Current Affairs JEE Main & Advanced

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. Hence kinetic energy (or internal energy) of 1 mole ideal gas \[E=\frac{1}{2}Mv_{rms}^{2}=\frac{1}{2}M\times \frac{3RT}{M}=\frac{3}{2}RT\]   Various Translational kinetic energies  

Quantity of gas	Kinetic energy
1 mole gas	\[\frac{3}{2}RT\]; R = Universal gas constant
\[\mu \] mole gas	\[\frac{3}{2}\mu RT\]
1 molecule	\[\frac{3}{2}k\,T\]; k = Boltzmann?s constant
N molecule	\[\frac{3}{2}N\,k\,T\]
1 gm gas	\[\frac{3}{2}rT\]; r = Specific gas constant
m gm gas

(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  \[E=\frac{3}{2}\,kT\,\]  or   \[E\propto T\] i.e. molecules of different gases say He, \[{{H}_{2}}\] and \[{{O}_{2}}\] etc. at same temperature will have same translational kinetic energy though their r.m.s. speed are different. (2) For two gases at the same temperature \[{{m}_{1}}({{v}_{rms}})_{1}^{2}={{m}_{2}}({{v}_{rms}})_{2}^{2}\] (3) Kinetic energy per mole of gas depends only upon the temperature of gas. (4) Kinetic energy per gram of gas depend upon the temperature as well as molecular weight (or mass of one molecule) of the gas. \[{{E}_{gram}}=\frac{3}{2}\frac{k}{m}T\,\] \[\Rightarrow \] \[{{E}_{gram}}\propto \frac{T}{m}\] (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.  

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. So the degree of freedom are of three types : (i) Translational degree of freedom (ii) Rotational degree of freedom (iii) Vibrational degree of freedom General expression for degree of freedom \[f=3A-B;\] 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 independent motions and hence 3 degrees of freedom (all translational) (2) Diatomic gas : Molecules of diatomic gas are made up of two atoms joined rigidly to one another through a bond. This cannot only move bodily, but also rotate about one of the three co-ordinate axes. However its moment of inertia about the axis joining the two atoms is negligible compared to that about the other two axes. 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 it has 6 degrees of freedom : 3 translational and 3 rotational.       Degree of freedom for different gases  

Atomicity of gas	Example	A	B	f = 3 A - B	Figure
Monoatomic	He, Ne, Ar	1	0	f = 3
Diatomic	\[{{H}_{2}},{{O}_{2}}{{,}_{{}}}{{N}_{2}},C{{l}_{2}}\] etc.	2	1	f = 5
Triatomic non linear	\[{{H}_{2}}O\]	3	3	f = 6
Triatomic linear	\[C{{O}_{2}},BeC{{l}_{2}}\]	3	2	f = 7

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 more...

(1) The distance travelled by a gas molecule between two successive collisions is known as free path. \[\lambda =\frac{\text{Total distance travelled by a gas molecule between successive collisions}}{\text{Total number of collisions}}\] During two successive collisions, a molecule of a gas moves in a straight line with constant velocity and Let \[{{\lambda }_{1}},{{\lambda }_{2}},{{\lambda }_{3}}.....\] be the distance travelled by a gas molecule during n collisions respectively, then the mean free path of a gas molecule is given by \[\lambda =\frac{{{\lambda }_{1}}+{{\lambda }_{2}}+{{\lambda }_{3}}+....+{{\lambda }_{n}}}{n}\] (2) \[\lambda =\frac{1}{\sqrt{2}\pi n{{d}^{2}}}\] where d = Diameter of the molecule, n = Number of molecules per unit volume (3) As \[PV=\mu RT=\mu NkT\Rightarrow \frac{N}{V}=\frac{P}{kT}=n=\]Number of molecule per unit volume so \[\lambda =\frac{1}{\sqrt{2}}\frac{kT}{\pi {{d}^{2}}P}\]. (4) From \[\lambda =\frac{1}{\sqrt{2}\pi n{{d}^{2}}}=\frac{m}{\sqrt{2}\pi (mn){{d}^{2}}}\]\[=\frac{m}{\sqrt{2}\pi {{d}^{2}}\rho }\] [As m = Mass each molecule, mn = Mass per unit volume = Density \[=\rho \]] (5) If average speed of molecule is v then \[\lambda =v\times \frac{t}{N}\]\[=v\times T\]    [As N = Number of collision in time t, T = time interval between two collisions]. (i) As  \[\lambda \propto \frac{1}{\rho }\] and \[\lambda \propto \,m\] i.e. the mean free path is inversely proportional to the density of a gas and directly proportional to the mass of each molecule. (ii) As \[\lambda =\frac{1}{\sqrt{2}}\frac{kT}{\pi {{d}^{2}}P}\]. For constant volume and hence constant number density n of gas molecules, \[\frac{P}{T}\] is constant so that l will not depend on P and T. But if volume of given mass of a gas is allowed to change with P or T then \[\lambda \propto T\] at constant pressure and \[\lambda \propto \frac{1}{P}\] at constant temperature.  

(1) The \[{{v}_{rms}}\] 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 \[{{v}_{rms}}\]. Many of the molecules have speed less than \[{{v}_{rms}}\] and many have speeds greater than \[{{v}_{rms}}\]. (2) Maxwell derived as equation given the distribution of molecules in different speed as follow \[dN=4\pi N\,{{\left( \frac{m}{2\pi kT} \right)}^{3/2}}{{v}^{2}}{{e}^{-\frac{m{{v}^{2}}}{2kT}}}dv\] where dN = Number of molecules with speeds between v and v + dv. (3) Graph between \[\frac{dN}{dv}\] (number of molecules at a particular speed) and v (speed of these molecules). From the graph it is seen that \[\frac{dN}{dv}\] is maximum at most probable speed. This graph also represent that \[{{v}_{rms}}>{{v}_{av}}>{{v}_{mp}}\] (Order remember trick RAM) \[\Rightarrow \]\[\sqrt{\frac{3RT}{M}}\,>\sqrt{\frac{8RT}{\pi M}}\,>\,\sqrt{\frac{2RT}{M}}\]\[\Rightarrow \]\[1.77\sqrt{\frac{RT}{M}}>1.6\sqrt{\frac{RT}{M}}\,>\,1.41\sqrt{\frac{RT}{M}}\] Area bonded by this curve with speed axis represents the number of molecules corresponds to that velocity range. This curve is asymmetric curve. Effect of temperature on velocity distribution : With temperature rise the \[\frac{dN}{dv}\,\text{vs}\,v\]. Curve shift towards right and becomes broader. (Because with temperature rise average molecular speed increases).

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 i.e. \[{{v}_{rms}}=\sqrt{\frac{v_{1}^{2}+v_{2}^{2}+v_{3}^{2}+v_{4}^{2}+....}{N}}=\sqrt{\,\overline{{{v}^{2}}}}\] (i) From the expression of pressure \[P=\frac{1}{3}\rho \,v_{rms}^{2}\] \[\Rightarrow \] \[{{v}_{rms}}=\sqrt{\frac{3P}{\rho }}=\sqrt{\frac{3PV}{\text{Mass of gas}}}=\sqrt{\frac{3RT}{M}}=\sqrt{\frac{3kT}{m}}\] \[\text{where }\rho =\frac{\text{Mass of gas}}{V}=\text{Density of the gas}\], \[M=\mu \times \](mass of gas), \[pV=\mu RT\], \[R=k{{N}_{A}},\] \[k=\] Boltzmannís constant, \[m=\frac{M}{{{N}_{A}}}=\]mass of each molecule. (ii) With rise in temperature rms speed of gas molecules increases as \[{{v}_{rms}}\propto \sqrt{T}\]. (iii) With increase in molecular weight rms speed of gas molecule decreases as \[{{v}_{rms}}\propto \frac{1}{\sqrt{M}}\]. e.g., rms speed of hydrogen molecules is four 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 \[({{v}_{rms}})=\sqrt{\frac{3RT}{M}}=\sqrt{\frac{3\times 8.31\times 273}{2\times {{10}^{3}}}}=1840\,m/s\]. (v) rms speed of gas molecules is \[\sqrt{\frac{3}{\gamma }}\] times that of speed of sound in gas, as  \[{{v}_{rms}}=\sqrt{\frac{3RT}{M}}\] and \[{{v}_{s}}=\sqrt{\frac{\gamma RT}{M}}\]\[\Rightarrow \]\[{{v}_{rms}}=\sqrt{\frac{3}{\gamma }}{{v}_{s}}\] (vi) rms speed of gas molecules does not depends on the pressure of gas (if temperature remains constant) because \[P\propto \rho \](Boyle?s law) if pressure is increased n times then density will also increases by n times but vrms remains constant. (vii) Moon has no atmosphere because \[{{v}_{rms}}\] of gas molecules is more than escape velocity \[({{v}_{g}})\]. A planet or satellite will have atmosphere only if  \[{{v}_{rms}}

Consider an ideal gas (consisting of N molecules each of mass m) enclosed in a cubical box of side L. (1) Instantaneous velocity : Any molecule of gas moves with velocity \[\vec{v}\] in any direction Where \[\vec{v}={{v}_{x}}\hat{i}+{{v}_{y}}\hat{j}+{{v}_{z}}\hat{k}\] \[\Rightarrow \] \[v=\sqrt{v_{x}^{2}+v_{y}^{2}+v_{z}^{2}}\]. Due to random motion of molecule \[{{v}_{x}}={{v}_{y}}={{v}_{z}}\] \[\Rightarrow \] \[{{v}^{2}}=3v_{x}^{2}=3v_{y}^{2}=3v_{z}^{2}\] (2) Time during collision : Time between two successive collision with the wall \[{{A}_{1}}\]. \[\Delta t=\frac{\text{Distance travelled by molecule between two successive collision}}{\text{Velocity of molecule}}\] \[=\frac{2L}{{{v}_{x}}}\] (3) Collision frequency (n) : It means the number of collision per second. Hence \[n=\frac{1}{\Delta t}=\frac{{{v}_{x}}}{2L}\] (4) Change in momentum : This molecule collides with the shaded wall  \[({{A}_{1}})\] with velocity \[{{v}_{x}}\] and rebounds with velocity \[-{{v}_{x}}\]. The change in momentum of the molecule \[\Delta p=(-m{{v}_{x}})-(m{{v}_{x}})=-2m{{v}_{x}}\] As the momentum remains conserved in a collision, the change in momentum of the wall \[{{A}_{1}}\] is \[\Delta p=2m{{v}_{x}}\] After rebound this molecule travel toward opposite wall \[{{A}_{2}}\] with velocity \[-{{v}_{x}}\], collide to it and again rebound with velocity \[{{v}_{x}}\] towards wall \[{{A}_{1}}\]. (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. i.e. \[{{F}_{\text{Single molecule}}}=\frac{\Delta p}{\Delta t}=\frac{2m{{v}_{x}}}{(2L/{{v}_{x}})}=\frac{mv_{x}^{2}}{L}\] The total force on the wall \[{{A}_{1}}\] due to all the molecules \[{{F}_{x}}=\frac{m}{L}\sum{v_{x}^{2}}\]\[=\frac{m}{M}(v_{{{x}_{1}}}^{2}+v_{{{x}_{2}}}^{2}+v_{{{x}_{3}}}^{2}+...)=\frac{mN}{L}\overline{v_{x}^{2}}\] \[\overline{v_{x}^{2}}=\]mean square of \[x\] component of the velocity. (6) Pressure :  Now pressure is defined as force per unit area, hence pressure on shaded wall \[{{P}_{x}}=\frac{{{F}_{x}}}{A}=\frac{mN}{AL}\overline{v_{x}^{2}}=\frac{mN}{V}\overline{v_{x}^{2}}\] For any molecule, the mean square velocity \[\overline{{{v}^{2}}}=\overline{v_{x}^{2}}+\overline{v_{y}^{2}}+\overline{v_{z}^{2}}\]; by symmetry \[\overline{v_{x}^{2}}=\overline{v_{y}^{2}}=\overline{v_{z}^{2}}\] \[\Rightarrow\]\ \overline{v_{x}^{2}}=\overline{v_{y}^{2}}=\overline{v_{z}^{2}}=\frac{\overline{{{v}^{2}}}}{3}\] Total pressure inside the container \[P=\frac{1}{3}\frac{mN}{V}\overline{{{v}^{2}}}=\frac{1}{3}\frac{m\,N}{V}v_{rms}^{2}\]              (where \[{{v}_{rms}}=\sqrt{\overline{{{v}^{2}}}}\]) (7) Relation between pressure and kinetic energy : As we know \[P=\frac{1}{3}\frac{m\,N}{V}v_{rms}^{2}\]\[=\frac{1}{3}\frac{M}{V}v_{rms}^{2}\]\[\Rightarrow \]\[P=\frac{1}{3}\rho \,v_{rms}^{2}\]    ... (i) [As M = mN = Total mass of the gas and \[\rho =\frac{M}{V}\]] \[\therefore \] K.E. per unit volume \[E=\frac{1}{2}\left( \frac{M}{V} \right)\,v_{rms}^{2}=\frac{1}{2}\rho \,v_{rms}^{2}\]    ...(ii)         From (i) and (ii), we get \[P=\frac{2}{3}E\] 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 :  \[P=\frac{1}{3}\frac{m\,N}{V}v_{rms}^{2}\] or \[P\propto \frac{(m\,N)T}{V}\] [As \[v_{rms}^{2}\propto T\]] (i) If volume and temperature of a gas are constant \[P\propto mN\] i.e. Pressure µ (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. (ii) If mass and temperature of a gas are constant. \[P\propto (1/V)\], i.e., if volume decreases, number of collisions per second will increase due to lesser effective distance between the walls resulting in greater pressure. (iii) If mass and volume of gas are constant, \[P\propto {{({{v}_{rms}})}^{2}}\propto T\] 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.  

(1) The gases actually found in nature are called real gases. (2) They do not obeys gas Laws. (3) For exactly one mole of an ideal gas \[\frac{PV}{RT}=1.\] Plotting the experimentally determined value of \[\frac{PV}{RT}\] for exactly one mole of various real gases as a function of pressure P, shows a deviation from identity. (4) The quantity \[\frac{PV}{RT}\] is called the compressibility factor and should be unit for an ideal gas. (5) Deviation from ideal behaviour as a function of temperature (6) A real gas behaves as ideal gas most closely at low pressure and 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 force. Hence the effective pressure becomes \[\left( P+\frac{a}{{{V}^{2}}} \right)\]. (iii) Vander Waal's gas equations For 1 mole of gas \[\left( P+\frac{a}{{{V}^{2}}} \right)\,(V-b)=RT\] For \[\mu \] moles of gas \[\left( P+\frac{a{{\mu }^{2}}}{{{V}^{2}}} \right)\ (V-\mu b)=\mu \,RT\] Here a and b are constant called Vander Waalís constant. Dimension : \[[a]=[M{{L}^{5}}{{T}^{-2}}]\] and \[[b]=[{{L}^{3}}]\] Units : \[a=N\times {{m}^{4}}\] and \[b={{m}^{3}}\]. (8) Andrews curves : The pressure (P) versus volume (V) curves for actual gases are called Andrews curves. (i) At \[{{350}^{o}}C,\] part AB represents vapour phase of water, in this part Boyle?s law is obeyed \[\left( P\propto \frac{1}{V} \right)\]. Part BC represents the co-existence of 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. (ii) At \[{{360}^{o}}C\] portion representing the co-existence of liquid vapour phase is shorter. (iii) At \[{{370}^{o}}C\] this portion is further decreased. (iv) At \[{{374.1}^{o}}C,\] it reduces to point (H) called critical point and the temperature \[{{374.1}^{o}}C\] is called critical temperature \[({{T}_{c}})\] of water. (v) The phase of water (at \[{{380}^{o}}C\]) 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 more...

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. For 1 mole of gas \[\frac{PV}{T}=R\] (constant)  \[\Rightarrow \] \[PV=RT\] where R = universal gas constant. Different forms of gas equation

Quantity of gas	Equation	Constant
1 mole gas	\[PV=RT\]	R = universal gas constant
\[\mu \] mole gas	\[PV=\mu RT\]
1 molecule of gas	\[PV=\left( \frac{R}{{{N}_{A}}} \right)\,T=kT\]	k = Boltzmann's constant
N molecules of gas	\[PV=NkT\]
1 gm of gas	\[PV=\left( \frac{R}{M} \right)\,T=rT\]	r = Specific gas constant
m gm of gas	\[PV=mrT\]

(1) Universal gas constant (R) : Universal gas constant signifies the work done by (or on) a gas per mole per kelvin. \[R=\frac{PV}{\mu T}=\frac{\text{Pressure }\times \text{Volume}}{\mu \times \text{ Temperature}}\]\[=\frac{\text{Work done}}{\mu \text{ }\times \text{ Temperature}}\] (i) At S.T.P. the value of universal gas constant is same for all gases \[R=8.31\frac{J}{mole\times kelvin}=1.98\frac{cal}{mole\times kelvin}\] \[\tilde{-}\,2\frac{cal}{mol\,\times kelvin}\] \[=0.8221\frac{\,litre\times atm}{mole\times kelvin}\]. (ii) Dimension : \[[M{{L}^{2}}{{T}^{-2}}{{\theta }^{-1}}]\] (2) Boltzman's constant (k) : It is represented by per mole gas constant i.e., \[k=\frac{R}{N}=\frac{8.31}{6.023\times {{10}^{23}}}\] \[=1.38\times {{10}^{-23}}\ J/K\] It's dimension : \[[M{{L}^{2}}{{T}^{-2}}{{\theta }^{-1}}]\] (3) Specific gas constant (r) : It is represented by per gram gas constant i.e., \[r=\frac{R}{M}\]. It's unit is \[\frac{Joule}{gm\times kelvin}\] and dimension \[[{{L}^{2}}{{T}^{-2}}{{\theta }^{-1}}]\] 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 \[{{r}_{{{H}_{2}}}}=\frac{R}{2}\]

(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. i.e. \[V\propto \frac{1}{P}\] or   PV = constant  \[\Rightarrow \] \[{{P}_{1}}{{V}_{1}}={{P}_{2}}{{V}_{2}}\] (i)  \[PV=P\,\left( \frac{m}{\rho } \right)=\]  constant  \[\Rightarrow \] \[\frac{P}{\rho }=\text{constant}\] or \[\frac{{{P}_{1}}}{{{\rho }_{1}}}=\frac{{{P}_{2}}}{{{\rho }_{2}}}\] (As volume \[=\frac{m}{\rho (\text{Density of the gas)}}\]and m = constant) (ii) \[PV=P\left( \frac{N}{n} \right)=\text{constant}\] \[\Rightarrow \] \[\frac{P}{n}=\text{constant}\] or  \[\frac{{{P}_{1}}}{{{n}_{1}}}=\frac{{{P}_{2}}}{{{n}_{2}}}\] (iii) As number of molecules per unit volume \[n=\frac{N}{V}\] \[\Rightarrow \] \[V=\frac{N}{n}\] also N = constant (iv) Graphical representation : If m and T are constant (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. i.e.,  \[V\propto T\] \[\Rightarrow \] \[\frac{V}{T}=\text{constant}\]\[\Rightarrow \]\[\frac{{{V}_{1}}}{{{T}_{1}}}=\frac{{{V}_{2}}}{{{T}_{2}}}\] (i)  \[\frac{V}{T}=\]\[\frac{m}{\rho T}=\text{constant}\]  (As volume \[V=\frac{m}{\rho }\]) or \[\rho T=\text{constant}\]\[\Rightarrow \]\[{{\rho }_{1}}{{T}_{1}}={{\rho }_{2}}{{T}_{2}}\]      (ii) If the pressure remains constant, the volume of the given mass of a gas increases or decreases by \[\frac{1}{273.15}\] of its volume at \[{{0}^{o}}C\] for each \[{{1}^{o}}C\] rise or fall in temperature. \[{{V}_{t}}={{V}_{0}}\left( 1+\frac{1}{273.15}t \right)\]. This is Charle?s law for centigrade scale. (v) Graphical representation: If m and P are constant (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. \[P\propto T\]   or   \[\frac{P}{T}=\text{constant}\]  \[\Rightarrow \] \[\frac{{{P}_{1}}}{{{T}_{1}}}=\frac{{{P}_{2}}}{{{T}_{2}}}\]               (i) The volume remaining constant, the pressure of a given mass of a gas increases or decreases by \[\frac{1}{273.15}\] of its pressure at \[{{0}^{o}}C\] for each \[{{1}^{o}}C\] rise or fall in temperature. \[{{P}_{t}}={{P}_{0}}\left[ 1+\frac{1}{273.15}t \right]\] This is pressure law for centigrade scale. (ii) Graphical representation : If m and V are constants (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 gas i.e. \[r\propto \frac{1}{\sqrt{\rho }}\] \[\propto \] \[\frac{1}{\sqrt{M}}\] (M is the molecular weight of the gas) \[\Rightarrow \] \[\frac{{{r}_{1}}}{{{r}_{2}}}=\sqrt{\frac{{{\rho }_{2}}}{{{\rho }_{1}}}}\]\[=\sqrt{\frac{{{M}_{2}}}{{{M}_{1}}}}\] If V is the volume of gas diffused in t sec then \[r=\frac{V}{t}\]\[\Rightarrow \]\[\frac{{{r}_{1}}}{{{r}_{2}}}=\frac{{{V}_{1}}}{{{V}_{2}}}\times \frac{{{t}_{2}}}{{{t}_{1}}}\] (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 individual pressures which each gases exert if it alone occupied the same volume at a given temperature. For n gases \[P={{P}_{1}}+{{P}_{2}}+{{P}_{3}}+.....{{P}_{n}}\] where P = Pressure exerted by mixture and \[{{P}_{1}},\,{{P}_{2}},\,{{P}_{3}},\,......{{P}_{n}}=\]Partial pressure of component gases.

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 : (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. (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 \[({{10}^{-9}}\,m)\] (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). (5) Molecules of a gas keep on moving randomly in all possible direction with all possible velocities. (6) The speed of gas molecules lie between zero and infinity (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.