Sunday, 28 December 2014

Solutions and Colligative properties - JEE Main Notes, key concepts and formulae


Notes, key concepts and formulae of the chapter Solutions and Colligative properties for JEE Main preparations


SOLUTION

A homogeneous mixture of two or more pure substances whose composition may be altered within
certain limits is termed as solution.

CHARACTERISTICS OF SOLUTION

(i) It is homogeneous in nature, yet retaining the properties of its constituents.

(ii) It is made of two parts i.e. a solute and a solvent.

(iii) The component which has the same physical state in pure form as the solution is called solvent
and the other is called solute. Example, in case of solution of sugar and water, sugar is the
solute and water is solvent.

(iv) If both the components have same state as the solution, the one component which is in excess
is called solvent and the other is called solute. Example, alcohol in water, benzene in toluene
etc.

TYPES OF SOLUTION

    Solvent            Solute                                                      Examples
1.   Gas                 Gas                                                 Mixture of gases, air.
2.   Gas               Liquid                                            Water vapour In air, mist.
3.   Gas                Solid                             Sublimation of a solid Into a gas, smoke storms.
4.  Liquid              Gas                          CO2 gas dissolve in water (aerated drink), soda water.
5.  Liquid            Liquid                            Mixture of miscible liquids e.g. alcohol in water.
6.  Liquid             Solid                                           Salt in water, sugar in water.
7.  Solid                Gas                       Adsorption of gases over metals, hydrogen over palladium.
8.  Solid              Liquid                           Mercury in zinc, mercury in gold i.e. all amalgams.
9.  Solid               Solid                       Homogeneous mixture of two or more metals (i.e. alloys)

CAUSE OF MISCIBILITY OF LIQUIDS

(i) Chemically alike liquids dissolve in one another. e.g. all alkanes are miscible in all proportions
with one another because they are non-polar. Similarly polar liquid dissolves in each other; e.g.
lower alcohol in water.

(ii) Dipole interactions i.e. water and ether, water and phenol.

(iii) Molecular size of liquids which are mutually soluble are also approximately same.

SOLUBILITY

The maximum amount of a solute that can be dissolved in 100 g of solvent at a given temperature
(usually 25°C) is known its solubility at that temperature.

Solubility =  (Amount of substance dissolved/Amount of solvent) x 100

The solubility of a solute in a liquid depends upon the following factors

(a) Nature of the solute i.e. ionic or covalent (polar or non-polar).

(b) Nature of the solvent i.e. polar with high dielectric constant or non-polar.

(c) Temperature - Usually the solubility of the solute increases with increase of temperature (e.g.
KNO3, NH4Br) but in some cases increase in solubility is negligible (e.g. NaCl) and in cases of some salts (e.g. Na2SO4 and CeSO4 solubility decreases with increase in temperature)
.



METHODS OF EXPRESSING THE CONCENTRATION OF A SOLUTION

There are many ways of expressing the concentration of a solution. These methods are as follows

(i) Mass percentage :

It may be defined as the number of parts by mass of solute per hundred parts by mass of
solution.

(ii) Volume percentage :

It may be defined as the number of parts by volume of solute per hundred parts by volume of
solution.

(iii) Normality :

Normality of a solution is defined as the number of gram equivalent of the solute dissolved per
litre of the solution. It is represented by N.

Mathematically.

Normality N = Mass of solute in grams per litre / Equivalent mass of the solute

A solution having normality equal to one is called “normal solution”. Such a solution contains
one gram equivalent of solute per litre of solution. A seminormal solution contains 1/2 gram
equivalent. A decinormal solution contains 1/10 gram equivalent and a centinormal solution
contains 1/100 gram equivalent of solute per litre of solution.

or

Normality = (Mass of the solute x 1000) / (Equivalent mass of the solute x V)

or
                   (Mass of solute in gram)/[(gm. eq. mass of solute)(volume of solution (L)]

Where V is the volume in millilitre.

(iv) Molarity (M) :

Molarity of a solution is defined as the number of gm moles of the solute dissolved per litre of
the solution. It is represented by capital M. Mathematically.

Molarity (M) = Mass of the solute in grams per litre  / Molecular mass of the solute
                    = gram moles of solute / volume of solution in litre
                    = (mass of solute x 1000) / (GMM of solute x vol. of solution in ml)

A solution having molarity “one” is called molar solution. It may be remembered that both
normality as well as molarity of a solution changes with change in temperature.

(v) Molality (m) :

Molality of a solution may be defined as the number of gm moles of the solute dissolved in 1000
gm (1 kg) of the solvent. It is represented by small ‘m’. Mathematically

Molality (m) = Mass of the solute in grams per kg of solvent / Molecular mass of the solute

A solution containing one mole of solute per 1000 gm of solvent (1 kg) has molality equal to one
and is called molal solution. Molality is expressed in units of moles per kilogram (mol kg–1). The
molality of a solution does not change with temperature.

(vi) Mole fraction :

Mole fraction may be defined as the ratio of the number of moles of one component to the total
number of moles of all the components (i.e. solute and solvent) present in the solution.

Let us suppose that a solution contains the components A and B and suppose that 'a' gram of
A and 'b' gram of B are present in it. Let the molecular masses of A and B are MA and MB
respectively.

Then number of moles of A are given by nA = a/MA

and number of moles of B are given by nB = b/MB

Total number of moles of A and B = nA + nB

Mole fraction of A, XA = nA /  (nA + nB)

Mole fraction of B, XB = nA / (nA + nB)

Sum of mole fractions of all components is always one.
i.e. XA + XB = 1

So if mole fraction of one component of a binary solution is known say XB. then the mole fraction
of  XA = 1 – XB.

It may be noted that the mole fraction is always independent of the temperature.

(vii) Mole percent :

Mole percent is the number of moles of a component in 100 mole of the solute and solvent.
Mole percent = Mole fraction x 100

(viii) Formality (F) :

It may be defined as the number of gram formula masses of the ionic solutes dissolved per litre
of the solution. Mathematically

Formality (F) = Mass of the ionic solute in gram per litre / Formula mass of the solute

(ix) Mass fraction :

If WA is the mass of component A and WB the mass of component B in the solution, then the
mass fraction of component A and B is written as

Mass fraction of A = WA / (WA + WB)

Mass fraction of B = WB / (WA + WB)

(x) Parts per million (ppm) :

When a solute is present in trace amounts, its concentration is expressed in parts per million.
It may be defined as the number of parts by mass of solute per million parts by mass of the
solution.

Parts per million (ppm) = (Mass of solute / Mass of solution) x 106

VAPOUR PRESSURE

The pressure exerted by the vapours above the liquid surface in equilibrium with the liquid at a given
temperature is called vapour pressure of the liquid.

VAPOUR PRESSURE OF SOLUTIONS AND RAOULT’S LAW

When a small amount of a non-volatile solute is added to the liquid (solvent) ,it is found that the vapour pressure of the solution is less than that of the pure solvent. This is due to the fact that the solute particles occupy a certain surface area and as the evaporation is to take place from the surface only the particles of the solvent will have a less tendency to change into vapour i.e. the vapour pressure of the solution will be less than that of the pure solvent and it is termed as lowering of vapour pressure.

The vapour pressure of the solutions of non-volatile solutes can be obtained by applying Raoult’s law.

According to this law. the vapour pressure of a solution containing non-volatile solute is
proportional to mole fraction of the solvent.

For a two component solution A (volatile) and B (non-volatile) the vapour pressure of solution is given by
Vapour pressure of solution = Vapour pressure of solvent in solution ∝ Mole fraction of solvent.

P = PA ∝ XA (or)

PA = KXA

Where K is proportionality constant.

For pure liquid XA = 1. then K becomes equal to the vapour pressure of the pure solvent which is
denoted by PºA.

Thus PA = PºAXA

(or)  Psolution =  Ppure solvent  x  mole fraction of solvent

In a solution of two miscible non-volatile liquids. A and B the partial vapour pressure PA of the liquid A is proportional to its mole fraction XA and the partial vapour pressure PB of liquid B is proportional to its mole fraction XB.

Thus PA ∝ XA


PA = PºAXA

Also PB ∝ XB

PB = PºBXB

Where PºA and PºB are the vapour pressures of pure components A and B respectively.
This relation is called Raoult’s law. It states that “For a solution of two or more miscible liquids, the
partial vapour pressure of each component is directly proportional to its mole fraction”.

Total vapour pressure of solution P = PºAXA + PºBXB

The solutions which obey Raoult’s law are called Ideal solutions. For such solution, vapour pressure
of the solution always lies between the vapour pressure of the pure components. Raoult’s law is
applicable only in case of dilute solutions.

RELATIVE LOWERING OF VAPOUR PRESSURE

For a solution of non-volatile solute with volatile solvent.

P = PA ∝ XA , P = vapour pressure of solution

PA = PºA XA  , PA = vapour pressure of solvent

If mole fraction of solute is XB, then XA = mole fraction of solvent and PºA = vapour pressure of pure solvent.
XA + XB  = 1 or  XA = 1–XB  or

PA = PºA (1–XB) = PºA – PºAXB or

A – PA = PºAXB or

XB = (PºA – PA) / PºA  

A – PA is the lowering of vapour pressure whereas (PºA – PA) / PºA   is called relative lowering of
vapour pressure. So Raoult’s law can also be defined as “For a solution containing non-volatile
solute, at a particular temperature. the relative lowering of vapour pressure is equal to mole fraction
of the solute”.

IDEAL SOLUTIONS

An ideal solution may be defined as the solution which obeys Raoult’s law over the entire range of
concentration and temperature and during the formation of which no change in enthalpy or volume
takes place. So for ideal solutions

(i) It should obey Raoult’s law.
i.e. PA = PºA XA and PB = PºBXB

(ii) Δ Hmixture = 0

(iii) Δ Vmixture = 0

There is no solution which behaves strictly as the ideal solution. However. the solution in which solvent - solvent and solute - solute interactions are almost of the same type as solvent-solute interactions behaves as ideal solutions.

Examples.
(a) n-heptane and n-hexane
(b) Chlorobenzene and bromobenzene
(c) Benzene and toluene
(d) Ethyl bromide and ethyl iodide

Only very dilute solutions behave as ideal solutions.

NON-IDEAL SOLUTIONS

For a non-ideal solutions, the conditions are

(i) Raoult’s law is not obeyed
i.e. PA ≠ PºA XA and PB  ≠ PºBXB

(ii) Δ Hmixture ≠ 0

(iii) Δ Vmixture ≠ 0

The non-ideal solutions are further classified into two categories

(a) Solutions with positive deviation and
(b) Solutions with negative deviation.

SOLUTION WITH POSITIVE DEVIATION

It has the following characteristics

(i) Solution in which solvent-solvent and solute–solute interactions are stronger than solvent-solute
interactions.

(ii) At intermediate composition vapour pressure of the solution is maximum. .

(iii) At intermediate composition boiling point is minimum.

So for the non-ideal solutions exhibiting positive deviations

(i) PA  > PºA XA and PB > PºBXB

(ii) Δ Hmixture = + ve

(iii) Δ Vmixture = + ve

AZEOTROPE OR AZEOTROPIC MIXTURE

A solution which distills without change in composition is called azeotropic mixture or azeotrope.
Example of positive deviation - A mixture of n-hexane and ethanol

SOLUTION WITH NEGATIVE DEVIATION

It has the following characteristics

(i) Solutions in which solvent-solvent and solutesolute interactions are weaker than solute-solvent
interactions.

(ii) At intermediate composition vapour pressure of the solution is minimum.

(iii) At intermediate composition boiling point is maximum.

So for non-ideal solutions exhibiting negative deviations

(i) PA  < PºA XA and PB < PºBXB

(ii) Δ Hmixture = – ve

(iii) Δ Vmixture = – ve

COLLIGATIVE PROPERTIES

The properties of dilute solutions containing nonvolatile solute, which depends upon relative number
of solute and solvent particles but do not depend upon their nature are called colligative properties.

Some of the colligative properties are

(i) Relative lowering of vapour pressure:
(ii) Elevation of boiling point;
(iii) Depression of freezing point; and’
(iv) Osmotic pressure.

RELATIVE LOWERING OF VAPOUR PRESSURE

As shown earlier the mathematical expression for relative lowering of vapour pressure is as follows

XB = (PºA – PA) / PºA ,  XB = Mole fraction of the solute

ΔP = PºA – PA = lowering of vapour pressure
PA = vapour pressure of pure solvent

Molecular mass of non-volatile substance can be determined from relative lowering of vapour pressure

(PºA – PA) / PºA  = (WB/MB) / [(WA/MA) + (WB/MB)]

For dilute solution WB/MB < WA/MA and hence WB/MB may be neglected in the denominator.
So

(PºA – PA) / PºA = (WB/MB) / (WA/MA) (or)

MB = [WB/(MA.WA)] PºA / (PºA – PA)

ELEVATION OF BOILING POINT

The boiling points elevates when a non-volatile solute is added to a volatile solvent which occurs due to lowering of vapour pressure.

The boiling point of a liquid may be defined as the temperature at which its vapour pressure
becomes equal to atmospheric pressure.

So when a non-volatile solute is added to a volatile solvent results lowering of vapour pressure and
consequent elevation of boiling point

ΔTb ∝ ΔP ∝ XB;

ΔTb = elevation in B.P.
ΔP = lowering of V.P.

or Δ Tb = KXB

XB = mole fraction of solute
K = elevation constant

(or)

ΔTb = K x (WB/MB) / (WA/MA)

If WA is the weight of solvent in kg. then nB/Wis equal to molality (m) of the solution.

ΔTb = KMAm (or) ΔTb = Kbm , (KM=  Kb)

Where Kb is molal elevation constant or molal ebullioscopic constant.

When molality of the solution is equal to one. then

ΔTb = Kb
Hence molal elevation constant of the solvent may be defined as the elevation in its boiling
point when one mole of non-volatile solute is dissolved per kg (1000 gm) of solvent. The unit of Kb
are K kg mol–1.

Because molality of solution m = (WB/MB).(1000/WA)

So ΔTb = Kb.(WB/MB).(1000/WA)

(or)

M= (Kx W x 1000)/(ΔTb x WA)

When the volume is taken as 1000 ml., then elevation in boiling point is known as molar elevation
constant.

DEPRESSION IN FREEZING POINT

The freezing point of a pure liquid is fixed. If a non-volatile solute is dissolved in a liquid the freezing point of the solution is lowered. The freezing point is that temperature at which the solvent has the same vapour pressure in two phases. liquid solution and solid solvent. Since the solvent vapour pressure in solution is depressed. its vapour pressure will become equal to that of the solid solvent at a lower temperature

ΔTf ∝ ΔP ∝ XB;

ΔT = depression in F.P.
ΔP = lowering of V.P.

or ΔTf = KXB

XB = mole fraction of solute
K = depression constant

(or)

ΔTf = K x (WB/MB) / (WA/MA)

If WA is the weight of solvent in kg. then nB/Wis equal to molality (m) of the solution.

ΔTf = KMAm (or) ΔTf = Km

Where Kf is molal depression constant. When molarity (m) of the solution is one. then
ΔTf = Kf

Hence molal depression constant or molal cryoscopic constant may be defined as “the depression
in freezing point when one mole of non-volatile solute is dissolved per kilogram (1000 gm) of solvent” and molar depression constant is defined as “the depression in freezing point when one mole of non-volatile solute is dissolved per litre (1000 ml) of solvent.”

The molecular mass of the non-volatile solute may be calculated by the use of following mathematical equation

M= (Kf x W x 1000)/(ΔTf x WA)

ANTIFREEZE SOLUTIONS

Water is used in radiators of vehicles as cooling liquid. If the vehicle is to be used at high altitudes
where temperature is sub-zero. water would freeze in radiators. To avoid this problem. a solution of
ethylene glycol in water is used in radiators which will lower the freezing point lower than zero.

OSMOTIC PRESSURE

When a solution is separated from the pure solvent with the help of a semipermeable membrane there
occurs the flow of solvent molecules from the pure solvent to the solution side. The flow of solvent
molecules from a region of higher concentration of solvent to the region of lower concentration of
solvent is termed as the phenomenon of osmosis. This also happens when two solution of different
concentrations are separated by a semipermeable membrane.

As a result of osmosis a pressure is developed which is termed as osmotic pressure. It is defined
in various methods.

(1) The excess hydrostatic pressure which builds up as a result of osmosis is called osmotic
pressure.

(2) The excess pressure that must be applied to the solution side to prevent the passage of solvent
into it through a semipermeable membrane.

(3) Osmotic pressure of a solution is equal to the negative pressure which must be applied to the
solvent in order to just stop the osmosis.

(4) The osmotic pressure of a solution may be defined as the extra pressure which should be
applied to the solution to increase the ‘chemical potential of solvent in solution equal to the
chemical potential of the pure solvent at the same temperature.


ISOTONIC SOLUTION

The two solutions having equal osmotic pressure are termed as isotonic solution.

Hypertonic solution - A solution having higher osmotic pressure than some’ other solution is said
to be called hypertonic solution.

Hypotonic solution - A solution having a lower osmotic pressure relative to some other solution is
called hypotonic solution.

Semipermeable Membrane

A membrane which allows the passage of solvent molecules but not that of solute. when a solution
is separated from the solvent by it is known as semipermeable membrane.

Some example of it are as follows

(a) Copper ferrocyanide Cu2[Fe(CN)6];
(b) Calcium phosphate membrane; and
(c) Phenol saturated with water.

THEORY OF DILUTE SOLUTIONS

The osmotic pressure of a dilute solution was the same as the solute would have exerted if it were
a gas at the same temperature as of the solution and occupied a volume equal to that of the solution.
This generalization is known as Van’t Hoff theory of dilute solutions.

The osmotic pressure is a colligative property. So the osmotic pressure is related to the number of
moles of the solute by the following relation

πV= nRT

π = (n/V)RT

π = CRT

Here C = concentration of solution in moles per litre;

R = solution constant;
T = temperature in Kelvin degree;
n = number of moles of solute; and
V = volume of solution.

.
VAN’T HOFF-BOYLE’S LAW

If the temperature remaining constant the osmotic pressure of the solution is directly proportional to
the molar concentration of the solute. ie.
π ∝ C (T constant)
(or)
π ∝ n/V (T constant)
For a solution containing same amount at different volume. we will have π ∝ 1/V (T constant)
i.e. osmotic pressure is inversely proportional to the volume.

VAN’T HOFF-CHARLE’S LAW

At constant volume. the osmotic pressure of a solution is directly proportional to its absolute
temperature. i.e.
π ∝ T (C constant)

Berkeley and Hartley’s method is used to determine the osmotic pressure.

DETERMINATION OF MOLECULAR MASS FROM. OSMOTIC PRESSURE

The molecular mass of a substance i.e. solute can be calculated by applying the following formula

M =WRT / πV

Accurate molecular mass will only be obtained under following conditions

(i) The solute must be non-volatile;
(ii) The solution must be dilute; and
(iii) The solute should not undergo dissociation or association in the solution.

Solution Constant R - The solution constant R has the same significance and value as the gas
canstant. i.e.

R = 0.0821 litre-atm K–1 mol–1 = 8.314 X 10–7 erg K–1mol–1
    = 8.314 JK–1 mol–1 = 8.314 Nm K–1 mol–1

Abnormal molecular masses - When the substances undergo dissociation or association in
solution then the value of observed colligative property and value of calculate colligative property are
quite different. As the dissociation and association cause changes in the number of particles as
compared to usual condition.

VAN’T HOFF FACTOR

In 1886, Van’t Hoff introduced a factor i called Van’t Hoff factor to express the extent of association
or dissociation of solute in solution.

Van’t Hoff factor i = Normal molecular mass / Observed molecular mass

In case of association, observed molecular mass being more than the normal the factor i has a value
less than 1. But in case of dissociation, the Van’t Hoff factor is more than 1 because the observed
molecular mass has a lesser magnitude. In case there is no association or dissociation the value i
becomes equal to 1. Therefore

i = Observed value of colligative property / Calculated value of colligative property

Introduction of the Van’t Hoff factor modifies the equations for the colligative properties as follows

Elevation of B.P. ΔTb  = iKm

Depression of F.P. ΔT= iKm

Osmotic pressure π = iCRT

From the value of’i it is possible to calculate degree of dissociation or degree of association of
substance in solution by the following formula

α = (i-1) / (n-1)

Where α is the degree of dissociation and n is the number of particles per molecule.

The ratio of Van’t Hoff factor i to the number of ions furnished by one molecule of the electrolyte ‘n’
is known as the osmotic coefficient. It is denoted by g. Mathematically

Osmotic coefficient ‘g’ = Van't Hoff factor 'i' / 'n'

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