas & 'b' is constant which depends on nature of real gas. 1 TC = 8a 27bR (b) Critical pressure (P C) - The minimum pressure required to liquefy a gas at its critical temperature is called critical pressure. PC = a 27b 2 (c) Critical volume (VC) - The volume occupied by one mole of a gas at the critical temperature and critical pressure is called critical volume. VC = 3 b 11. LIQUIFICATION OF GASES : (a) On increasing the pressure gas can be easily liquified because intermolecular attraction between gaseous molecule increases. (b) On decreasing the temperature K.E. of gaseous molecules decrease That's why gas can be easily liquified. (c) Temperature of gas must be lower than critical temperature (TC). 12. BOYLE’S TEMPERATURE : The temperature at which a real gas behaves like an ideal gas over an appreciable pressure range. TB  a Rb 13. INVERSION TEMPERATURE : It is the temperature at which any real gas is expanded adiabatically and no effect of cooling or heating is observed. Ti  2a Rb