physics notes for neet chapter 13.pdf

60 Kinetic Theory of Gases 593 Chapter E3 13 Kinetic Theory of Gases (4) The volume of molecules is negligible in comparison to the volume of gas. (The volume of molecules is only 0.014% of the volume of the gas). ID Gas In gases the intermolecular forces are very weak and its molecule may fly apart...

60 Kinetic Theory of Gases 593 Chapter E3 13 Kinetic Theory of Gases (4) The volume of molecules is negligible in comparison to the volume of gas. (The volume of molecules is only 0.014% of the volume of the gas). ID Gas In gases the intermolecular forces are very weak and its molecule may fly apart in all directions. So the gas is characterized by the following properties. (ii) It expands indefinitely and uniformly to fill the available space. D YG (iii) It exerts pressure on its surroundings. (6) The speed of gas molecules lie between zero and infinity U (i) It has no shape and size and can be obtained in a vessel of any shape or size. (5) Molecules of a gas keep on moving randomly in all possible direction with all possible velocities. (iv) Intermolecular forces in a gas are minimum. (v) They can easily compressed and expand. U Assumption of Ideal Gases (or Kinetic Theory of Gases) ST Kinetic theory of gases relates the macroscopic properties of gases (such as pressure, temperature etc.) to the microscopic properties of the gas molecules (such as speed, momentum, kinetic energy of molecule etc.) Actually it attempts to develop a model of the molecular behaviour which should result in the observed behaviour of an ideal gas. It is based on following assumptions : (7) The gas molecules keep on colliding among themselves as well as with the walls of containing vessel. These collisions are perfectly elastic. (8) The time spent in a collision between two molecules is negligible in comparison to time between two successive collisions. (9) The number of collisions per unit volume in a gas remains constant. (10) No attractive or repulsive force acts between gas molecules. (11) Gravitational attraction among the molecules is ineffective due to extremely small masses and very high speed of molecules. (12) Molecules constantly collide with the walls of container due to which their momentum changes. The change in momentum is transferred to the walls of the container. Consequently pressure is exerted by gas molecules on the walls of container. (13) The density of gas is constant at all points of the container. Gas Laws (1) Boyle’s law : For a given mass of an ideal gas at constant temperature, the volume of a gas is inversely proportional to its pressure. P1 P2 (1) Every gas consists of extremely small particles known as molecules. The molecules of a given gas are all identical but are different than those of another gas. Increase pressure (2) The molecules of a gas are identical, spherical, rigid and perfectly elastic point masses. (3) Their size is negligible in comparison to intermolecular distance (10 m) –9 V1 1 i.e. V  (A) P Decrease pressure V2 (B) or PV = constant  P1 V1  P2 V2 Fig. 13.1 594 Kinetic Theory of Gases (As volume  P   constant or m  (Densityof the gas) P1 1  P2 2 and m = constant) P P P N (ii) PV = P    constant   constant or 1  2 n1 n2 n n N (iii) As number of molecules per unit volume n  V N V  also N = constant n (iv) Graphical representation : If m and T are constant P P  T or P V V (B) (C) P P P  constant  1  2 T T T 1 2 (i) The volume remaining constant, the pressure of a given mass of a increases or decreases by Pt 1 of its pressure at 0°C for 273.15 P0 each 1°C rise or fall in temperature. 1/P (D) (E) ID 1   Pt  P0 1  t 273.15   1/V This is pressure centigrade scale. (2) Charle's law : If the pressure remaining constant, the volume of the given mass of a gas is directly proportional to its absolute temperature. U P D YG V1 V V V   1  2  constant Fig. 13.3 T1 T2 T ST This is Charle’s centigrade scale. law P or 1/P T or 1/T (B)P (A) (C) – 273.15 t(°C) O for T Fig. 13.4 1/T (D) (B) or T  constant  1 T1   2 T2 (ii) If the pressure remains constant, the volume of the given mass of a gas increases or decreases by Vt 1 of its volume at 0°C for 273.15 V0 each 1°C rise or fall in temperature. 1   Vt  V0 1  t. 273.15   T 1/P V2 m m V  constant (As volume V  )  T  T (E) Fig. 13.6 (4) Avogadro’s law : Equal volume of all the gases under similar conditions of temperature and pressure contain equal number of molecules i.e. N 1  N 2. (5) Grahm’s law of diffusion : When two gases at the same pressure and temperature are allowed to diffuse into each other, the rate of diffusion of each gas is inversely proportional to the square root of the density of the 1 1 gas i.e. r   (M is the molecular weight of the gas)  M  r1  r2 2  1 M2 M1 If V is the volume of gas diffused in t sec then (v) Graphical representation: If m and P are constant V P/T T2 > T1 Decrease volume U (i) Fig. 13.5 for P/T Increase volume i.e., V  T law O P P (A) t(°C) – 273.15 (ii) Graphical representation : If m and V are constants Fig. 13.2 T1 V or 1/V (3) Gay-Lussac’s law or pressure law : The volume remaining constant, the pressure of a given mass of a gas is directly proportional to its absolute temperature. E3 V (E) Fig. 13.5 gas P (A) T or 1/T (D) PV PV V/T V/T 60 m  (i) PV = P    constant   r V 1/V r V t V  1  1 2 r V t1 t 2 2 (6) Dalton’s law of partial pressure : The total pressure exerted by a mixture of non-reacting gases occupying a vessel is equal to the sum of the T T (A) (B) 1/T (C) Kinetic Theory of Gases 595 individual pressures which each gases exert if it alone occupied the same volume at a given temperature. For n gases P  P1  P2  P3 ..... Pn and PV for exactly one mole of various RT real gases as a function of pressure P, shows a deviation from identity. experimentally determined value of The equation which relates the pressure (P) volume (V) and temperature (T) of the given state of an ideal gas is known as ideal gas equation or equation of state. (4) The quantity PV is called the compressibility factor and should RT be unit for an ideal gas. PV  R (constant)  PV = RT T 60 Equation of State or Ideal Gas Equation 2 Table 13.1 : Different forms of gas equation  mole gas PV = RT 1 molecule of gas  R   T  kT PV     NA  N molecules of gas PV = NkT 1 gm of gas R PV    T  rT M m gm of gas PV = mrT E3 Equation PV = RT 1.5 Constant PV RT R = universal gas constant CO2 1 0.5 k = Boltzmann's constant 0 r = Specific gas constant 400 600 800 1000 Fig. 13.7 U 2 D YG PV Pressure  Volume Work done R   T   Temperatu re   Temperatu re (i) At S.T.P. the value of universal gas constant is same for all gases R cal J cal ~ 2 = 8.31  1.98 mol  kelvin mole  kelvin mole  kelvin litre atm. mole  kelvin U (ii) Dimension : [ML2T 2 1 ] ST (2) Boltzman's constant (k) : It is represented by per mole gas R 8.31  1.38  10 23 J / K constant i.e., k   N 6.023  10 23 It's dimension : [ML2 T 2 1 ] (3) Specific gas constant (r) : It is represented by per gram gas Joule R constant i.e., r . It's unit is and dimension M gm  kelvin [L2 T 2 1 ] Since the value of M is different for different gases. Hence the value of r is different for different gases. e.g. It is maximum for hydrogen rH 2  Real Gases 200 (atm) (5) Deviation from ideal behaviour as a function ofPtemperature (1) Universal gas constant (R) : Universal gas constant signifies the work done by (or on) a gas per mole per kelvin.  0.8221 Ideal gas ID Quantity of gas CH4 H2 N2 where R = universal gas constant. 1 mole gas PV  1. Plotting the RT (3) For exactly one mole of an ideal gas where P = Pressure exerted by mixture P1 , P2 , P3 ,...... Pn  Partial pressure of component gases. For 1 mole of gas (1) The gases actually found in nature are called real gases. (2) They do not obeys gas Laws. R 2 200 K 1.5 PV RT 500 K 1000 K 1 Ideal gas 0.5 0 300 600 900 1200 P(atm) (6) A real gas behaves as ideal gas most closely at low pressure and Fig. 13.8 high temperature. Also can actual gas can be liquefied most easily which deviates most from ideal gas behaviour at low temperature and high pressure. (7) Equation of state for real gases : It is given by Vander Waal's with two correction in ideal gas equation. The it know as Vander Waal's gas equation. (i) Volume correction : Due to finite size of molecule, a certain portion of volume of a gas is covered by the molecules themselves. Therefore the space available for the free motion of molecules of gas will be slightly less than the volume V of a gas. Hence the effective volume becomes (V – b). (ii) Pressure correction : Due to intermolecular force in real gases, molecule do not exert that force on the wall which they would have exerted in the absence of intermolecular force. Therefore the observed pressure P of the gas will be less than that present in the absence of intermolecular a   force. Hence the effective pressure becomes  P  2 . V   (iii) Vander Waal's gas equations a   For 1 mole of gas  P  2  (V  b)  RT V   596 Kinetic Theory of Gases Consider an ideal gas (consisting of N molecules each of mass m) enclosed in a cubical box of side L.  a 2  For  moles of gas  P  2  (V  b)   RT V   Here a and b are constant called Vander Waal’s constant. Y Y Dimension : [a] = [ML5 T 2 ] and [b] = [L ] 3 v Units : a = N  m and b = m. (8) Andrews curves : The pressure (P) versus volume (V) curves for actual gases are called Andrews curves. 4 G v Z 370°C E 360°C C B 350°C Vapour A Fig. 13.9 (i) At 350°C, part AB represents vapour phase of water, in this part (B) (2) Time during collision : Time between two successive collision with the wall A. 1 Distancetravelled by molecule between two successivecollision t  Velocityof molecule 2L vx  (iii) At 370°C this portion is further decreased. (iv) At 374.1°C, it reduces to point (H) called critical point and the temperature 374.1°C is called critical temperature (T ) of water. c (3) Collision frequency (n) : It means the number of collision per v 1 second. Hence n   x t 2 L (4) Change in momentum : This molecule collides with the shaded wall (A ) with velocity v and rebounds with velocity v x. U (ii) At 360°C portion representing the co-existence of liquid vapour phase is shorter. D YG (v) The phase of water (at 380°C) above the critical temperature is called gaseous phase. (9) Critical temperature, pressure and volume : The point on the P-V curve at which the matter gets converted from gaseous state to liquid state is known as critical point. At this point the difference between the liquid and vapour vanishes i.e. the densities of liquid and vapour become equal. (i) Critical temperature (T ) : The maximum temperature below which a gas can be liquefied by pressure alone is called critical temperature and is characteristic of the gas. A gas cannot be liquefied if its temperature is more than critical temperature. c CO (31.1°C), O (–118°C), N (–147.1°C) and H O (374.1°C) U (A) ID 1  Boyle’s law is obeyed  P  . Part BC represents the co-existence of V  vapour and liquid phases. At point C, vapours completely change to liquid phase. Part CD is parallel to pressure axis which shows that compressibility of the water is negligible. 2 Z motion of molecule v x  vy  vz  v 2  3v x2  3vy2  3vz2 V Andrews curve for water 2 1 x The change in momentum of the molecule p  (mv x )  (mv x )  2mv x As the momentum remains conserved in a collision, the change in momentum of the wall A is p  2mv x 1 After rebound this molecule travel toward opposite wall A with velocity v x , collide to it and again rebound with velocity v x towards wall A. (5) Force on wall : Force exerted by a single molecule on shaded wall is equal to rate at which the momentum is transferred to the wall by this molecule. 2 1 i.e. FSingle molecule  2 (ii) Critical pressure (P ) : The minimum pressure necessary to liquify a gas at critical temperature is defined as critical pressure CO (73.87 bar) and O (49.7atm) p 2mv x mv x2   t (2 L / v x ) L The total force on the wall ST c 2 2 (iii) Critical volume (V ) : The volume of 1 mole of gas at critical pressure and critical temperature is defined as critical volume CO (95 10 m) c –6 2 3 (iv) Relation between Vander Waal’s constants and T , P , V Tc  c c a 8a , Pc  , Vc  3b , 27 Rb 27b 2 RT 27 R 2 Tc2 a , b   c 8  Pc 64 Pc Pressure of an Ideal Gas X  Fig. 13.10 (1) Instantaneous velocity : Any molecule of gas moves with velocity v in any direction  where v  v x ˆi  v y ˆj  v z kˆ  v  v x2  vy2  v z2. Due to random 374.1°C F vx vz 60 Liquid vapour region H Liquid X x L Gas 380°C 2 v E3 D P vy x 3  PV 3  and c c  R  T 8 c  c m Fx  L  A1 due to all the molecules m 2 mN 2 v x2 = (v x  v x22  v x23 ...)  vx M 1 L v x2  mean square of x component of the velocity. (6) Pressure : Now pressure is defined as force per unit area, hence F mN 2 mN 2 pressure on shaded wall Px  x  vx  vx A AL V For any molecule, the mean square velocity v 2  v x2  vy2  v z2 ; by symmetry v x2  vy2  v z2  v x2  v y2  v z2  v2 3 Total pressure inside the container P 1 mN 2 1 m N 2 v  vrms 3 V 3 V (where vrms  v 2 ) Kinetic Theory of Gases 597 (7) Relation between pressure and kinetic energy : As we know 1 mN 2 1 M 2 1 2... (i) P v rms  v rms  P   v rms 3 V 3 V 3 M ] [As M = mN = Total mass of the gas and   V...(ii) in gas, as v rms  2 E 3 i.e. the pressure exerted by an ideal gas is numerically equal to the two third of the mean kinetic energy of translation per unit volume of the gas. (8) Effect of mass, volume and temperature on pressure : (m N )T 1 mN 2 2 or P  [As v rms T] P v rms V 3 V (i) If volume and temperature of a gas are constant P  mN i.e. Pressure  (Mass of gas). i.e. if mass of gas is increased, number of molecules and hence number of collision per second increases i.e. pressure will increase. From (i) and (ii), we get P  (iii) If mass and volume of gas are constant, P  (v rms )2  T i.e. vrms  D YG The motion of molecules in a gas is characterised by any of the following three speeds. (1) Root mean square speed : It is defined as the square root of mean of squares of the speed of different molecules  v 22  v32  v42 .... N  3P 3 PV  Mass of gas U   3 RT  M 3kT m ST Mass of gas where    Densityof the gas , M =   (mass of V gas), pV  RT , R = kN A , k  Boltzmann’s constant, m= rms 3  vs (vii) Moon has no atmosphere because v of gas molecules is more than escape velocity (v ). rms e A planet or satellite will have atmosphere only if vrms  ve (viii) At T = 0; v = 0 i.e. the rms speed of molecules of a gas is zero at 0 K. This temperature is called absolute zero. rms (2) Most probable speed : The particles of a gas have a range of speeds. This is defined as the speed which is possessed by maximum fraction of total number of molecules of the gas. e.g., if speeds of 10 molecules of a gas are 1, 2, 2, 3, 3, 3, 4, 5, 6, 6 km/s, then the most probable speed is 3 km/s, as maximum fraction of total molecules possess this speed. 2P Most probable speed vmp    2 RT  M 2kT m v1  v 2  v 3  v 4 ..... N and according to kinetic theory of gases v av  Average speed vav  v2 1 2 (i) From the expression of pressure P   v rms 3  vrms   v rms  (3) Average speed : It is the arithmetic mean of the speeds of molecules in a gas at given temperature. Various Speeds of Gas Molecules v12 M (vi) rms speed of gas molecules does not depends on the pressure of gas (if temperature remains constant) because P   (Boyle’s law) if pressure is increased n times then density will also increases by n times but v remains constant. U i.e., if temperature increases, the mean square speed of gas molecules will increase and as gas molecules are moving faster, they will collide with the walls more often with greater momentum resulting in greater pressure. RT 3 RT and v s  M ID (ii) If mass and temperature of a gas are constant. P  (1/V), i.e., if volume decreases, number of collisions per second will increase due to lesser effective distance between the walls resulting in greater pressure. 3 times that of speed of sound  (v) rms speed of gas molecules is 60 1M 2 1 2   vrms   vrms 2V  2 3  8.31  273  1840 m / s. 2  10 3 E3  K.E. per unit volume E  3 RT  M (vrms )  M = mass of each molecule. NA 8P   8 RT  M  8 kT  m Maxwell’s Law (or the Distribution of Molecular Speeds (1) The v gives us a general idea of molecular speeds in a gas at a given temperature. This doesn't mean that the speed of each molecule is v. Many of the molecules have speed less than v and many have speeds greater than v. rms rms rms rms (2) Maxwell derived as equation given the distribution of molecules in different speed as follow  m  dN  4N    2kT  3/2 2 v e  mv 2 2 kT dv where dN = Number of molecules with speeds between v and v + dv. (ii) With rise in temperature rms speed of gas molecules increases as v rms  T. (iii) With increase in molecular weight rms speed of gas molecule 1 decreases as v rms . e.g., rms speed of hydrogen molecules is four M times that of oxygen molecules at the same temperature. (iv) rms speed of gas molecules is of the order of km/s e.g., at NTP for hydrogen gas dN At a particular temperature dv (Number of molecules at a particular speed) vmp vav vrms Fig. 13.11 v(m/s) 598 Kinetic Theory of Gases This graph also represent that vrms  vav  vmp (Order remember trick RAM) (5) If average speed of molecule is v then   v  3 RT 8 RT 2 RT    M M M RT RT RT  1.6  1.41 M M M  1 and   m i.e. the mean free path is inversely  proportional to the density of a gas and directly proportional to the mass of each molecule.   Area bonded by this curve with speed axis represents the number of molecules corresponds to that velocity range. This curve is asymmetric curve. T1 dN dv (A) (ii) As   1 kT 2 2 d P  U (Because with temperature rise average molecular speed increases). Mean Free Path.  D YG (1) The distance travelled by a gas molecule between two successive collisions is known as free path. Total distance travelled by a gas molecule between successivecollisions  Total number of collisions During two successive collisions, a molecule of a gas moves in a straight line with constant velocity and Let 1 , 2 , 3..... be the distance travelled by a gas molecule during n collisions respectively, then the mean free path of a gas molecule is given by U 1   2   3 ....  n n ST 1 2nd 2  N P   n  Number of molecule V kT. m 2 (mn )d The term degree of freedom of a system refers to the possible independent motions, systems can have. or The total number of independent modes (ways) in which a system can possess energy is called the degree of freedom (f). The independent motions can be translational, rotational or vibrational or any combination of these. General expression for degree of freedom Fig. 13.13 volume 2 2 d P Degree of Freedom (iii) Vibrational degree of freedom n = Number of molecules per unit kT Fig. 13.15 (ii) Rotational degree of freedom where d = Diameter of the molecule, 1 (B) (i) Translational degree of freedom 2nd 2 (3) As PV =  RT =  NkT  T P (A) So the degree of freedom are of three types : 1 (4) From  . For constant volume and hence constant ID v per unit volume so   Fig. 13.14 P is constant so that  will not T depend on P and T. But if volume of given mass of a gas is allowed to 1 change with P or T then   T at constant pressure and   at P constant temperature. Fig. 13.12 (2)   (B) number density n of gas molecules, T2 T2 > T1  m  E3 Effect of temperature on velocity distribution : With temperature rise dN the vs v. Curve shift towards right and becomes broader. dv t  vT N [As N = Number of collision in time t, T = time interval between two collisions]. (i) As  1.77 [As m = Mass each molecule, mn = Mass per unit volume = Density = ] 60 dN (number of molecules at a particular speed) dv dN and v (speed of these molecules). From the graph it is seen that is dv maximum at most probable speed. (3) Graph between 2  f = 3A – B ; where A = Number of independent particles, B = Number of independent restriction (1) Monoatomic gas : Molecule of monoatomic gas can move in any direction in space so it can have three y independent motions and hence 3 degrees of vy freedom (all translational) v m 2d  z vx x vz 2 Fig. 13.16 Kinetic Theory of Gases 599 Fig. 13.17 Hence it can have only two rotational motion. Thus a diatomic molecule has 5 degree of freedom : 3 translational and 2 rotational. (3) Triatomic gas (Non-linear) : A non-linear molecule can rotate about any of three co-ordinate axes. Hence y it has 6 degrees of freedom : 3 translational and 3 rotational. x A B Figure He, Ne, Ar 1 0 f=3 Diatomic H2, O2, N2, Cl2 etc. 2 1 f=5 H2O 3 3 f=6 B A non B A D YG Triatomic linear CO2, BeCl2 3 2 f=7 A 1 gm gas 3 rT ; r = Specific gas constant 2 3 m rT 2 (1) Kinetic energy per molecule of gas does not depends upon the mass of the molecule but only depends upon the temperature of the gas. As 2 A B B U A diatomic molecule that is free to vibrate (in addition to translation and rotation) will have 7 (2  3  2) degrees of freedom. Kinetic Energy of Ideal Gas ST (5) From the above expressions it is clear that higher the temperature of the gas, more will be the average kinetic energy possessed by the gas molecules at T = 0, E = 0 i.e. at absolute zero the molecular motion stops. A The above degrees of freedom are shown at room temperature. Further at high temperature, in case of diatomic or polyatomic molecules, the atoms with in the molecule may also vibrate with respect to each other. In such cases, the molecule will have an additional degrees of freedom, due to vibrational motion. An object which vibrates in one dimension has two additional degree of freedom. One for the potential energy and one for the kinetic energy of vibration. In ideal gases, the molecules are considered as point particles. For point particles, there is no internal excitation, no vibration and no rotation. The point particles can have only translational motion and thus only translational energy. For an ideal gas the internal energy can only be tranlational kinetic energy. Law of Equipartition of Energy According to this law, for any system in thermal equilibrium, the total energy is equally distributed among its various degree of freedom. And each 1 degree of freedom is associated with energy kT (where 2 k  1.38  10 23 J / K , T = absolute temperature of the system). (1) At a given temperature T, all ideal gas molecules no matter what their mass have the same average translational kinetic energy; namely, 3 kT. When measure the temperature of a gas, we are also measuring the 2 average translational kinetic energy of it' s molecules. (2) At same temperature gases with different degrees of freedom (e.g., He and H ) will have different average energy or internal energy namely 2 f kT. (f is different for different gases) 2 (3) Different energies of a system of degree of freedom f are as follows Hence kinetic energy (or internal energy) of 1 mole ideal gas (i) Total energy associated with each molecule = E 1 1 3 RT 3 2 Mvrms  M  RT 2 2 M 2 (ii) Total energy associated with N molecules = Table 13.3 : Various Translational kinetic energies Quantity of gas 2 (4) Kinetic energy per gram of gas depend upon the temperature as well as molecular weight (or mass of one molecule) of the gas. T 3 k E gram  T  E gram  m 2m B B A 3 NkT 2 U A Triatomic linear N molecule (3) Kinetic energy per mole of gas depends only upon the temperature of gas. A A 3 k T ; k = Boltzmann’s constant 2 (2) For two gases at the same temperature m1 (vrms )12  m 2 (vrms )22 A–B Monoatomic 1 molecule 3 kT or E  T i.e. molecules of different gases say He, H and O 2 etc. at same temperature will have same translational kinetic energy though their r.m.s. speed are different. Fig. 13.18 f=3 3 RT 2 E3 Example  mole gas ID Atomicity of gas z 3 RT ; R = Universal gas constant 2 m gm gas E Table 13.2 ; Degree of freedom for different gases 1 mole gas 60 (2) Diatomic gas : Molecules of diatomic gas are made up of two atoms joined rigidly to one another y through a bond. This cannot only move bodily, but also rotate about one of the three co-ordinate axes. However its x moment of inertia about the axis joining the two atoms is negligible compared to that about the other two axes. z Kinetic energy (iii) Total energy associated with  mole = f kT 2 f NkT 2 f RT 2 600 Kinetic Theory of Gases (iv) Total energy associated with  molen = Specific Heat in Terms of Degree of Freedom f RT 2 (1) C : For a gas at temperature T, the internal energy f f U  RT  Change in energy U  RT... (i) 2 2 Also, as we know for any gas heat supplied at constant volume... (ii) (Q)V  CV T  U V f (v) Total energy associated with each gram = rT 2 f mrT 2 1 fR 2 (2) C : From the Mayer’s formula C p  Cv  R From equation (i) and (ii) CV  Specific Heat (CP and CV) of a Gas The specific heat of gas can have many values, but out of them following two values are very important (1) Specific heat at constant volume (C ) : The specific heat of a gas at constant volume is defined as the quantity of heat required to raise the temperature of unit mass of gas through 1°C or 1 K when its volume is kept (Q)V constant, i.e., cV  m T P  C P  CV  R  V v M (Q)V 1 (Q)V  m T  T m   As   M    (2) Specific heat at constant from (C ) : The specific heat of a gas at constant pressure is defined as the quantity of heat required to raise the temperature of unit mass of gas through 1 K when its pressure is kept P (Q) p D YG M (Q) p m T 1 (Q) p   T m   As   M    Mayer's Formula (1) Out of two principle specific heats of a gas, C is more than C because in case of C , volume of gas is kept constant and heat is required only for raising the temperature of one gram mole of the gas through 1°C or 1 K. Hence no heat, what so ever, is spent in expansion of the gas. P and number of molecules of second gas N 2   2 N A (1) Total mole fraction   (1   2 ). (2) If M 1 is the molecular weight of first gas and M 2 that of second gas. Then molecular weight of mixture M  V U It means that heat supplied to the gas increases its internal energy …..(i) only i.e. (Q)V  U  CV T (2) While in case of C the heat is used in two ways P ST (ii) In doing work, due to expansion at constant pressure (W) (Q)P  U  W   CP T From equation (i) and (ii)  …..(ii) m1 CVmix C P  CV  PV  T C  m2 V1 1CV1   2 CV2 M M2  1  m1 m 2 1   2  CV2 M1 M 2 (4) Specific heat of the mixture at constant pressure will be C Pmix 1CP1  2CP2   1  2  1     R  2  2 R    1    1  1   2  1  1   2   1     R    2  2    1     1  2    1  1    2  1   CP T   CV T  W  T (CP  CV )  PV  1 M1   2 M 2 1  2 (3) Specific heat of the mixture at constant volume will be (i) In increasing the temperature of the gas by T So A 2 Number of molecules of first gas N 1  1 N A p V V If two non-reactive gases are enclosed in a vessel of volume V. In the mixture  moles of one gas are mixed with  moles of another gas. If N is Avogadro’s number then 1 m T P Gaseous Mixture If instead of unit mass, 1 mole of gas is considered, the specific heat is called molar specific heat at constant pressure and is represented by C. C p  MC p  v U constant, i.e., c P  p ID CV  McV  f    1 R CP  2  1 2 (3) Ratio of C and C () :    f CV f R 2 (i) Value of  is different for monoatomic, diatomic and triatomic 5 7 4 gases.  mono   1.6,  di   1.4 ,  tri   1.33 3 5 3 (ii) Value of  is always more than 1. So we can say that always C > C. E3 If instead of unit mass, 1 mole of gas is considered, the specific heat is called molar specific heat at constant volume and is represented by capital C. f f  R  R    1 R 2 2  60 (iv) Total energy associated with m gram = =R  [For constant pressure, W = PV also from PV = RT, PV = RT] This relation is called Mayer’s formula and shows that C P  CV i.e. molar specific heat at constant pressure is greater than that at constant volume.  m1   1  m 2   2   R      m1 m 2  M1   1  1  M 2   2  1     M1 M 2 (1C P1   2 C P2 ) (5)  mixture  C Pmix CVmix  1   2 (1CV1   2 CV2 ) 1   2 Kinetic Theory of Gases 601    1       R   2  2  R  1     1     1 1     2    R   R    2   1      1     1  1   2    mixture  v :v :v = 3: rms av mp 8  : 2  3 : 2.5 : 2  For oxygen gas molecules v = 461 m/s, v = 424.7 m/s and v = rms av rms 376.4 m/s 1 1   2 2   1  2  1 1 1 ( 2  1)   2 2 ( 1  1)  1  1 2 1 ( 2  1)   2 ( 1  1)  1  1  2  1  An atom in a solid though has no degree of freedom for translational and rotational motion, due to vibration along 3 axes has 3  2 = 6 degrees of freedom (and not like an ideal gas molecule). When a diatomic or polyatomic gas dissociates into atoms it behaves as monoatomic gas whose degree of freedom are changed accordingly 60 1C P1   2 C P2  1CV1   2 CV2  In General a polyatmic molecule has 3 translational, 3 rotational degree of freedom and a certain number of vibration mode fvib. Hence  The cooking gas cylinder contains L.P.G. (Liquid Petroleum gas) which is saturated. And as pressure 4  fvib. 3  fvib E3  poly   Only average translational kinetic energy of a gas contributes to its temperature. Two gases with the same average translational kinetic energy have the same temperature even if one has grater rotational energy and thus greater internal energy. of saturated vapours is independent of volume (at constant temperature). the pressure of gas coming out of the cylinder remains constant  Unsaturated vapours obey gas laws while saturated vapours don’t. ID till the cylinder becomes empty.  If the number of molecules in a gas increases, then the 4  b  4 N A  r 3  ; r = radius of each molecule and N = avogrado 3   number. A U temperature, kinetic energy and pressure of the gas increases because P  n, T  n and kinetic energy  T  n.  For real gases effective volume is considered as (V – b) where  At constant volume if T increases then v , v , P and collision  Variation of degree of freedom of a diatomic gas (H ) with frequency increases. temperature. At very low temperature only translation is possible. as the temperature increases rotational motion can begin. At still higher temperatures vibratory motion can begin. rms D YG  If two gases are filled in vessel then nothing can be predicted about the pressure of gases. However their mean molecular energies will be same but their rms velocities will be different.  The average distance between two gas molecules at NTP is 10 m.  The space available for a single gas molecule at NTP is 37.2  10 –9 m. 2 –2 2 CV 4R 7R/2 Vibration 5R/2 Rotation 3R/2  The molecules of gases will escape out from a planet if the Mve2 temperature of planet T  ; where v = escape velocity from the 3R planet, R = universal, gas constant and M = Molecular mass of the gas. U e  As f (degree of freedom) increases then C , C  and  . P V ST  The number of molecules present in 1 gm mole of a gas is defined as Avogadro number (N ). A N A  6.023  10 23 per gm mole  6.023  10 26 per kg mole. At S.T.P. or N.T.P. (T = 273 K and P = 1 atm) 22.4 litre of each gas has 6.023  10 23 molecule  One mole of any gas at S.T.P. occupy 22.4 litre of volume e.g. 32 gm oxygen, 28 gm nitrogen and 2gm hydrogen occupy the same volume at S.T.P.  For any gas 1 mole = M gram = 22.4 litre = 6.023  10 molecule. 23 Translation 100 1000 10000 Temperature(K)