Gaseous State PDF

225 60 Gaseous State E3 Chapter 6 Gaseous state The volume, V, of the gas. (ii) Its pressure, P (iii) Its temperature, T (iv) The amount of the gas (i.e., mass or number of moles). U Characteristics of gases (i) ID The state of matter in which the molecular forces of attraction between the particles...

225 60 Gaseous State E3 Chapter 6 Gaseous state The volume, V, of the gas. (ii) Its pressure, P (iii) Its temperature, T (iv) The amount of the gas (i.e., mass or number of moles). U Characteristics of gases (i) ID The state of matter in which the molecular forces of attraction between the particles of matter are minimum, is known as gaseous state. It is the simplest state and shows great uniformity in behaviour. (1) Gases or their mixtures are homogeneous in composition. D YG (2) Gases have very low density due to negligible intermolecular forces. (3) Gases have infinite expansibility and high compressibility. (4) Gases exert pressure. (5) Gases possess high diffusibility. (2) Volume : (i) Since gases occupy the entire space available to them, the measurement of volume of a gas only requires a measurement of the container confining the gas. (ii) U ST (8) Gaseous molecules collide with one another and also with the walls of container with perfectly elastic collisions. (9) Gases can be liquified, if subjected to low temperatures (below critical) or high pressures. (10) Thermal energy of gases >> molecular attraction. (11) Gases undergo similar change with the change of temperature and pressure. In other words, gases obey certain laws known as gas laws. Measurable properties of gases (1) The characteristics of gases are described fully in terms of four parameters or measurable properties : is expressed in litres (L), metres (m 3 ). (iii) 1L  1000 mL ; 1 mL  10 3 L ; 1 L  1 dm 3  10 3 m 3 (6) Gases do not have definite shape and volume like liquids. (7) Gaseous molecules move very rapidly in all directions in a random manner i.e., gases have highest kinetic energy. Volume millilitres (mL) or cubic centimetres (cm 3 ) or cubic 1 m 3  10 3 dm 3  10 6 cm 3  10 6 mL  10 3 L (3) Mass : (i) The mass of a gas can be determined by weighing the container in which the gas is enclosed and again weighing the container after removing the gas. The difference between the two weights gives the mass of the gas. (ii) The mass of the gas is related to the number of moles of the gas i.e. moles of gas (n)  M ass in grams m  M olar mass M (4) Temperature : (i) Gases expand on increasing the temperature. If temperature is increased twice, the square of the velocity (v 2 ) also increases two times. (ii) degree Temperature is measured in centigrade ( o C) or celsius degree with the help of 226 Gaseous State thermometers. Temperature Fahrenheit (Fo). is also measured in (iii) S.I. unit of temperature is kelvin (K) or absolute degree. K  o C  273 Absolute pressure = Gauge pressure + Atmosphere pressure. (xi) When the pressure in a system is less than atmospheric pressure, the gauge pressure becomes negative, but is frequently designated and called vacuum. For example, 16 cm vacuum will be C F o  32  5 9 o (iv) Relation between F and o C is 76  16  1.013  0.80 bar. 76 (xii) If ‘h’ is the height of the fluid in a column or the difference in the heights of the fluid columns in the two limbs of the manometer, d is the density of the fluid (ii) Pressure exerted by a gas is due to kinetic 1 energy (KE  mv 2 ) of the molecules. Kinetic energy of 2 the gas molecules increases, as the temperature is increased. Thus, Pressure of a gas  Temperature (T). then pressure is given by, Pgas  Patm  h dg (iv) Commonly two types of manometers are used, (b) Closed end D YG 1 Pa  1 Nm 2  1 kg m 1 s 2 (vi) C.G.S. unit of pressure is dynes cm 2. (vii) M.K.S. unit of pressure is kgf / m 2. The unit sometime called ata (atmosphere technical absolute). (viii) Higher unit of pressure is bar, KPa or MPa. U 1 bar  10 5 Pa  10 5 Nm 2  100 KNm 2  100 KPa (ix) Several other units used for pressure are, Name Symbol Value bar 1bar  10 Pa atmosphe re atm 1 atm  1.01325  10 5 Pa Torr Torr 101325 1 Torr  Pa  133. 322 Pa 760 ST bar millimetr e of mercury mm Hg E3 Condition T P Vm (Molar volume) S.T.P./N.T. P. 273.15 K 1 atm 22.414 L S.A.T.P*. 298.15 K 1 bar 24.800 L * Standard ambient temperature and pressure. Boyle's law (v) The S.I. unit of pressure, the pascal (Pa), is defined as 1 newton per metre square. It is very small unit. kgf / cm 2 (xiii) Two sets of conditions are widely used as 'standard' values for reporting data. U (a) Open end manometer; manometer (Hg  13.6  10 3 Kg / m 3  13.6 g / cm 3 ) and g is the gravity, ID (iii) Pressure of a pure gas is measured by manometer while that of a mixture of gases by barometer. 60 (5) Pressure : (i) Pressure of the gas is the force exerted by the gas per unit area of the walls of the container in all directions. Thus, Pressure Force( F) Mass(m)  Accelerati on (a) (P)   Area( A) Area(a) 5 (1) In 1662, Robert Boyle discovered the first of several relationships among gas variables (P, T, V). (2) It states that, “For a fixed amount of a gas at constant temperature, the gas volume is inversely proportional to the gas pressure.” Thus, P  1 / V at constant temperature and mass or P  K / V (where K is constant) or PV  K or P1 V1  P2 V2  K (For two or more gases) (3) Graphical representation of Boyle's law : Graph between P and V at constant temperature is called isotherm and is an equilateral (or rectangular) hyperbola. By plotting P versus 1 / V , this hyperbola can be converted to a straight line. Other types of isotherms are also shown below, T3 T2 T1 < T 2 < T3 P 1 mm Hg  133.322 Pa O (x) The pressure relative to the atmosphere is called gauge pressure. The pressure relative to the perfect vacuum is called absolute pressure. T1 P T3 T2 T1 V or 1/d T1 < T2 < T3 O 1/V or d T1 < T2 < T 3 T3 PV T2 T1 O P log P O log Gaseous State 227 1/d or V 1/d or V Thus, d  P  or 1 V mass    V  d    O – 273.15oC T(k 0o C t(oC) ) (5) At constant mass and pressure density of a gas is inversely proportional to it absolute temperature. Thus, d  d1 P V  1  2 .......  K d 2 P2 V1 or mass    V  d    d1 T2 V2   ......  K d 2 T1 V1 E3 (5) At altitudes, as P is low d of air is less. That is why mountaineers carry oxygen cylinders. 1 1  T V 60 (4) At constant mass and temperature density of a gas is directly proportional to its pressure and inversely proportional to its volume. 22.4 L mol–1 = V0 Charle's law (6) Use of hot air balloons in sports and meteorological observations is an application of (1) French chemist, Jacques Charles first studied variation of volume with temperature, in 1787. Charle's law. ( o C  273 ) at constant pressure”. Thus, V  T at constant pressure and mass (where k is constant), V V V or 1  2  K (For two or more gases) T T1 T2 D YG K (3) If t  0 o C , then V  V0 V0  K  273.15 hence,  K V0 273.15 V0 t   V [t  273.15 ]  V0 1    V0 [1   v t] 273.15 273. 15   U where  v is the volume coefficient, ST V  V0 1 v    3.661  10  3 o C 1 tV0 273.15 Thus, for every 1 o change in temperature, the volume of a gas changes by (1) In 1802, French chemist Joseph Gay-Lussac studied the variation of pressure with temperature and extende the Charle’s law so, this law is also called Charle’s-Gay Lussac’s law. U or V  KT  K (t( o C)  273.15 ) , Gay-Lussac's law (Amonton's law) ID (2) It states that, “The volume of a given mass of a gas is directly proportional to the absolute temperature 1 1     of the 273. 15  273  a (2) It states that, “The pressure of a given mass of gas is directly proportional to the absolute temperature ( o C  273 ) at constant volume.” Thus, P  T at constant volume and mass or P  KT  K(t(o C)  273.15 ) K P P P or 1  2  K (For two or more gases) T T1 T2 (3) If t  0 o C , then P  P0 Hence, P0  K  273.15  K P of V versus t( o C ) at constant pressure is a straight line cutting the temperature axis at  273.15 o C. It is the lowest possible temperature. P0 273.15 P0 t   [t  273. 15 ]  P0 1    P0 [1  t] 273. 15 273. 15   where  P is the pressure coefficient, volume at 0 o C. (4) Graphical representation of Charle's law : Graph between V and T at constant pressure is called isobar or isoplestics and is always a straight line. A plot (where K is constant) P  P  P0 1   3.661  10  3 o C 1 tP0 273.15 Thus, for every 1 o change in temperature, the 1 1   pressure of a gas changes by   of the 273. 15  273  pressure at 0 o C. 228 Gaseous State (4) This law fails at low temperatures, because the volume of the gas molecules be come significant. If all the above law's combines, then V (5) Graphical representation of Gay-Lussac's law : A graph between P and T at constant V is called isochore. V 1 < V 2< V 3 < V 4 V1 P P V or nT P nRT ( R  Ideal gas constant) P PV  nRT or This is called ideal gas equation. R is called ideal gas constant. This equation is obeyed by isothermal and adiabatic processes. V2 V3 V4 O T(k ) T Force  Volume Force  Length Area   mole  Temperatur e mole  Temperatur e Avogadro's law (1) According to this law, “Equal volumes of any two gases at the same temperature and pressure contain the same number of molecules.” Thus, V  n (at constant T and P) ID  8. 3143 Nm mol 1 K 1 L mol 1 at S.T.P. or N.T.P. U (3) This law can also express as, “The molar gas volume at a given temperature and pressure is a specific constant independent of the nature of the gas”. specific constant  22.4 L mol 1 at ST S.T.P. or N.T.P.  8.3143 KPa dm 3 mol 1 K 1  8. 3143 MPa cm 3 mol 1 K 1  5.189  10 19 eV mol 1 K 1  1.99 cal mol 1 K 1 (3) Gas constant, R for a single molecule is called Boltzmann constant (k) k R 8.314  10 7  ergs mole 1 degree 1 N 6.023  10 23  1.38  10 16 ergs mol 1 de gree 1 or 1.38  10 23 joule mol 1 degree 1 (4) Calculation of mass, molecular weight and density of the gas by gas equation Ideal gas equation (1) The simple gas laws relating gas volume to pressure, temperature and amount of gas, respectively, are stated below : 1 1 or V  (n V P and T constant) Charle's law : constant).  8.3143 joule mol 1 K 1 (S.I. unit) number of molecules (Avogadro's number  6.02  10 ) and by this law must occupy the same volume at a given temperature and pressure. The volume of one mole of a gas is called molar volume, Vm which is 22.4 P 1 U 2 moles 2 volumes 2 litres 1 litre 1n litre D YG 1 mole 1 volume 1 litre 1 / 2 litre 1 / 2 n litre 23 Boyle's law : K R  0.0821 L atm mol 1 K 1 (2) One mole of any gas contains the same Vm  R is expressed in the unit of work or energy 1 : Example, 2 H 2 (g) O 2 (g)  2 H 2 O(g) Thus, Work or energy. mole  Temperatur e Since different values of R are summarised below V V or 1  2 .......  K n1 n 2 2 moles 2 volumes 2 litres 1 litre 1n litre  mol or V  Kn (where K is constant) E3 O 60 (2) Nature and values of R : From the ideal gas PV Pressure  Volume equation, R   nT mole  Temperatur e PV  nRT    mass of the gas (m)  n   Molecular weight of the gas (M )    VT Avogadro's law : V  n constant) (n (T and and m RT M M P P d mRT PV PM RT m   d   V  Gaseous State (4) Applications : This law is used in the calculation of following relationships, dT M M ,   Constant P R R or (i) (  M and R are constant for a particular gas) d T d T dT or 1 1  2 2 = Constant P1 T2 P of gas  (5) Gas densities differ from those of solids and liquids as, Gas densities are generally stated in g/L instead of g / cm 3. (ii) Gas densities are strongly dependent on pressure and temperature as, d  P  1 / T Densities of liquids and solids, do depend somewhat on temperature, but they are far less dependent on pressure. molar mass (iv) Density of a gas at STP  22. 4 32  1.43 g L1 22.4 in mixture (iii) Pressure of dry gas collected over water : When a gas is collected over water, it becomes moist due to water vapour which exerts its own partial pressure at the same temperature of the gas. This partial perssure of water vapours is called aqueous tension. Thus, Pdry gas  Pmoist gas or PTotal  Pwater vapo ur Pdry gas  Pmoist gas  or tension is temperature) directly Aqueous tension (Aqueous proportional to absolute (iv) Relative humidity (RH) at a given temperature is given by, RH  Partial pr essure of water in air. Vapour pre ssure of water (5) Limitations : This law is applicable only when the component gases in the mixture do not react with each other. For example, N 2 and O 2 , CO and CO 2 , N 2 and Cl 2 , CO and N 2 etc. But this law is not applicable D YG d(O2 ) at STP  gas U 28  1.25 g L1 , d( N 2 ) at STP  22.4 (ii) % of a Partial pr essure of a gas (P1 )   100 PTotal ID (iii) The density of a gas is directly proportional to its molar mass. No simple relationship exists between the density and molar mass for liquid and solids. Partial pr essure of a gas (P1 ) PTotal 60 (For two or more different temperature and pressure) (i) Mole fraction of a gas (X 1 ) in a mixture E3 Thus, 229 to gases which combine chemically. For example, H 2 Dalton's law of partial pressures and Cl 2 , CO and Cl 2 , NH 3 , HBr and HCl, NO and O 2 (1) According to this law, “When two or more gases, which do not react chemically are kept in a closed vessel, the total pressure exerted by the mixture is equal to the sum of the partial pressures of individual gases.” etc. Thus, Ptotal  P1  P2  P3 ......... Where P1 , P2 , P3 ,...... are partial pressures of gas number 1, 2, 3......... U (2) Partial pressure is the pressure exerted by a gas when it is present alone in the same container and at the same temperature. ST Number of moles of the gas (n1 )  PTotal  Mole fraction (X 1 )  PTotal Total number of moles (n) in the mixture (3) If a at V1 , V2 , V3...... number of gases having volume pressure P1 , P2 , P3........ are mixed together in container of volume V, then, PTotal  P1 V1  P2 V2  P3 V3..... V  (n1  n 2  n 3.....) or or  n RT V RT V ( n  n1  n2  n3.....) Thus, VTotal  V1  V2  V3 ...... Where V1 , V2 , V3 ,...... Partial pressure of a gas (P1 )  (6) Another law, which is really equivalent to the law of partial pressures and related to the partial volumes of gases is known as Law of partial volumes given by Amagat. According to this law, “When two or more gases, which do not react chemically are kept in a closed vessel, the total volume exerted by the mixture is equal to the sum of the partial volumes of individual gases.” ( PV  nRT ) are partial volumes of gas number 1, 2, 3..... Graham's law of diffusion and Effusion (1) Diffusion is the process of spontaneous spreading and intermixing of gases to form homogenous mixture irrespective of force of gravity. While Effusion is the escape of gas molecules through a tiny hole such as pinhole in a balloon.  All gases spontaneously diffuse into one another when they are brought into contact.  Diffusion into a vacuum will take place much more rapidly than diffusion into another place. 230 Gaseous State  Both the rate of diffusion of a gas and its rate of effusion depend on its molar mass. Lighter gases diffuses faster than heavier gases. The gas with highest rate of diffusion is hydrogen. (iv) The volume of gas effused through a given surface per unit time is also called rate of effusion. (2) According to this law, “At constant pressure and temperature, the rate of diffusion or effusion of a gas is inversely proportional to the square root of its vapour density.” 1 Thus, rate of diffusion (r)  (T and P d constant) For two or more gases at constant pressure and temperature, (i) To determine molecular weights of gases. vapour (ii) To prepare indicator, used in mines. Ausell’s d2  2  d1  2 d2  d1 M2 M1 where, M 1 and M 2 are the molecular weights of then, d2 d1 diffusion (a) When equal volume of the two gases diffuse, i.e. V1  V2 r1 t 2   r2 t1 d2 d1 (b) When volumes of the two gases diffuse in the same time, i.e. t1  t 2 d2 d1 U r1 V  1  r2 V2 r1 P  1 r2 P2 (4) Rate determined as, of M2 M1 diffusion  P   r    M   and 60 (1) Kinetic theory was developed by Bernoulli, Joule, Clausius, Maxwell and Boltzmann etc. and represents dynamic particle or microscopic model for different gases since it throws light on the behaviour of the particles (atoms and molecules) which constitute the gases and cannot be seen. Properties of gases which we studied earlier are part of macroscopic model. (2) Postulates (i) Every gas consists of a large number of small particles called molecules moving with very high velocities in all possible directions. effusion (ii) The volume of the individual molecule is negligible as compared to the total volume of the gas. (iii) Gaseous molecules are perfectly elastic so that there is no net loss of kinetic energy due to their collisions. (iv) The effect of gravity on the motion of the molecules is negligible. (v) Gaseous molecules are considered as point masses because they do not posses potential energy. So the attractive and repulsive forces between the gas molecules are negligible. (vi) The pressure of a gas is due to the continuous bombardment on the walls of the containing vessel. (when p is not constant) ST (iii) Since, r  p then, (iii) Atmolysis : The process of separation of two gases on the basis of their different rates of diffusion due to difference in their densities is called atmolysis. It has been applied with success for the separation of isotopes and other gaseous mixtures. U rate D YG r1 V /t w /t  1 1  1 1  r2 V2 / t 2 w 2 / t 2 then, gas ID r1  r2 of the two gases. (ii) Since, Volume of a gas diffused (r)  Time taken for diffusion then, marsh Kinetic theory of gases d2 d1 (3) Graham's law can be modified in a number of ways as, (i) Since, 2  vapour density (V.D.) = Molecular weight then, densities and E3 r1  r2 (5) Applications : Graham's law has been used as follows, (vii) At constant temperature the average K.E. of all gases is same. can be (viii) The average K.E. of the gas molecules is directly proportional to the absolute temperature. (i) Rate of diffusion is equal to distance travelled by gas per unit time through a tube of uniform cross-section. (3) Kinetic gas equation : On the basis of above postulates, the following gas equation was derived, (ii) Number of moles effusing per unit time is also called rate of diffusion. (iii) Decrease in pressure of a cylinder per unit time is called rate of effusion of gas. PV  1 2 mnu rms 3 where, P = pressure exerted by the gas V = volume of the gas m = average mass of each molecule Gaseous State n = number of molecules (ii) u = root mean square (RMS) velocity of the gas. The total number 231 of collision per unit time are given by, Z AA  bimolecular 1 (4) Calculation of kinetic energy K.E. of one molecule  n 1 mu 2 2  molecules 1 3 mnu 2  PV 2 2 1,  A  B Then K.E. of 1 mole gas  60 = (iv) (a) At particular temperature; Z  p 2 3 RT 2 3  8.314  T  12.47 T Joules. 2 Average K.E. per mole 3 RT 3   KT N (Avogadro number ) 2 N 2 (b) At particular pressure; Z  T 3 / 2 (c) At particular volume; Z  T 1 / 2 (3) During molecular collisions a molecule covers a small distance before it gets deflected. The average distance travelled by the gas molecules between two successive collision is called mean free path (). Average distance travelled per unit time( u av ) . No. of collisions made by single molecule per unit time (Z A ) ID R    Boltzmann constant  K  N    (i) If T  0 K (i.e.,  273.15 o C) then, average K.E. = 0. D YG Thus, absolute zero (0K) is the temperature at which molecular motion ceases. Molecular collisions (1) The closest distance between the centres of two molecules taking part in a collision is called molecular or collision diameter (). The molecular diameter of all the gases is nearly same lying in the U order of 10 8 cm. ST  Molecular diameter (2) The number of collisions taking place in unit time per unit volume, called collision frequency (z). (i) The number of collision made by a single molecule with other molecules per unit time are given by, Z A  2 2uav.n where n is the number of molecules per unit molar volume, Avogadro number( N 0 ) 6.02  10 23 3  m Vm 0.0224 u av 2 2 u avr.n  1 2n  2 (4) Based on kinetic theory of gases mean free T path,  . Thus, P U This equation shows that K.E. of translation of a gas depends only on the absolute temperature. This is known as Maxwell generalisation. Thus average K.E.  T. n 1/ 2 2 M A , M B are molecular weights (M  mN 0 ) ( PV  RT )  where,  AB  1 mnu 2 ) 3 n  (M A  M B )  2  Z AB   AB 8RT  MAMB   E3 ( PV  of  2 u av. n 2 (iii) If the collisions involve two unlike molecules, the number of bimolecular collision are given by, We know that, K.E. 2 Larger the size of smaller the mean free path, i.e.,   the 1 molecules, (radius) 2 (ii) Greater the number of molecules per unit volume, smaller the mean free path. (iii) Larger the temperature, larger the mean free path. (iv) Larger the pressure, smaller the mean free path. (5) Relation between collision frequency (Z) and u mean free path () is given by, Z  rms  Molecular speeds or velocities (1) At any particular time, in the given sample of gas all the molecules do not possess same speed, due to the frequent molecular collisions with the walls of the container and also with one another, the molecules move with ever changing speeds and also with ever changing direction of motion. (2) According to Maxwell, at a particular temperature the distribution of speeds remains constant and this distribution is referred to as the Maxwell-Boltzmann distribution and given by the following expression, dn 0  M   4   n  2RT  3/2.e  Mu 2 / 2 RT.u 2 dc 232 Gaseous State where, dn 0  Number of molecules out of total (ii) (i) E3 (ii) Relation Relation v av : between  mp and u rms :  mp between and v av : (iv) Relation between  mp , v av and u rms 1800 K (T3)  mp U D YG u12  u 22  u 32 ..... u n2  n U and v av  1.128   mp where k = Boltzmann constant  R N0 ST (a) For the same gas at two different temperatures, the ratio of RMS velocities will be, T1 T2 (b) For two different gases at the same temperature, the ratio of RMS velocities will be, M2 M1 (c) u rms between or urms  1.085  vav (iii) 3 RT 3kT 3P 3 PV 3 RT      (mN 0 )  M (mN 0 )  M M m d u1  u2 Relation vav  0.9213  urms (i) Root mean square velocity (urms) : It is the square root of the mean of the squares of the velocity of a large number of molecules of the same gas. u1  u2 2P d (5) Relation between molecular speeds or velocities, (4) Types of molecular speeds or Velocities u rms 2 PV  M or u rms  1.224   mp 300 K (T1) T1 SO3 > PCl3 is order of rate of diffusion.  Vapour density is independent of temperature and has no unit while absolute density is dependent of temperature and has unit of gm–1  The isotherms of CO2 were first studied by Andrews. 60 (5) Joule-Thomson effect : When a real gas is allowed to expand adiabatically through a porous plug or a fine hole into a region of low pressure, it is accompanied by cooling (except for hydrogen and helium which get warmed up).  1 Cal = 4.2 Joule, 1 Kcal = 4200 Joule E3 (v) Compressed oxygen is used for welding purposes. (vi) Compressed helium is used in airships.