GASEOUS STATE
Introduction :
Matter as we know broadly exists in three states.
There are always two opposite tendencies between particles of matter which determine the state of matter nter molecular attractive forces.
The molecular motion / random motion.
In this chapter the properties and behaviour of the gases will be analysed and discussed in detail. These properties are measured with the help of the gas laws as proposed Boyle,Charles,Gay lussac etc
Boyle’s law and measurement of pressure :
Statement :
For a fixed amount of gas at constant temperature, the volume occupied by the gas is inversely proportional to the pressure applied on the gas or pressure of the gas.
V P
1
hence PV = constant
this constant will be dependent on the amount of the gas and temperature of the gas. P1V1= P2V2
P
v
P
2P
1V
2V
1B
A
Application of Boyles Law : For the two parts ‘A’ and ‘B’ P1V1= K & P2V2= K hence it follows that P1V1= P2V2.
Units
Volume Pressure : Temperature
Volume of the gas is the Pressure = N/m2= Pa S.I. unit Kelvin scale Boiling point = 373 K Volume of the container C.G.S unit = dyne-cm2 ice point = 273 K
S.I. unit m3 Convert 1N/m2into dyne/cm2 Fahrenheit scale B.P. = 212º F ice point = 32º F C.G.S. unitcm3 2 4 5 2 10 cm dyne 10 m 1 N 1 Celcius scale B.P. = 100ºC
1 = 10–3m3 1N/m2= 10 dyne/cm2 ice point = 0ºC
1 = 103cm9 1dm3= 1 = 10–3m3 1 atm = 1.013 × 105N/m2 32 212 32 F 273 373 273 K 0 100 0 C 1ml = 10–3 = 1 cm3 = ) 0 ( R ) 100 ( R ) 0 ( R R
where R = Temp. on unknown scale.
Atmospheric pressure :
The pressure exerted by atmosphere on earth’s surface at sea level is called 1 atm. 1 atm = 1.013 bar
1 atm = 1.013 × 105N/m2= 1.013 bar = 760 torr
Example-1 A rubber balloon contains some solid marbles each of volume 10 ml. A gas is filled in the balloon at a pressure of 2 atm and the total volume of the balloon is 1 litre in this condition. If the external pressure is increased to 4atm the volume of Balloon becomes 625 ml. Find the number of marbles present in the balloon.
Solution : Let the no. of marbles be = n . volume of marble = 10 n ml. volume of balloon earlier = 1000 ml. later = 625 ml.
Now for the gas inside the balloon temperature and amount of the gas is constant, hence boyles law can be applied P1V1= P2V2 4× (625 – 10n) = 2 × (1000 – 10n) 625 × 4 = 2000 – 20n + 40n 625 × 4 – 2000 = 20n 20 2000 – 4 625 = n. 5 125 = n ; n = 25
MEASUREMENT OF PRESSURE
Barometer :
A barometer is an instrument that is used for the measurement of pressure.The construction of the barometer is as follows P0 P0 P0 P = P0 atm Mg Perfect Vaccum
A mercury barometer is used to measure atmospheric pressure by determining the height of a mercury column supported in a sealed glass tube. The downward pressure of the mercury in the column is exactly balanced by the outside atmospheric pressure that presses down on the mercury in the dish and pushes it up the column.
A thin narrow calibrated capillary tube is filled to the brim, with a liquid such as mercury, and is inverted into a trough filled with the same fluid.Now depending on the external atmospheric pressure, the level of the mercury inside the tube will adjust itself, the reading of which can be monitored. When the mercury column inside the capillary comes to rest, then the net forces on the column should be balanced.
Applying force balance, we get,
Patm× A= m×g (‘A’ is the cross-sectional area of the capillary tube) If ‘’ is the density of the fluid, then m = × v
hence, Patm× A = (× g × h) × A (v = A × h)
(‘h’ is the height to which mercury has risen in the capillary) or, Patm=gh
Normal atmospheric pressure which we call 1 atmosphere (1 atm), is defined as the pressure exerted by the atmosphere at mean sea level. It comes out to be 760 mm of Hg = 76 cm of Hg. (at mean sea level the reading shown by the barometer is 76 cm of Hg)1 atm = (13.6 × 103) × 9.8 × 0.76 = 1.013 × 105Pascal. 1 torr = 1 mm of Hg.
1 bar = 105N/m2(Pa)
Faulty Barometer : An ideal barometer will show a correct reading only if the space above the mercury
column is vacuum, but in case if some gas column is trapped in the space above the mercury column, then the barometer is classified as a faulty barometer. The reading of such a barometer will be less than the true pressure.
For such a faulty barometer
mg Patm× A Pgas× A
P0A = Mg + PgasA
P0=gh + Pgas or gh = P0– Pgas
Example-2 The reading of a faulty barometer is 700 mm of Hg. When actual pressure is 750 mm of Hg. The length of the air column trapped in this case is 10 cm .Find the actual value of the atmospheric pressure when reading of this barometer is 750 mm of Hg. Assume that the length of the Barometer tube above mercury surface in the container remains constant.
Solution : P0= Pgas+ 700g
Pgas= 750g – 700 g = 50 g
Now for the gas column in the capillary, amount and temperature are constant hence P1V1 = P2V2 (50g) (100 A) =Pgas × (50 A)
Pgas = 100g
Now, applying force balance in the new conditions :
atm
P =Pgas + 750g = 100 g + 750 g = 850 g
Hence, the atmospheric pressure is now, 850 cm of Hg.
Example-3 In each of the following examples, find the pressure of the trapped gas.
Solution : Total pressure of gas column = 75 + 10 = 85 cm of Hg.
Example-4
Solution : Pgas= 65 cm of Hg.
Example-5
Pg= 75 + 10 cos.
Solution : From the above problem, it can be generalised that, applying force balance every single time is not necessary. If we are moving up in a fluid, then substract the vertical length, and while moving down add the vertical length.
Charles law :
For a fixed amount of gas at constant pressure volume occupied by the gas is directly proportional to temperature of the gas on absolute scale of temperature.
V T or V = kT t
tan cons T
V where ‘k’ is a proportionality constant and is dependent on amount of gas and pressure.
2 2 1 1 T V T
V Temperature on absolute scale, kelvin scale or ideal gas scale. V = a + bt Temperature on centigrade scale.
V
T
Relation : T = t + 273
Since volume is proportional to absolute temperature. The volume of a gas should be theoretically zero at absolute zero temperature.
Infact no substance exists as gas at a temperature near absolute zero, though the straight line plots can be extra plotted to zero volume. Absolute zero can never be attained practically though it can be approached only.
By considering –273.15°C as the lowest approachable limit, Kelvin developed temperature scale which is known as absolute scale.
Example-6 If the temp. of a particular amount of gas is increased from 27ºC to 57ºC, find final volume of the gas, if initial volume = 1 lt and assume pressure is constant.
Solution : 2 2 1 1 T V T V ) 57 273 ( V ) 27 273 ( 1 2 So V2= 1.1 lt.
Example-7 An open container of volume 3 litre contains air at 1 atmospheric pressure. The container is heated from initial temperature 27ºC or 300 K to tºC or (t + 273) K the amount of the gas expelled from the container measured 1.45 litre at 17ºC and 1 atm.Find temperature t.
Solution : T0
1= 300 K
It can be assumed that the gas in the container was first heated to (t + 273), at which a volume ‘V’ escaped from the container
hence applying charles law :
300 3 = 273 t V 3
Now, this volume ‘V’ which escapes when the container get cooled 273 t V = 290 45 . 1
Solve the two equations and get the value ofV and t. determine V & calculate t that will be the answer.
Calculation of pay load :
Pay load is defined as the maximum weight that can be lifted by a gas filled balloon.
M Buoyancy
balloon
For maximum weight that can be lifted, applying force balance Fbuoyancy= Mballoon× g + Mpay load× g
air v.g. =gasv.g + Mg + mg. mass of balloon = m net force on volume of balloon = v balloon = 0
density of air = air (at equilibrium / when balloon is incoming density of gas inside the with constant speed)
balloon =gas
Example-8 A balloon of diameter 20 m weights 100 kg. Calculate its pay-load, if it is filled with He at 1.0 atm and 27ºC. Density of air is 1.2 kg m–3. [R = 0.0082 dm3atm K–1mol–1]
Solution : Weight of balloon = 100 kg = 10 × 104g
Volume of balloon = 3 3 100 2 20 7 22 3 4 r 3 4 = 4190 × 106cm3= 4190 × 103litre
Weight of gas (He) in balloon = RT PVM RT M w PV = 300 082 . 0 4 10 4190 1 3 = 68.13 × 104g Total weight of gas and balloon
= 68.13 × 104+ 10 × 104= 78.13 × 104g
Weight of air displaced = 3
6 10 10 4190 2 . 1 = 502.8 × 104g Pay load = wt. of air displaced – (wt. of balloon + wt. of gas) Pay load = 502.8 × 104– 78.13 × 104= 424.67 × 104g
Gay-lussac’s law :
For a fixed amount of gas at constant volume, pressure of the gas is directly proportional to temperature of the gas on absolute scale of temperature.
P T T
P
= constant dependent on amount and volume of gas
2 2 1 1 T P T
P temperature on absolute scale
originally, the law was developed on the centigrade scale, where it was found that pressure is a linear function of temperature P = P0+ bt where ‘b’ is a constant and P0is pressure at zero degree centigrade.
P
T
O P P0 t (–273°C) Example : PV = K V = K1/p T V = K2 V = K2T P K1 = K2T PT = 2 1 K K = const. P = T 1 ?
where are we wrong ?This is wrong because we are varying temperature & K1= f(1) thus K1will change according to temperature
So
2 1
K K
will be a function of temp & not constant.
Example-9 The temperature of a certain mass of a gas is doubled. If the initially the gas is at 1 atm pressure. Find the % increase in pressure ?
Solution : 1 1 T P = 2 2 T P ; T 1 = T 2 P2 % increase = 1 1 – 2 x 100 = 100%
Example-10 The temperature of a certain mass of a gas was increased from 27°C to 37°C at constant volume. What will be the pressure of the gas.
Solution : 1 1 T P = 2 2 T P ; 300 P = 310 P2 ; P2= 30 31 P
Avogadro’s Hypothesis :
For similar values of pressure & temperature equal number of molecules of different gases will occupy equal volume.
N1 V (volume of N1molecules at P & T of one gas) N1 V (volume of N1molecules at P & T of second gas)
Molar volume & volume occupied by one mole of each and every gas under similar conditions will be equal. One mole of any gas or a combination of gases occupies 22.413996 L of volume at STP.
The previous standard is still often used, and applies to all chemistry data more than decade old, in this definition Standard Temperature and Pressure STP denotes the same temperature of 0°C (273.15K), but a slightly higher pressure of 1 atm (101.325 kPa) .
Standard Ambient Temperature and Pressure (SATP), conditions are also used in some scientific works.
SATP conditions means 298.15 K and 1 bar (i.e. exactly 105Pa) At SATP (1 bar and 298.15 K), the molar volume of an ideal gas is 24.789 L mol–1 (Ref. NCERT )
Equation of State :
Combining all the gas relations in a single expression which discribes relationship between pressure, volume and temperature, of a given mass of gas we get an expression known as equation of state.
T PV
= constant (dependent on amount of the gas (n)).
1 1 1 T V P = 2 2 2 T V P
Ideal gas Equation :
nT PV
= constant [universal constant]
= R (ideal gas constant or universal gas constant)
R = 8.314 J/Kmole 25/3 = 1.987 cal/mole 2 = 0.08 Latm/K/mole 1/12
Example-11 Some spherical balloons each of volume 2 litre are to be filled with hydrogen gas at one atm & 27°C from a cylinder of volume 4 litres. The pressure of the H2gas inside the cylinder is 20 atm at 127°C. Find number of balloons which can be filled using this cylinder. Assume that temperature of the cylinder is 27°C.
Solution : No. of moles of gas taken initially =
400 R 4 20 = 2.43 L
No. of moles of gas left in cylinder =
300 R 4 1 = 0.162L
No. of moles of gas to be filled in balloons = 2.43 – 0.162 = 2.268 Let we have 'n' balloons that we can fill
No. of moles of gas that can be filled in 1 balloon =
300 082 . 0 2 1 = 0.081 0.081 × n = 2.268 n = 28 balloons.
Daltons law of partial pressure :
Partial pressure :
In a mixture of non reacting gases partial pressure of any component of gas is defined as pressure exerted by this component if whole of volume of mixture had been occupied by this component only.
Partial pressure of first component gas
v RT n P1 1 ; v RT n P2 2 ; v RT n P 3 3 Total pressure = P1+ P2+ P3.
Daltons law :
For a non reacting gaseous mixture total pressure of the mixture is the summation of partial pressure of the different component gases.
PTotal= P1+ P2+ P3 v RT ) n n n ( 1 2 3 1 T 1 T 1 x n n P P
2 T 2 T 2 x n n P P
(mole fraction of second component of gas)
3 T 3 T 3 x n n P P
(mole fraction of third component of gas)
Example-12 The stop cock connecting the two bulbs of volume 5 litre and 10 litre containing as ideal gas at 9 atm and 6 atm respectively, is opened. What is the final pressure if the temperature remains same.
Solution : After the opening of the stop cock the pressure of the each bulb will remain same.
At the beginning, the no. of moles of gas in A = RT
6 x 10
At the beginning, the no. of moles of gas in B = RT
9 x 5
total no. of moles at the beginning = RT 105
Total no. of moles of gas before opening the stop cock
= total no. of moles of gas after opening stop cock = RT 105
pressure after the opening of the stop cock P = RT 105 x total V RT = 5 10 105 = 7 atm
Example-13 A mixture of NO2& CO having total volume of 100 ml contains 70 ml of NO2at 1 atm, mixture is left for some time and same NO2get dimerised to N2O4such that final volume of the mixture become 80 ml at 1 atm, calculate the mole fraction of NO2in final equilibrium mixture.
Solution : Initial volume of NO2= 70 ml
Initial volume of CO = 100 – 70 = 30 ml Final volume of mixture = 80 ml
Let the volume of NO2in final mixture be x Let ‘v’ ml NO2 be converted to N2O4
2NO2 N2O4
V V/2
Hence final volume
= volume of CO + volume of NO2left + volume of N2O4 formed = 30 + 70 – V + V/2 = 80
V = 40 ml
Hence volume of NO2left = 70 – V = 30 ml Now as volume moles
mole fraction = volume fraction = 80 30
= 8 3
Analysis of gaseous mixture :
Vapour density :
Vapour density of any gas is defined as the density of any gas with respect to density of the H2gas under identical conditions of temperature T and pressure P.
vapour density = densitydensityof HofundergasatsameT&PP&T
2 P = M RT . V m P = M RT = RT PM
vapour density = 2 H gas PM RT RT PM = 2 H gas M M = 2 Mgas
Mgas= 2 × vapour density
Average molecular mass of gaseous mixture :
total mass of the mixture divided by total no. of moles in the mixture
Mmix= TotalTotalno.massof molesof mixtureinmixture
If we have
‘n1’ , ‘n2’ and ‘n3’ are moles of three different gases having of molar mass ‘M1’, ‘M2’ and ‘M3’ respectively.
Mmin= 3 2 1 3 3 2 2 1 1 n n n M n M n M n
Example-14 Calculate the mean molar mass of a mixture of gases having 7 g of Nitrogen, 22 g of CO2and 5.6 litres of CO at STP.
Solution : Moles of N2= 7/28 = 1/4
Moles of CO2= 22/44 = 1/2
Moles of CO = 5.6 / 22.4 = 1/4
mean molar mass = Mmin=
3 2 1 3 3 2 2 1 1 n n n M n M n M n = ( 7 + 7 + 22 ) / 1 = 36
Graham’s Law of Diffusion/Effusion :
Diffusion :
Net spontaneous flow of gaseous molecules from region of high concentration (higher partial pressure) to the region of lower concentration or lower partial pressure
flow will be from both sides, N2will try to equalise its partial pressure in both the vessels, and so will O2.
Graham’s Law :
“Under similar conditions of pressure (partial pressure) the rate of diffusion of different gases is inversely proportional to square root of the density of different gases.”
rate of diffusion r d 1 d = density of gas 2 1 r r = 1 2 d d = 1 2 M M = 1 2 D . V D . V V.D is vapour density
r = volume flow rate = dt dVout
r = moles flow rate = dt dnout
r = distance travelled by gaseous molecules per unit time = dt dx
The general form of the grahams law of diffusion can be stated as follows, when one or all of the parameters are varied.
rate
TM P
A
P – Pressure, A – area of hole, T – Temp. , M – mol. wt.
If partial pressure of gases are not equal.
Then rate of diffusion is found to be proportional to partial pressure & inversely proportional to square root of molecular mass.
r P r M 1 2 1 r r = 2 1 P P 1 2 M M
Selective diffusion :
If one or more than one components of a mixture are allowed to diffuse and others are not allowed then it is selective diffusion of those components.
Platinum allows only H2gas to pass throughEffusion : (forced diffusion) a gas is made to diffuse through a hole by the application of external pressure.
Example-15 In a tube of length 5 m having 2 identical holes at the opposite ends. H2& O2are made to effuse into the tube from opposite ends under identical conditions. Find the point where gases will meet for the first time.
Solution : 2 1 r r = dt dx × dx dt = 1 2 M M 2 1 dx dx = 2 32 2 1 dx dx = 4 2 2 O by travelled ce tan dis H by travelled ce tan dis = 4 ) x 5 ( x = 4 x = (5 – x) 4 ; x = 20 – 4x ; 5x = 20 ; x = 4 from H2side
Example-16 Assume that you have a sample of hydrogen gas containing H2, HD and D2that you want to separate into pure components (H =1H and D =2H). What are the relative rates of diffusion of the three molecules according to Graham’s law ?
Solution : Since D2is the heaviest of the three molecules, it will diffuse most slowly, and let we call its relative rate 1.00. We can then compare HD and H2with D2.
Comparing HD with D2, we have
diffusion D of Rate diffusion HD of Rate 2 = HD of mass Molecular D of mass Molecular 2 = amu 0 . 3 amu 0 . 4 = 1.15
Comparing H2with D2we have
diffusion D of Rate diffusion H of Rate 2 2 = 2 2 H of Mass D of Mass = 24..00amuamu = 1.41
Thus, the relative rates of diffusion are H2(1.41) > HD (1.15) > D2(1.00).
Kinetic Theory of Gases :
Postulates / assumptions of KTG :
A gas consists of tiny spherical particles called molecules of the gas which are identical in shape & size (mass)
The volume occupied by the molecules is negligible in comparision to the total volume of the gas. For an ideal gas, volume of the ideal gas molecule ~ 0.
Gaseous molecules are always in random motion and collide with other gaseous molecules & with the walls of the container.
Pressure of the gas is due to these molecular collisions among themselves and with walls of the container
These collisions are elastic in nature
Molecular attraction forces are negligible. Infact, for an ideal gas attractive or repulsive forces are equal to zero.
Newton’s laws of motion are applicable on the motion of the gaseous molecules. Effect of gravity is negligible on molecular motion.
The average K.E. of gaseous molecules is proportional to the absolute temperature of the gas. 2
1
M (u2) T (bar is for average) Kinetic equation of gaseous state (expression for pressure of gas).
Derivation :
m = mass of one molecule U = Uxiˆ + Uyjˆ + Uzkˆ
Consider collision with face ABCD
inital Pi = mUx iˆ ; final Pf = – mUxiˆ change in momentum due to collision = 2 Uxm
time taken between two successive collision with face ABCD = t =
x U 2 frequency of collisions (f) = t 1 = 2 Ux
change in momentum in one sec. = force = 2 m 2 U Ux x = 2 x U m
force due to all the molecules
= m } Ux ... U U { 2x 2x N2 2 1 average value ofUN2 = 2 N U = N U . ... U Ux21 2x 2x N 2 Fx= M } U N { 2x
all the three directions are equal as the motion is totally random in all directions, hence
2 x U = U2y = 2 z U 2 U = U2xU2yU2z = 3 2 x U Fx= M . N 3 1 2 U Pressure = 2 x F = 3 1 3 N
U2 The volume of the container ‘V’ =
3
PV =
3 1
mNU2 Kinetic equation of gases
where 2
U is mean square speed
root mean square speed = Urms= U2 =
N U ... U U U12 22 23 N2
Verification of Gaseous Laws Using Kinetic Equation :
From postulates ; PV = 3 1 mNU2 2 1m U2 T = T Where ’‘ is a proportionality constant PV = 3 2 mU2 2 1 N ; PV = 3 2
NT (N = Total number of molecules) Boyle’s Law : N : constant
T : constant PV = constant Charles law : N : constant
P : constant V T
Kinetic energy of gaseous molecule (translational K.E.)
To calculate we have to use ideal gas equation (experimental equation) PV = nRT Kinetic equation PV = 3 2 nRT = 3 2
(nNA) T (n = number of moles of gas) on comparing = 2 3 × A N R = 2 3 K where K = A N R = Boltzmann constant
Average K.E. of molecules = 2 1 mU2 =T Average K.E. = 2 3
K T (only dependent on temperature not on nature of the gas.)
Average K.E. for one mole = NA 2 U m 2 1 = 2 3 K NAT = 2 3 RT
Root mean square speed : Urms= U2 = m kT 3 = A N m T R 3
Where m-mass of one molecule
Dependent on nature of gas i.e mass of the gas
Urms= M T R 3 M = molar mass Average speed : Uav= U1+ U2+ U3+ ... UN Uav= M RT 8 = m KT 8 K is Boltzmman constant
Most probable speed :
The speed possessed by maximum number of molecules at the given temperature
UMPS= M RT 2 = m KT 2
Example-17 In a container of capacity 1 litre there are 1023molecules each of mass 10–22gms. If root mean square speed is 105cm/sec then calculate pressure of the gas.
Solution : PV = 3 1 MN U2 P = ? V = 10–3m3 m = 10–25kg N = 1023 2 U = 105cm/sec = 103m/sec 2 U = 106m2/sec2 P × 10–3 = 3 1 × 10–25× 1023× 106 P = 3 1 × 10–2× 106× 103 P = 3 1 × 107pascals
Maxwell’s distributions of molecular speeds :
Postulates/Assumptions of speed distributions It is based upon theory of probability. It gives the statistical averages of the speed of the whole collection of gas molecules.
Speed of gaseous molecules of may vary from 0 to. The maxwell distribution of speed can be plotted against fraction of molecules as follows.
The area under the curve will denote fraction of molecules having speeds between zero and infinity
Total area under the curve will be constant and will be unity at all temperatures.
Area under the curve between zero and u1will give fraction of molecules racing speed between 0 to u1. This fraction is more at T1and is less at T2.
The peak corresponds to most probable speed.
At higher temperature, fraction of molecules having speed less than a particular value decreases.
Real Gases :
Real gases do not obey the ideal gas laws exactly under all conditions of temperature and pressure. Real gases deviates from ideal behaviour because
Real gas molecules have a finite volume.
{since on liquefaction real gases occupy a finite volume}
Inter molecular attractive forces between real gas molecules is not zero. {Real gases can be converted into liquid where as ideal gases cant be}
Deviation of real gases from ideal behaviour can be measured by using compresibility factor : (Z) Z = ideal real ) PV ( ) PV ( (PV)ideal= nRT Z = nRT PV = RT PVm
, VMis volume of one mole of gas or molar volume.
Z = ideal m m V V
Variation of Z with pressure at constant temperature :
Z=1 ideal
He/H2
N2
NH /CO3 2
P
Variation of Z with pressure at different temperature (for a gas ) :
Conclusions :
Z = 1 for ideal gas ; Z > 1 at all pressures for He/H2
Z < 1 at low pressure (for all other gases) ; Z > 1 at high pressure (for all other gases)
Vander Waal Equation of real gases :
The ideal gas equation does not consider the effect of attractive forces and molecular volume. vander Waal's corrected the ideal gas equation by taking the effect of
(a) Molecular volume (b) Molecular attraction
Volume correction : Ideal gas equation :
PiVi= nRT ; In the equation ‘Vi’ stands for the volume which is available for free movement of the molecules. Videal = volume available for free movement of gaseous molecule
hence, Vi= V – {volume not available for free movement} For an ideal gas Vi= V {V = volume of container}
but for a real gas
Molecules have finite volume :
Excluded volume per molecule =2 1 (2r)3 3 4
= Co-volume per molecule.
The volume that is not available for free movement is called excluded volume. let us see, how this excluded volume is calculated.
r r
1 2
Excluded volume
(not available for free momement)
For above example, the entire shaded region is excluded, as its centre of mass cannot enter this region. If both molecules were ideal, then they would not have experienced any excluded volume but not in the case, of real gas as the centre of mass of ‘2’ cannot go further.
Hence for this pair of real gas molecules,
Excluded volume per molecule = 2 1 3 ) r 2 ( 3 4 = 4 r3 3 4
excluded volume per mole of gas (b) = NA4
3 r 3 4
= 4 x NA x Volume of individual molecule for n moles, excluded volume = nb
Vi= V – nb volume correction
Pressure correction or effect of molecular attraction forces :
Due to these attraction, speed during collisions will be reduced Momentum will be less
Force applied will be less Pressure will be less. Pideal= P + {correction term}
Correction term no. of molecules attracting the colliding molecule (n/v). Correction term density of molecules (n/v).
no. of collision density of molecules v n
net correction term v n v n = 2 2 v an
‘a’ is constant of proportionality
and this is dependent on force of attraction
Stronger the force of attraction greater will be ‘a’ (Constant)
Pi= P + 2
2
v an
Vander waal’s equation is
2 2 v an P (v – nb) = nRT
VERIFICATION OF VANDER WAAL’S EQUATIONS :
Variation of Z with P for vander waals' equation at any temp.Vander waal equation for 1mole
2 m V a P (V m– b) = RT
AT LOW PRESSURE (at separate temp.)
At low pressure Vmwill be high.
Hence b can be neglected in comparision to Vm. but 2 m
V a
can't be neglected as pressure is low Thus equation would be
2 m V a P V m= RT PVm+ m V a = RT RT PVm + RT V a m = 1 Z = 1 – VaRT m Z < 1
Real gas is easily compressible as compared to an ideal gas. AT HIGH PRESSURE (moderate temp.)
Vmwill be low
so b can't be neglected in comparision to Vm
but 2 m
V a
can be neglected as compared to much higher values of P.. Then vander Waals' equation will be
P(Vm– b) = RT PVm– Pb = RT RT PVm = RT Pb + 1 Z = RT Pb + 1 (Z > 1)
If Z > 1, then gas is more difficult to compress as compared to an ideal gas. At low pressure and very high temperature.
Vmwill be very large
hence ‘b’ can't be neglected and 2 m
V a
can also be neglected as Vmis very large PVm= RT (ideal gas condition)
For H2or He a ~ 0 because molecules are smaller in size or vander Wall's forces will be very weak, these are non polar so no dipole-dipole interactions are present in the actions.
P(Vm– b) = RT so Z = 1 + RT Pb
‘a’ factor depends on inter molecular attractive forces.
Example-18 Arrange following in decreasing ’a’ factor (H2O, CO2, Ar) H2O > CO2 > Ar
polar
For non polar molecules :Greater the size or surface area, greater will be vander waals' forces, so greater will be ’a’ constant.
Example-19 Arrange following gases according to ‘a’ He , Ar, Ne, Kr.
aKr> aAr> aNe> aHe
More ‘a’ factor means higher will be boiling point. liquification pressure :
Is the pressure required to convert gas into liquid. for easy liquefaction a and P
When Z < 1, Vm< Vm, ideal easily liquifiable
Z > 1, Vm> Vm, ideal more difficult to compress.
Example-20 Arrange the following according to liquification pressure. n-pentane ; iso-pentane , neo pentane.
an-pentene> aiso-pentane> aneo-pentane liquification pressure = LP e tan pen n P
L < LPisopentane < LPneopentane
b is roughly related with size of the molecule. (Thumb rule)
b = NA4 r3 3 4
Example-21 Two vander waals gases have same value of b but different a values. Which of these would occupy greater volume under identical conditions ?
Solution : If two gases have same value of 'b' but different values of 'a', then the gas having a larger value of 'a' will occupy lesser volume. This is because the gas with a larger value of 'a' will have larger force of attraction and hence lesser distance between its molecules.
Virial Equation of state :
It is a generalised equation of gaseous state. All other equations can be written in the form of virial equation of state.
Z is expressed in power series expansion of P or m V 1 Z = 1 + m V B + 2 m V C + 3 m V D + ...
B – second virial coefficient C – third virial coefficient D – fourth virial coefficient
Vander waals' equation in virial form :
2 m V a P (V m– b) = RT P = ) b V ( RT m – 2 m V a Z = RT PVm = (VV b) m m – V RT a m = (1 b/V ) 1 m – V RT a m x 1 1 = 1 + x + x2+ x3+ ... Z = ... V b V b V b 1 3 m 3 2 m 2 m – V RT a m = 1 + m V 1 RT a b + 2 m 2 V b + 3 m 3 V b + ...
comparing vander waals equation with virial equation
B = b – RT
a
, C = b2, D = b3 at low pressure : Vmwill be larger
hence 2 m V 1 , 3 m V 1 ... can be neglected Z = 1 + m V 1 RT a b If RT a b = 0 at T = Rb a ; Z = 1 so at T = Rb a
, gas will behave as an ideal gas (or follows Boyles law)
But at constant temperature, ideal gas equation is obeying Boyles law as T = Rb
a
, so the temperature is called Boyles' temp.
TB= Rb a Z = 1 – RT V a m
for a single gas, if we have two graphs as above, we must conclude T2< T1. At Boyles' temperature ‘a / RT’ factor is compensated by 'b' factor, so Z = 1.
Critical constant of a gas :
When pressure increases at constant temperature volume of gas decreases
AB gas
BC vapour + liquid CD liquid
critical point : At this point, all the physical properties of liquid phase will be equal to physical properties in vapour such that
density of liquid = density of vapour
TCor critical temp :
Temperature above which a gas can not be liquified
PCor critical pressure :
minimum pressure which must be applied at critical temperature to convert the gas into liquid.
VCor critical volume :
volume occupied by one mole of gas at TC& PC
Critical constant using vander waals' equations :
2 m V a P (V m– b) = RT (PVm2 a) (Vm– b) = RT Vm2 PVm3+ aV m– PbVm 2– ab – RTV m 2= 0 Vm3– V m 2 P RT b + P a Vm– P ab = 0
Given equation is cubic, hence there will be three roots of equation at any temperature and pressure. At critical point, all three roots will coincide and will give single value of V = VC
At critical point, Vander Waals' equation will be
Vm3– V m 2 C C P RT b + C P a Vm– C P ab = 0 ...(1)
But, at critical point, all three roots of the equation should be equal, hence equation should be : Vm3– 3V m 2V C+ 3VmVC 2– V C 3= 0 ...(2)
comparing with equation (1)
b + C C P RT = 3VC ....(i) C P a = 3 VC2 ...(ii) C P ab = VC3 ...(iii) PC= 2 C V 3 a On substituting value of VC PC= 3(3b)2 a = 2 b 27 a
by (i) C C P RT = 3 VC– b = 9b – b = 8b TC= 278Rba
At critical point, the slope of PV curve (slope of isotherm) will be zero
C T m V P = 0 ...(i)
at all other point slope will be negative 0 (zero) is the maximum value of slope.
m V C T m V P = 0 ....(ii)
{Mathematically such points are known as point of inflection (where first two differentiation becomes zero)} using the two, TC, PCand VCcan be calculated
by
By any two a can be calculated but a calculated by VCand TCand a calculated by TCand PCmay differ as these values are practical values and VCcan’t be accurately calculated. So when we have VCTC& PCgiven, use PC& TCto deduce ‘a’ as they are more reliable.
Reduced Equation of state :
Reduced Temp : Temperature in any state of gas with respect to critical temp of the gas
Tr= C T T Reduced pressure : Pr= C P P Reduced volume : Vr= C m V V
Vander waals' equation, 2 m V a P (V m– b) = RT Substitute values : 2 C 2 r C r V V a P P (V rVC– b) = R TrTC
Substitute the value of PCTCand VC
2 2 r 2 r ) b 3 ( V a b 27 a P (3b V r– b) = RTr 27Rb a 8 r r V 1 3 P (3 Vr– 1) = 3 T R 8 r 2 r r V 3 P (3V
r– 1) = 8 Tr (Reduced equation of state)
Above equation is independent from a, b and R, so will be followed by each and every gas, independent of its nature.
Example-22 The vander waals constant for HCI are a = 371.843 KPa.dm6mol–2and b = 40.8 cm3mol–1find the
critical constant of this substance.
Solution : The critical pressure, PC= 2
b 27 a = 2 6 3 10 ) 8 . 40 ( 27 10 843 . 371 = 2 9 ) 8 . 40 ( 27 10 843 . 371 = 8.273 x 106 PC= 8.273 MPa
The critical pressure, TC= Rb 27 a 8 R = 8.314 KPa dm3K–1mol–1 TC= Rb 27 a 8 = 3 10 8 . 40 27 314 . 8 843 . 371 8 = 324.79 = 324.8 K
The critical volume, VC= 3b = 3 x 40.8 = 122.4 cm3
Example-23 The vander waals constant for gases A, B and C are as follows : Gas a/dm6KPa mol–2 b/dm3mol–1
A 405.3 0.027
B 1215.9 0.030
C 607.95 0.032
Which gas has
(i) Highest critical temperature (ii) The largest molecular volume (iii) Most ideal behaviour around STP ?
Solution : TC= Rb 27
a 8
Since, R is constant, higher the value of a/b, higher will be critical temperature. VC= 3b and VC Vm(for a particular gas) therefore higher the value of VC, higher will be molar volume of the gas.
If the critical temperature is close to 273 K, gas will behave ideally around the STP. Let us illustrate the result in a tabular form.
Gas a/dm6KPa mol–2 b/dm3mol–1 T
C VC a/b
A 405.3 0.027 534.97 K 0.081 1.501 x 104
B 1215.9 0.030 1444.42 K 0.09 4.053 x 104
C 607.95 0.032 677.07 K 0.096 1.89 x 104
(i) B gas has the largest critical temperature. (ii) C gas has the largest molecular volume.
(iii) A gas has the most ideal behaviour around STP
Vapour pressure of a liquid (aqueous Tension of water) :
Vapour pressure depends on (a) Temperature (T VP ) (b) Nature of the liquid
Vapour pressure is independent of amount of liquid & surface area of liquid.
Vapour pressure of the liquid is independent of pressure of any gas in the container, Ptotal= Pgas+ Pwater vapour
Example-24 In a container of capacity 1 litre, air and some liquid water is present in equilibrium at total pressure of 200 mm of Hg. This container is connected to another one litre evacuated container. Find total pressure inside the container when equilibrium is again stablised (aqueous tension or vapour pressure at this temp. is 96 mm Hg).
Solution : Total pressure = 200 mm of Hg = Pgas+ Pvapour water Pgas+ 96 = 200
Pgas= 104 mm of Hg Initially when second container is connected
P1= 104 mm of Hg P2= ?
V1= 1 V2= 2 litre
P1V1= P2V2 104 × 1 = P2× 2 52 = P2
After equilibrium is established
Ptotal= 52 + 96 = Pgas+ Pwater = 148 mm of Hg at equilibrium.
Eudiometry :
The analysis of gaseous mixtures is called eudiometry. The gases are identified by absorbing them in specified and specific reagents.
Some Common Facts :
Liquids and solutions can absorb gases.
If a hydrocarbon is burnt, gases liberated will be CO2& H2O. [H2O is seperated out by cooling the mixture & CO2by absorption by aqueous KOH]
If organic compound contains S or P, then these are converted into SO2& P4O10by burning the organic compound.
If nitrogen is present, then it is converted into N2.
[The only exception : if organic compound contains – NO2group then NO2is liberated]
If mixture contains N2gas & this is exploded with O2gas, do not assume any oxide formation unless specified.
Ozone is absorbed in turpentine oil and oxygen in alkaline pyragallol.
Example-25 Carbon dioxide gas (CO2) measuring 1 litre is passed over heated coke the total volume of the gases coming out becomes 1.6 litre. Find % conversion of CO2into carbon monoxide.
Solution : CO2+ C 2CO CO2 CO
1 0 at time t 1 – x 2x
Initial volume = 1 litre
final volume = 1.6 litre Final volume = (1 + x) litres 1 + x = 1.6 x = 0.6 x = 0.6 1 6 . 0
× 10 = 60% of CO2will be converted into CO
Example-26 100 ml of hydrocarbon is mixed with excess of oxygen and exploded. On cooling, the mixture was reported to have a contraction of 250 ml. The remaining gas when passed through a solution of aqueous KOH, the mixture shows a further contraction of 300 ml. Find molecular formula of the hydrocarbon. Solution : CxHy+ 4 Y x O 2
x
CO2 + 2H O y 2 100 mlx
100 2 y 100. mixture contains CO2, H2O & excess O2on cooling, H2O is separated, volume of H2O = 250 ml
2 y
as KOH absorbs CO2, hence 300 ml contraction is because of CO2that has been absorbed. Volume of CO2= 100 x = 300 ; x = 3
Empirical formula = C3H5 ; molecular formula = C6H10.
Note :
If water is already condensed out then total contraction in reaction mixture = {volume of reactants} – {volume of products + volume of unused species excluding volume of H2O}.Example-27 100 ml of an hydrocarbon is burnt in excess of oxygen in conditions so that water formed gets condensed out the total contraction in volume of reaction mixture was found to be 250 ml when the reaction mixture is further exposed to aqueous KOH a further contraction of 300 ml is observed find molecular formula of hydrocarbon.
Solution : CxHy + O2 CO2 + H2O 100 ml excess 300 ml By POAC on ‘C’ atoms x × 100 = 300 x = 3 POAC on ‘H’ atoms y × 100 = 2 × moles of H2O POAC on O atoms 2 × v = 2 × 300 + 1 × H2O {V = volume of O2consumed} 2 × v = 600 + 50 y v = 2 y 50 600 volume of O2consumed The total volume contraction is 250 ml. Hence, 100 + V – 300 = 250 – 200 + V = 250 2 × 450 – 600 = 50 y 50 300 = y = 6 Hydro carbon will be C3H6
Alternative :
Using balanced chemical equation
CxHy + 4 y x O2 xCO2 + 2 y H2O t = 0 100 ml V 0 0 t 0 V – 100 x 4y 100 x ml 2 y 100 volume remained V–100 4 y x + 100x + 50y –100 – V = 250 –25y + 50y = 150 25y = 150 y = 6
Summary
A gas is a collection of atoms or molecules moving independently through a volume that is largely empty space. Collisions of the randomly moving particles with the walls of their container exert a force per unit area that we perceive as pressure. The SI unit for pressure is the pascal, but the atmosphere and the millimeter of mercury are more commonly used. The physical condition of any gas is defined by four variables ; pressure (P), temperature (T), volume (V) and molar amount(n). The specific relationship among these variables are called the gas laws :
Boyle's law : The volume of a gas varies inversely with its pressure. That is, V 1/P or PV = k at constant n. T.
Charles Law : The volume of a gas varies directly with its Kelvin temperature. That is, V T or V/T = k at constant n, P
Avogadro's Law : The volume of a gas varies directly with its molar amount. That is, V n or V/n = k at constant T,P.
The three individual gas laws can be combined into a single ideal gas law, PV = nRT. If any three of the four variables P, V, T and n are known, the fourth can be calculated. The constant R in the equation is called the gas constant and has the same value for all gases. At standard temperature and pressure (STP ; 1 atm and 0°C), the standard molar volume of an ideal gas is 22.414 L.
The gas laws apply to the mixture of gases as well as to pure gases According to Dalton's law of partial pressures, the total pressure exerted by a mixture of gases in a container is equal to the sum of the partial pressure of each individual gas would exert alone.
The behaviour of gases can be accounted for using a model called the kinetic-molecular theory, a group of five postulates :
1. A gas consists of tiny particles moving at random.
2. The volume of the gas particles is negligible as compared with the total volume. 3. There are no forces between particles, either attractive or repulsive.
4. Collisions of gas particles are elastic.
5. The average kinetic energy of gas particles is proportional to their absolute temperature.
The connection between temperature and kinetic energy obtained from the kinetic- molecular theory makes it possible to calculate the average speed of a gas particle at any temperature. An important practical consequence of this relationship is Graham' law, which states that the rate of a gas effusion or spontaneous passage through a pinhole in a membrane depends inversely on the square root of the molar mass of gas. Real gases differ in their behaviour from that predicted by the ideal gas law, particularly at higher pressure, where gas particles are forced close together and intermolecular forces become significant.
MISCELLANEOUS SOLVED PROBLEMS (MSPs)
1. The diameter of a bubble at the surface of a lake is 4 mm and at the bottom of the lake is 1 mm. If atmospheric pressure is 1 atm and the temperature of the lake water and the atmosphere are equal. what is the depth of the lake ?
(The density of the lake water and mercury are 1 g/ml and 13.6 g/ml respectively. Also neglect the contribution of the pressure due to surface tension)
Sol. P1V1 = P2V2 (760 mm × 13.6 × g) 3 4 (4 mm/2)3= (760 mm × 13.6 × g + h × 1 × g) 3 4 (1 mm/2)3 760 × 13.6 × 64 = (760 × 13.6 + h) h = 64 × 760 × 13.6 – 760 × 13.6 h = 63 × 760 × 13.6 mm h = 1000 1000 6 . 13 760 63 km = 0.6511 km = 651.1 m Ans.
2. A gas is initially at 1 atm pressure. To compress it to 1/4 th of initial volume, what will be the pressure required ? Sol. P1= 1 atm V1= V P2= ? V2= 4 V P1V1 = P2V2 at const. T & n P2= 2 1 1 V V P = 4 V V atm 1 = 4 atm Ans.
3. A gas column is trapped between closed end of a tube and a mercury column of length (h) when this tube is placed with its open end upwards the length of gas column is (1), the length of gas column becomes (2) when open end of tube is held downwards. Find atmospheric pressure in terms of height of Hg column.
Hg h P0 case P1 1 Hg h P0 case P2 2
Sol. for gas P1= (PO+ h) P2= (PO– h)
V1 = r21 V2 = r22 at const T. and moles.
P1V1= P2V2; (PO+ h)r21 = (PO– h)r22 PO2+ h1 = PO2– h2 PO2– Po1 = h, + h2 P0= ) ( ) ( h 1 2 2 1 cm of Hg column Ans.
4. If water is used in place of mercury then what should be minimum length of Barometer tube to measure normal atmospheric pressure.
Sol. PHg = PH2O = Patm.
0.76 m × 13.6 × g = hH2O × 1 × g
O H2
5. A tube of length 50 cm is initially in open atmosphere at a pressure 75 cm of Hg. This tube is dipped in a Hg container upto half of its length. Find the level of mercury column in side the tube.
Sol. If after dipping the tube, the
length of air column be x cm (situation shown in the adjoining figure).
Then by using, PiVi = PfVf We have. 75 cm Hg ×A = Pf× x × A ... (1) ( = 50 cm) & also, Pf = 75 cm Hg + (x – 2 ) ... (2) (2) & (1) [75 + (x – 25)] × x = 75 × 50 x2+ 50 x – 3750 = 0 x = 41.14 or – 91.14 But, x can't be – ve x = 41.14
mercury column inside the tube = (50 – 41.14) cm = 8.86 cm Ans.
6. An open container of volume V contains air at temperature 27ºC or 300 K.The container is heated to such a temperature so that amount of gas coming out is 2/3 of
(a) amount of gas initially present in the container. (b) amount of gas finally remaining in the container.
Find the temperature to which the container should be heated.
Sol. (a) Here , P & V are constant, n & T are changing. Let, initially the amount of gas present be n & temp
is 27ºC or 300K. Finally amount of gas present in container = n – 3 2 n = n 3 1
& final temp. be T.T.
Then using n1T1= n2T2, we have,
n × 300 = 3 n
× T2 T2= 900K i.e., final temp = 900K Ans.
Sol. (b) Let there be x moles of gas remaining in the container, 3 2 of x come out (x + 3 2 x) = n 3 x 5 = n x = 5 n 3 Using n1T1= n2T2 n × 300 K = 5 n 3 × T2 T2= 500K
final temperature = 500 K Ans.
7. Find the lifting power of a 100 litre balloon filled with He at 730 mm and 25°C. (Density of air = 1.25 g /L).
Sol. Since, PV = nRT PV = M W RT W = RT PVM = 760 730 × 298 082 . 0 4 100 g i.e., Wt. of He = 15.72 g Wt. of air displaced = 100 × 1.25 g/L = 125 g
Lifting power of the balloon = 125 g – 15.72 g = 109.28 g Ans.
8. A weather balloon filled with hydrogen at 1 atm and 300 K has volume equal to 12000 litres. On ascending it reaches a place where temperature is 250 K and pressure is 0.5 atm. The volume of the balloon is : (A) 24000 litres (B) 20000 litres (C) 10000 litres (D) 12000 litres
Sol. Using 1 1 1 T V P = 2 2 2 T V P ; K 300 L 12000 atm 1 = K 250 V atm 0.5 2 V2= 20,000 L Hence Ans. (B)
9. Four one litre flasks are separately filled with the gases, O2, F2, CH4and CO2under the same conditions. The ratio of number of molecules in these gases :
(A) 2 : 2 : 4 : 3 (B) 1 : 1 : 1 : 1 (C) 1 : 2 : 3 : 4 (D) 2 : 2 : 3 : 4
Sol. According to avogadro’s hypothesis.
All the flasks contains same no. of molecules
Ratio of no. of molecules of O2, F2, CH4& CO2 = 1 : 1 : 1 : 1 Ans (B)
10. A sample of water gas has a composition by volume of 50% H2, 45% CO and 5% CO2. Calculate the volume in litres at STP at which water gas which on treatment with excess of steam will produce 5 litre of H2. The equation for the reaction is :
CO + H2O CO2+ H2
Sol. If x L CO in needed then
volume of H2in water gas = 0.x4550%L
= L 9 . 0 x L 2 1 45 . 0 x
But, from equation : CO + H2O CO2+ H2 & Gay-Lussac’s law, we get, that the volume of H2 produced = volume of CO taken.
Volume of H2due to reaction = x L Total volume of H2= x 9 . 0 x L = 5 L 9 . 0 x 9 . 1 = 5 L x = 9 . 1 5 9 . 0
Volume of water gas = L
45 . 0 9 . 1 5 9 . 0 L 45 . 0 x = 5.263 L Ans.
11. A car tyre has a volume of 10 litre when inflated. The tyre is inflated to a pressure of 3 atm at 17°C with air. Due to driving the temperature of tyre increases to 47°C.
(a) What would be pressure at this temperature ?
(b) How many litres of air measured at 47°C and pressure of 1 atm should be left out to restore the tyre to 3
atm at 47°C ?
Sol. V = 10L P, = 3 atm T, = 17oC = 290K T2= 47oC = 320K
(a) Here v & n are constant
Using 1 1 T P = 2 2 T P P2= 1 2 1 T T P = 290 320 3 atm i.e. P2= 3.3103 atm
i.e. Pressuse at 47oC = 3.3103 atm Ans.
(b) Now, P1= 3 atm ; V1= 10L ; T1= 17oC = 290K Final P2= 3atm ; V2= ? ; T2= 47oC = 320 K 1 1 T V = 2 2 T V V2= 1 2 2 T T V = K 290 K 320 L 10 = 11.035L
Volume of gas should be removed at 47oC & 3atm = (11.035 – 10) L = 1.035L
Now this vol at 1atm be v then 1.0351 × 3atm = v × 1atm v = 3.105L
12. The partial pressure of hydrogen in a flask containing two grams of hydrogen and 32 gm of sulphur dioxide is :
(A) 1/16th of the total pressure (B) 1/9th of the total pressure (C) 2/3 of the total pressure (D) 1/8th of the total pressure
Ans. (C) Sol. nH2 = mol / g 2 g 2 = 1mol. nSO2 = mol / g 64 g 32 = 0.5 mol PH2=
2 2 2 SO H H n n n × PT= ) 5 . 0 1 ( 1 × PT= 3 2 PT. (C)13. Equal volume of two gases which do not react together are enclosed in separate vessels. Their pressures are 10 mm and 400 mm respectively. If the two vessels are joined together, then what will be the pressure of the resulting mixture (temperature remaining constant) :
(A) 120 mm (B) 500 mm (C) 1000 mm (D) 205 mm
Ans. (D)
Sol. Let, vol of containers be V & temperature be T
P1= 10mm P2= 400mm n1= RT V P1 & n2= RT V P2 n1+ n2= RT V ) P P ( 1 2
After joining two containers final vol = (V+V) = 2V(for gases)
Pfinal =
final 2 1 V RT n n = RT V ) P P ( 1 2 × V 2 RT =
2 P P1 2 =
2 mm 400 10 = 205 mm.14. 5 ml of H2gas diffuses out in 1 sec from a hole. Find the volume of O2that will diffuse out from the same hole under identical conditions in 2 sec.
Sol. Rate of diffusion of H2= sec 1 ml 5 = 5ml/s = rH2 (say) ro2 = rH2 × 4 1 = 5ml/s × 4 1
Volume of O2diffused in 2.0 seconds =
4 5
× 2 ml = 2.5 ml Ans.
15. A vessel contains H2& O2 in the molar ratio of 8 : 1 respectively. This mixture of gases is allowed to diffuse through a hole, find composition of the mixture coming out of the hole.
Sol. Here,nH2 : nO2 = 8 : 1 &
2 2 O H r r = 2 2 O H n n 2 2 H O M M 2 2 O H r r = 1 8 × 2 32 = 1 32
t / out g min co O of moles of . no t / out g min co H of moles of . no 2 2 = 1 3216. The rate of diffusion of a sample of a ozonized oxygen is 0.98 times than that of oxygen. Find the % (by volume of ozone in the ozonized sample).
Sol. Let, rate of diffusion of ozonized oxygen be rg & Let, reat of diffusion of oxygen berO2
2 O g r r = 0.98 (given) ... (1) but 2 O g r r = 2 / 1 g O M M 2
(mean molar mass of ozonized oxygen = Mg)
Form (1) 0.98 = g M 32 (0.98)2= g M 32 Mg=
2 98 . 0 32 = 33.32Let % of O3be x % O2= (100 – x) in ozonized oxygen.
100 32 x – 100 x 48 = 33.32 3200 + 16x = 33.32 x = 13216 = 8.25 % i.e. % of O3= 8.25 % Ans.17. (i) Calculate the pressure exerted by 1023gas molecules, each mass 10–22g in a container of volume one
litre. The rms velocity is 105cm/sec.
(ii) What is the total kinetic energy (in cal) of these particles ? (iii) What must be the temperature ?
Sol. (i) Here N = 1023molecules m = 10–22g , V = 1L = 1000 cm3
rms velocity = 105cm/second PV = 3 1 Nm 2 U P = 3 1 Nm 2 U = 3 cm 1000 3 1 × 1023× 10–22g × (105)2cm2/s2 = 3.33 × 107dyne/cm2 Ans.
(ii) Total K.E of the molecules
= 2 1 Nm 2 U = 2 1 × 1023× 10–25kg × (103m/s)2 = 2 J 10000 = 5000 J 1195.0 Cal Ans.
(iii) Avg K.E of one molecule =
2 1 m 2 U = 2 3 kT T = 3 1 × k U m 2 = 3 1 ×
314 . 8 10 023 . 6 s / m 10 Kg 10–25 3 2 23 K kNAR = 2414.8 K Ans.18. If for two gases of molecular weights MA and MBat temperature TA and TB; TAMB= TBMA, then which property has the same magnitude for both the gases.
(A) Density (B) Pressure (C) KE per mol (D) RMS speed
Sol. Given that TAMB= TBMA
A A M T = B B M T But, r.m.s. = M RT 3 r.m.sA= A A M RT 3 & r.m.sB = B B M RT 3 r.m.sA= r.m.sB Ans. (D)
19. It has been considered that during the formation of earth H2gas was available at the earth. But due to the excessive heat on the earth this had been escaped. What was the temperature of earth during its formation? (The escape velocity is 1.1 x 106cm/s)
Sol. Escape velocity of H2should be equal to average velocity of H2. Avg velocity of H2= 1.1 × 106cm/s = 1.1 × 104m/s
But, avg. velocity = M RT 8 1.1 × 104= 3 – 10 2 T 314 . 8 8
kg 10 2 g 2 M –3 H2 T =
314 . 8 8 10 2 10 1 . 1 4 2 –3 K = 11430.5 K = 11157.5oC Ans.20. Under critical states of a gas for one mole of a gas, compressibility factor is
(A) 8 3 (B) 3 8 (C) 1 (D) 4 1
Sol. For 1 mole of gas Z =
C C C RT V P
(Under critical condition)
But, PC= 2 b 27 a , VC= 3b , TC= Rb 27 a 8 Z = 2 b 27 a × R b 3 × a 8 Rb 27 = 8 3 Ans. (A)
21. One litre of oxygen gas is passed through a ozonizer & the final volume of the mixture becomes 820 ml. If this mixture is passed through oil of turpentine. Find the final volume of gas remaining.
Sol. The reaction that takes place in ozonizer as 3O22O3. If Vml of O2out of 1000 ml is ozonized then vol of O3obtained =
3 2
V Final vol of ozonized oxygen
= (1000 – V + 3 2 V) mL = 820 mL 1000 – 3 1 V = 820 3 V = 180 V = 540
O2remaining = (1000 – 540) mL ( O3is absorbed in oil of turpentine) = 460 mL Ans.
22. A gaseous mixture containing CO, methane CH4 & N2 gas has total volume of 40 ml. This mixture is exploded with excess of oxygen on cooling this mixture a contraction of 30 ml is observed & when this mixture is exposed to aqueous KOH a further contraction of 30 ml is observed .find the composition of the mixture.
Sol. Let vol of CO be x mL vol of CH4be y mL vol of N2be z mL
On explosion with excess of oxygen the following reactions takes place
CO(g) + 2 1
O2(g) CO2(g) (By Gaylussac's law of combing volume)
x mL x mL
CH4(g) + 2O2(g) CO2(g) + 2H2O(g)
y mL y mL 2 ymL
N2remains unreacted
On cooling H2O (g) liquifies hence volume reduction of 30 mL is observed
2y = 30 y = 15
But, vol of CO2obtained = (x + y) mL
This is absorbed in KOH & vol reduction of 30 mL is observed. x + y = 30 x = 30 – y = (30 – 15) = 15 and, x + y + z = 40 z = 40 – x – y = 40 – 15 – 15 = 10 Composition of mixture is vol of CO = 15 mL Ans. vol of CH4= 15 mL Ans. vol of N2= 10 mL Ans.
