Solutions: Overview, Questions, Preparation

A 2.1 Mass of Solution = Mass of Benzene + Mass of Carbon Tetrachloride

= 22 g + 122 g = 144 g

Mass percentage of Benzene = Mass of Benzene / Mass of Solution X 100 = 22/144 X 100 = 15.28%

Mass percentage of CCl₄ = Mass of CCl₄ / Mass of Solution X 100 = 122/144 X 100 = 84.72%

Let the total mass of the solution be 100 g and the mass of benzene be 30 g.

∴ Mass of carbon tetrachloride = (100 - 30) g = 70 g

Molar mass of benzene (C₆H₆) = (6 × 12 + 6 × 1) g mol ^-1

= 78 g mol ^-1

∴ Number of moles of C₆H₆ =30/78 mol

= 0.3846 mol

Molar mass of carbon tetrachloride (CCl₄) = 1 × 12 + 4 × 35.5

= 154 g mol ^-1

∴ Number of moles of CCl₄ = 70/154 mol

= 0.4545 mol

Thus, the mole fraction of C₆H₆ is given as:

Number of moles of C₆H₆ / Number of moles of C₆H₆  + Number of moles of CCl₄

= 0.3846 / (0.3846 + 0.4545)

• A 2.3 Molarity = Moles of Solute / Volume of Solution in liter
• (a) Given, In 4.3 L of solution there is 30 g of Co(NO₃)₂. 6H₂O

Molar mass of Co(NO₃)₂.6H₂O = (1 × 59 + 2 × (1 × 14 + 3 × 16) + 6 × 18)

= 291 g/mol.

∴ Moles = Given Mass / Molar Mass = 30/291 = 0.103 mol.

Now, Molarity = 0.103 mol / 4.3 L

= 0.023 M

• (b) Given, 30 mL of 0.5 M H₂SO₄ diluted to500 mL.

In 1000 mL of 0.5 M H₂SO₄, number of moles present is 0.5 mol.

∴ In 30 mL of 0.5 M H₂SO₄, number of moles present = 30X 0.5 / 1000 mol.

= 0.015 mol.

∴ Molarity = 0.015 mol / 0.5L

= 0.03 M.

A 2.4, 2.5 kg of 0.25 molal aqueous solution.
Molar mass of urea (NH2CONH2) = (2 (1 × 14 + 2 × 1) + 1 × 12 + 1 × 16)
 = 60 g/mol
1000 g of water contains 0.25 mol = (0.25 × 60) g of urea.
 = 15 g of urea.
Means, 1015 g of solution contains 15 g of urea

Therefore,
2500 g of solution contains = 15 X 2500 / 1015
 = 36.95 g
Hence, mass of urea required is 37 g (approx).

A 2.5 Molality, also called molal concentration, is a measure of the concentration of a solute in a solution in terms of amount of substance in a specified amount of mass of the solvent. Molar mass of KI = 39 + 127 = 166 g/mol.
20% aqueous solution of KI means 200 g of KI is present in 1000 g of solution. Therefore, 

Molality = Moles of KI / Mass of Water in kg

= (200/166) / (0.8) = 1.506 m

(b) Molarity is the concentration of a solution expressed as the number of moles of 
solute per litre of solution.
Given,
Density of the solution = 1.202 g/mL
Volume of 100 g solution = mass/ density
 = 100/1.202 
 = 83.19 mL
Therefore, molarity = 20/166 mol/ 83.19 × 10^-3 L 
 = 1.45 M
∴ Molarity of KI = 1.45M

(c) Molar mass of KI = 39 + 127 = 166 g/mol.
Moles of KI = 20/166 = 0.12 mol
Moles of water = 80/18 = 4.44 mol

Therefore,
Mole fraction of KI = Moles of KI / Moles of KI+ Moles of Water
 = 0.12 / 0.12+ 4.44
 = 0.0263
∴ Mole fraction of KI = 0.0263

A 2.6 According to Henry's law,"At a constant temperature, the amount of a given gas that dissolves in a given type and volume of liquid is directly proportional to the partial pressure of that gas in equilibrium with that liquid."
Stated as, 
p = K_Hx 
Where, P = partial pressure of the solute above the solution
K_H = Henry's constant
x = concentration of the solute in the solution
Given,
Solubility of H₂S in water at STP is 0.195 m
We know,
At STP pressure p = 0.987 bar
0.195 mol of H₂S is dissolved in 1000g of water

Moles of water = 1000/18
 = 55.56 g/mol
∴ the mole fraction of H₂S = Moles of H₂S / Moles of H₂S + Moles of water
 = 0.195 / 0.195+55.6
 = 0.0035
According to Henry's law, p = K_Hx 
K_H = p/x
K_H = 0.987 / 0.0035
K_H = 282 bar 
∴ The Henry’s law constant is 282 bar

A 2.7 According to Henry's law, "At a constant temperature, the amount of a given gas that dissolves in a given type and volume of liquid is directly proportional to the partial pressure of that gas in equilibrium with that liquid."
Stated as,
p = K_Hx 
Where, 
P = partial pressure of the solute above the solution
K_H = Henry's constant
x = concentration of the solute in the solution
Given,

K_H = 1.67x10⁸ Pa 

P_CO2  2.5 atm = 2.5 × 1.01325 × 10⁵Pa

According to Henry's law,

p = K_Hx

x = P/K_H

x = 2.5 X 1.01325 X 10⁵ / 1.67 X 10⁸

x = 0.00152

In 500 ml of soda water there is 500 ml of water (neglecting soda)

Mole of water = 500/18

= 27.78 mol

Now,

x = n_co2 / n_co2 + n_H2o

n_co2  = x X n_H2o

 = 0.00152 X 27.78

= 0.042 mol

Hence, quantity of CO₂ in 500mL of soda water  0.042 × 44 = 1.848 g

Given, P_A^o = 450 mm Hg

^P_Bo = 700 mm Hg

p_total = 600 mm of Hg

By using Rault's law,

p_total = P_A + P_B

p_total = P_A^ox_A + P_B^ox_B

p_total = P_A^ox_A + P_B^o( 1 - xA )

p_{total = (}P_A^o- P_B^o)x_{A +} P_B^o

600 = (450 - 700) x_A + 700

-100 = -250 x_A

x_A = 0.4

∴ x_B = 1 - x_A

x_B = 1 – 0.4

x_B = 0.6

Now,

P_A = P_A^ox_A

P_A = 450 × 0.4

P_A = 180 mm of Hg and

P_B = P_B^ox

P_B = 700 × 0.6

P_B = 420 mm of Hg

Composition in vapour phase is calculated by

Mole fraction of liquid,

A =P_A / P_A + P_B

= 180/180+420

= 0.30

Mole fraction of liquid,

B =P_B / P_A + P_B

= 420 / 180+420

= 0.70

A 2.9 Given,

Vapour pressure of water, P_I^o = 23.8 mm of Hg

Weight of water, w₁ = 850 g

Weight of urea, w₂ = 50 g

Molecular weight of water, M₁ = 18 g/mol

Molecular weight of urea, M₂ = 60 g/mol

n₁ = w₁/M₁ = 850/18 = 47.22 mol

n₂ = w₂/M₂ = 50/60  = 0.83 mol

We have to calculate vapour pressure of water in the solution p₁

By using Raoult's therom,

P_I = 23.4 mm of Hg Hence,

The vapour pressure of water in the solution is 23.4mm of Hg and its relative lowering is 0.0173.

A 2.10 Given,

Mass of water, w_l = 500 g

Boiling point of water = 99.63°C (at 750 mm Hg).

Molal elevation constant, K_b = 0.52 K kg/mol

Molar mass of sucrose (C₁₂H₂₂O₁₁), M₂  (11 × 12 + 22 × 1 + 11 × 16) = 342 g/mol

Elevation of boiling point ΔT_b = (100 + 273) - (99.63 + 273) = 0.37 K

We know that,

ΔT_b = K_b X 1000 X W₂ / M₂ X W₁

0.37 = 0.52 X 1000 X W₂ / 340 X 500

w₂ = 0.37 X 342 X 500 / 0.52 X 1000

w₂ = 121.67 g

Hence,

121.67 g (approx) Sucrose is added to 500g of water so that it boils at 100°C.

A 2.11 Given

Mass of acetic acid, w₁ = 75 g

Lowering of melting point, ΔT_f = 1.5 K

K_f = 3.9 K kg/mol

Molar mass of ascorbic acid (C₆H₈O₆), M₂ 6 × 12 + 8 × 1 + 6 × 16 = 176 g/mol[1]

We know that,

= 5.08 g

Hence,

5.08 g of ascorbic acid is needed to be dissolved.

A 2.12 Given,

Volume of water, V = 450 mL = 0.45 L

Temperature, T=(37 + 273)K = 310 K

1.0 g of polymer of molar mass 185,000

Number of moles of polymer, n = 1 / 185,000 mol

We know that,

Osmotic pressure, π = nRT/V

= 1 X 8.314 X 10³ X 310 / 185000 X 0.45

= 30.98 Pa

= 31 Pa (approx)

A 2.1 A solution is a homogeneous mixture of two or more than two substances on molecular level whose composition can vary within certain limits.

The part or component of the mixture present in a lesser amount is called

the SOLUTE and the one present in larger amount is called the SOLVENT. For eg- small amount of salt [solute] dissolved in water [solvent].

There are nine types of solutions formed. They are:

Out of these nine types solution, solid in liquid, liquid in liquid & gas in liquid are very common. When the components of the solution are mixed, the resulting solution may exist in any of the three possible states of matter that is solid, liquid or gaseous.

They are:

• (1) Gaseous solution: In such solutions solvent is Since the solvent is gas,the solute can be solid, liquid or gas. For example, a mixture of oxygen and nitrogen gas is a gaseous solution.
• (2) Liquid solution: In such type of solutions liquid acts as the solvent. The solute in these solutions may be gas, liquid, or solid.
• (3) Solid solutions: As the name suggests, in such solutions solid acts as the solvent. The solute in these solutions may be a gas, liquid or solid. For example, a solution of copper in gold is a solid solution.

A 2.2 As the name signifies, a solid solution is one in which solvent is solid.So considering this aspect absorption of hydrogen over platinum or palladium is an example of such solution. Platinum or palladium is used as a catalyst in hydrogenation processes.

A 2.3 (1) Mole fraction - The mole fraction of a particular component in a solution is the ratio of the number of moles of that component to the total number of moles of all the components present in the solution.

Mathematically,

Mole Fraction of component = Number of moles of given component / Total number of moles in solution

Mole Fraction is independent of temperature.

• (2) Molality - Molality of a solution is defined as the number of moles of solute dissolved per 1000g [1kg] of the It is represented by m.

Molality actually represents the concentration of solution in mol / kg.

Mathematically,

Molality = Number of moles of solute/ Mass of solvent in kg

It is represented by m.

• (3) Molarity- The number of moles of solute dissolved per litre of the solution at a particular temperature is called the molarity of the solution at that

Molarity actually represents the concentration of a solution in mol / L.

Mathematically,

Molarity = Number of moles of solute/ Volume of solution in litres

• (4) Mass percentage - Mass percentage is defined as the mass of the solute in grams dissolved per 100g of the It is also referred to as weight percentage [w/w].

For example, 10% [by mass] urea solution means that 10 g of urea are present in 100 g of solution, the solvent being only 100-10 = 90 g.

Mathematically, the mass percentage of a solute in a solution is given by:

Mass Percentage of Solute = Mass of solute / Mass of solute + Mass of solvent X 100

Or

Mass Percentage of Solute = Mass of solute / Mass of solution X 100

A 2.4 Given:

Concentration of Nitric Acid, HNO₃ = 68%

Density of solution, d = 1.504 g/ml

To find: Molarity, M_o

Formula:

Density, d = Mass (M) / volume (V)

Molarity, M_o  = Number of moles of solute/ Volume of solution in litres

Solution:

68% of Nitric acid by mass in aqueous solution means that 68g [[68 × 100]/100] of Nitric acid present in 100g of solution.

⇒ Molecular mass of Nitric Acid, HNO₃ = [1 × 1] + [1 × 14] + [16 × 3]

= 63g

⇒ Number of moles of Nitric Acid = [68/63]

= 1.079 moles

⇒ Given Density, d = 1.504 g/ml

⇒ Volume, v = [100/1.504]

= 66.489 ml

⇒ Molarity, M_o = [1.079/66.489] × 1000

= 16.23 M

Therefore the molarity of the sample is 16.24 M.

A 2.5 Given:

10% w/w solution means that if solution is of 100g then 10g of glucose is present in it [[10 × 100]/100] and the amount of water present in it is [100-10], M_w = 90g.

Density, d = 1.2g/ml

To find: Molality and Mole Fraction of Each component Formula:

Density, d = Mass (M) /volume (V)

Molality = Number of moles of solute/ Mass of solvent in kg

Mole Fraction of component  = number of moles of given component / total number of moles in solution

Molarity, M_o = Number of moles of solute/ Volume of solution in litres

Solution:

Calculation of Molality:

⇒ Molecular mass of Glucose, [C₆H₁₂O₆] = [6 × 12] + [12 × 1] + [6 × 16]

= 72 + 12 + 96

= 180g

⇒ Number of moles of Glucose, N_g = [10/180]

= 0.0556 moles

⇒ Molality = N_g/M_W

⇒ Molality = 0.0556/ 0.09

= 0.6177 m

≈ 0.62 m

Calculation of Mole Fraction:

⇒ Molecular Mass of Water, [H₂O] = [2 × 1] + [16 × 1]

= 2 + 16

= 18g

⇒ Number of moles of water, N_w = [90/18]

= 5 moles

⇒ Mole Fraction of Glucose,

x_g = N_G / N_W + N_G

= 0.0556 / 5+0.0556

= 0.011

⇒ Mole Fraction of Water, x_w = (1-x_g)

= 1 – 0.011

= 0.989

Calculation of Molarity:

From density we can find out volume.

⇒ Volume, V = [100/1.2]

= 83.33 ml

⇒ Molarity  = 0.0556/ 83.33 X 1000

Molarity, M = 0.667 M

≈ 0.67 M

Therefore the answers obtained in the numerical can be summarized in following table:

A 2.6 Given:

Molarity of HCl, = 0.1 M

Mass of Mixture = 1g

To find: Volume of HCl to react completely with mixture

Formula:

Molarity, M_o = number of moles of solute/ Volume of solution in litres

Solution:

Calculation of Amount of each component in mixture:

⇒ Let the amount of Na₂CO₃ be X g

⇒ And Let amount of NaHCO₃ be [1-X] g

⇒ Molecular Weight of Na₂CO₃ = [23 × 2] + [12] + [3 × 16]

= 106 g

⇒ Molecular Weight of NaHCO₃ = [23] + [1] + [12] + [3 × 16]

= 84 g

⇒ Number of moles of NaHCO₃ = 1-x / 84

⇒ Number of moles of Na₂CO₃ = x / 106

Now it is given in the question that the mixture is equimolar, so

⇒ Number moles of Na₂CO₃ = Number of moles of NaHCO₃

1-x/84 = x/106

106 – 106X = 84X

190X = 106

x= 106/190

X = 0.5579

≈ 0.558

Amount of Na₂CO₃ = 0.558 g

⇒ Amount of NaHCO₃ = 1- 0.558 g = 0.442 g

⇒ Number moles of Na₂CO₃ = [0.558/106]

= 0.00526 moles

⇒ Number of moles of NaHCO₃ = 0.00526 moles …. [Mixture is equimolar]

Calculation of volume of HCl required

⇒ In the question it is given that HCl reacts with the mixture.

Na₂CO₃ + 2HCl → 2NaCl + H₂O + CO₂

NaHCO₃ + HCl → NaCl + H₂O + CO₂

From the above chemical equations it can be found that 2 moles of HCl is required to react with one mole of Na₂CO₃ and 1 moles of HCl is required to react with one mole of NaHCO₃.

So total number of moles of HCl required to completely react with mixture is as follows:

⇒ Total Number of Moles of HCl = [2 × 0.00526] + 0.00526

= 0.01578 moles

From Molarity Formula we have,

Molarity, M_o = number of moles of solute/ volume in litres (V)

⇒ Volume in litres, V = [0.01578/0.1]

= 0.1578 L

⇒ Volume in ml, V = 0.1578 × 1000

= 157.8 ml

Therefore the volume of 0.1 M HCl are required to react completely with 1 g mixture of Na₂CO₃ and NaHCO₃ containing equimolar amounts of both is 157.8 ml.

A 2.7 Let the first solution be solution A and second solution be solution B. Given:

⇒ Mass of Solute in solution A, M_a = [25 × 300]/100

= 75g

⇒ Mass of Solute in solution B, M_b = [40 × 400]/100

= 160g

To find: Mass Percentage of Solute

Formula:

Mass Percentage of Solute = mass of solute/ mass of solution X 100

Solution:

⇒ Total mass of Solute in mixture, M = M_a + M_b

= [75 + 160]

= 235g

⇒ Total mass of solution = 300 + 400

= 700g

⇒ Mass Percentage of Solute = 235/700 X 100

= 33.5%

⇒ Percentage of Water in final solution = 100-33.5

= 66.5%

Therefore mass percentage of solute is 33.5% and that of water is 66.5%.

A 2.8 Given:

Mass of ethylene glycol (C₂H₆O₂) = 222.6 g

Mass of water = 200 g

Density, d = 1.072 g/ml

To find: Molality and Molarity of solution

Formula:

Molality = Number of moles of solute/ Mass of solvent in kg

Molarity, M_o = number of moles of solute/ volume of solution in litres

Density, d = Mass (M) / volume (V)

Calculation of Molality:

⇒ Molecular Mass of ethylene glycol (C₂H₆O₂) = [12 × 2] + [6 × 1] + [16 × 2]

= 24 + 6 + 32

= 62 g

⇒ Number of moles of ethylene glycol (C₂H₆O₂) = [222.6/62]

= 3.59 moles

⇒ Mass of water = 200 g

⇒ Molality = numebr of moles of solute/ Mass of solvent in kg

= 3.59 / 200 X 1000

= 17.95 m

Calculation of Molarity:

⇒ Total mass of solution = [222.6 + 200]

= 422.6 g

From density formula we can find out the volume required.

⇒ Volume of solution, V = [422.6 /1.072]

= 394.216 ml

⇒ Molarity, M_o = number of moles of solute/ volume of solution in litres

⇒ Molarity, M_o =3.59 / 394.216 X 1000

⇒ Molarity = 9.1067

⇒ Molarity ≈ 9.11 M

Therefore the Molality and Molarity of the solution is as follows:

Molality = 17.95 m

Molarity = 9.11 M

A 2.9 Given:

Level of contamination = 15 ppm [by mass]

To find: Mass Percentage and Molality

Formula:

Molality = Number of moles of solute / Mass of solvent in kg

Mass Percentage of Solute  = Mass of solute  / Mass of solution X 100

Solution:

Calculation of Mass Percentage:

15 ppm means 15 parts of Chloroform in 10⁶ parts of drinking water

⇒ Mass Percentage = Mass of choloroform / Total mass X 100

= 15 / 10⁶ X 100

= 1.5 × 10^-3

Calculation of Molality:

⇒ Molecular Mass of Chloroform, CHCl₃ = [12] + [1] + [35.5 × 3]

= 119.5 g

⇒ Number of Moles of Chloroform = [15 / 119.5]

= 0.1255 moles

Molality = Number of moles of solute / Mass of solvent in kg

= 0.1255 / 10⁶ X 1000

= 1.255 × 10^-4

Therefore the Mass Percentage is = 1.5 × 10^-3 and the Molality of the solution is = 1.255 × 10^-4 m.

A 2.10 The lower members of alcohols are completely miscible [highly soluble] with water but the solubility decreases with increase in the molecular weight. The lower members of the alcohol group have the capability to form intermolecular hydrogen bonding with water molecules as alcohols are polar molecules in nature.

Alkyl groups are hydrophobic [prevents formation of hydrogen bonds with water] in nature. In lower alcohols, the alkyl group is small and the –OH group of alcohol is effective in making hydrogen bonds with water.

But with the increase in the size of alkyl group, the hydrophobic [water hating] nature of alkyl group dominates over the hydrophilic [water liking] nature of –OH group making the molecule less soluble in water. So solubility of alcohols decreases with increase in its molecular mass.

Since molecular interaction is weaker between higher alcohols and water, as a result, when alcohol and water are mixed, the intermolecular interactions become weaker and the molecules can easily get liberated off. This increases the vapour pressure of the solution, which in turn lowers the boiling point of the resulting solution.

A 2.11 Whenever a gas is dissolved in a liquid, a small amount heat is liberated in the process. So dissolving a gas in liquid is overall an exothermic process.

So according to the LeChatelier principle, whenever the temperature is increased for a reaction which is exothermic in nature, the equilibrium shifts backwards and the reaction proceeds in backward direction that means the solution gets dissociated and will give off gas and hence solubility of gas decreases.

So with the increase in temperature, the solubility of the gases in liquids decreases.

A 2.12 The solubility of a gas in water depends on following three parameters:

1. Nature of gas
2. Temperature
3. Pressure

The solubility decreases with increase in temperature. Temperature and pressure follow inverse proportionality. So solubility increases with increase with pressure. A quantitative relation between pressure and solubility of a gas in a solvent was given by W. Henry [1803]. This relationship is known as Henry’s law.

Statement:

Henry’s law can be expressed as follows.

At constant temperature, the solubility of a gas in a liquid is directly proportional to the pressure of the gas.

Mathematically,

Solubility ? Pressure of the gas

Some of the important applications of Henry’s law are as follows.

• (i) Since the solubility of a gas in water increases with pressure, soft drink bottles are sealed under high pressure to accommodate more CO₂ in the soft drink making the drink fizzier.
• (ii) If deep sea divers [scuba divers] use air for respiration, they develop a medical condition known as bends with involves the blocking of capillaries. This is because air is mainly a mixture of oxygen and According to Henry’s Law, the solubility of gases increases with increase in pressure. When diver breathes air under high pressure in water, nitrogen dissolves in his blood. When the diver comes towards surface, the pressure gradually decreases. This releases the dissolved gases and leads to the formation of bubbles in the blood. This is quite painful and dangerous to life. In order to avoid bends and toxic effects of dissolved nitrogen in blood, the tanks used by sea divers are filled with air diluted with helium [ 11.7% H_e + 56.2% N₂ + 32.1% O₂ ].
• (iii) The pressure of oxygen in air decreases in going up the mountains. At very high altitudes the partial pressure of oxygen in air is much less than that at the ground Therefore, people living at high altitude or climbers have low concentrations of oxygen in the blood and tissues. This leads to weakness and loss in the clarity of thinking. These symptoms create a condition known as anoxia.
• (iv) Henry’s law is also related to biology as it explains the supply of inhaled oxygen to tissues. When air is inhaled, it combines with haemoglobin [oxygen carrier of RBCs] of the blood in lungs to form oxyhaemoglobin because in lungs the partial pressure of oxygen is high. Partial pressure of oxygen is low in tissues. Hence, oxygen is released from oxyhaemoglobin and is utilised by the cells to carry out their tasks.

A 2.13 Given:

Mass of ethane in solution A = 6.56 × 10^–3 g

Partial pressure of solution A = 1 bar

Mass of ethane in solution B = 5.00 × 10^–2 g

To find: Partial Pressure of gas

Formula:

By Henry’s law:

Mass of dissolved gas M = k × P Where

k = proportionality constant

P = Partial Pressure

Solution:

⇒ M₁ = k × P₁................. [1]

⇒ M₂ = k × P₂................. [2]

Dividing the [2] by [1], we get

M₂/M₁ = P₂/P₁

P₂ = M₂ X P₁ / M₁

P₂ = 5 X 10^–2 X 1  / 6.56 X 10^–3

P₂ = 7.62 bar

Therefore the partial pressure of the gas is 7.62 bar.

A 2.14 Raoult’s law states that at a given temperature, the vapour pressure of a solution containing non volatile solute is directly proportional the mole fraction of the solvent.

Non ideal solutions show positive and negative deviations from ideal behaviour.

Non ideal solutions showing positive deviations from Raoult’s law-

A solution is said to show positive deviation from Raoult’s Law when at any composition, its vapour pressure is higher than that given by Raoult’s Law.

The positive deviation is shown by those liquid pairs in which the A-B molecular interaction forces are weaker than the corresponding A-A or B-B molecular interaction forces. When the A-B molecular interaction forces are weaker, then molecules of liquid A find it easier to escape as compared to pure solution thereby increasing the vapour pressure of solution.

As a result, each component of solution has a partial vapour pressure greater than expected on the basis of Raoult’s law. This is called positive deviations from Raoult’s law, that is P_A>P_A^oX_A and P_B>P_B^oX_B

∆_mixH [Change in Enthalpy] is positive because energy is required to break A-A & B-B attractive forces. Hence it is an endothermic process.

Examples of liquid Pairs:

1. Water + methanol
2. Water + ethanol
3. Acetone + benzene
4. Carbon tetrachloride + benzene

The graph of vapour pressure and mole fraction for liquids showing positive deviation from Raoult’s Law is shown below:

From the diagram it is clear that the composition curve of such a solution lies above the composition curve obtained on basis of Raoult’s Law

Non ideal solutions showing Negative deviations from Raoult’s law –

When the vapour pressure of the solution is lower than that calculated on the basis of Raoult’s Law, the solution is said to show negative deviation from Raoult’s Law. Even the vapour pressure of individual components show negative deviation from Raoult’s Law

The negative deviation is shown by those liquid pairs in which the A-B molecular interaction forces are stronger than the corresponding A-A or B-B molecular interaction forces. When the A-B molecular interaction forces are stronger, then molecules of liquid A find it easier to escape from pure solution as compared to the mixture thereby decreasing the vapour pressure of solution.

Consequently, each component of solution has a partial vapour pressure less than expected on the basis of Raoult’s law. This is called negative deviations from Raoult’s law, i.e. P_A A ^oX _A & P _B ^oX _B.

Examples of liquid pairs:

1. Water + hydrochloric acid
2. Water + nitric acid
3. Chloroform + benzene
4. Acetone + aniline

∆_mixH is negative because energy is released due to increase in attractive forces. Hence it is an exothermic process.

The graph of vapour pressure and mole fraction for liquids showing negative deviation from Raoult’s Law is shown below:

A 2.15 given:

2% non-volatile solute in an aqueous solution means that if the solution is of mass 100 g then the mass of solute in the solution is,

Mass of solute = [2 × 100]/100

= 2 g

∴ Mass of Solvent = 100-2

= 98 g

Vapour pressure of solution at normal boiling point, P₁ = 1.004 bar

Vapour pressure of pure water at normal boiling point, P₁° = 1 atm = 1.013 bar

To find: Molar mass of solute

Formula:

From Raoult’s Law, we have:

Now we know that number of moles of solute is given by the following relationship,

⇒ Number of moles = Mass of solute / Molecular mass of solute

Using the above relationship the equation [1] can be modified as follows:

Where

W₁ = mass of solvent

W₂ = mass of solute

M₁ = molecular mass of solvent

M₂ = molecular mass of solute

Solution:

(1.013 - 1.004) / 1.013 = 2 X 18 / 98 X M₂

0.009 / 1.013 = 36 / 98M₂

M₂ = 36 X 1.013 / 98 X 0.009

M₂ = 41.3469 g

∴ M₂ ≈ 41.35 g

Therefore the molar mass of the solute is 41.35 g.

A 2.16 Given: Temperature = 373k

Vapour pressure of pure heptane (p₁⁰ ) = 105.2 kpa and that of octane (p₂⁰ ) = 46.8 kpa

Mass of heptane = 26 g

Mass of octane = 35 g

Molecular weight of heptane = C₇H₁₆ = 12 × 7 + 1 × 16 = 100 gmol^-1

Molecular weight of octane = C₈H₁₈ = 114 gmol^-1

Moles of heptane, n₁ = given mass /molecular weight = 26/100

⇒ n₁ = 0.26mol

Moles of octane, n₂ = given mass /molecular weight = 35/114

⇒ n₂ = 0.307mol

∴ Partial pressure of heptane, p₁ = χ₁ × p₁⁰

⇒ p₁ = 0.456 × 105.2 = 47.97kpa

∴ Partial pressure of octane, p₁ = χ₂ × p₂⁰

⇒ p₂ = 0.544 × 46.8 = 25.46 kpa

∴ Total pressure exerted by solution = p₁ + p₂

= 47.97 + 25.46

= 73.43 kpa

A 2.17 Given: 1 molal solution means 1 mole of solute present in 1000g of water solvent)

Molecular weight of water = H₂O = 1 × 2 + 16 = 18g/mol

No. of moles of water, n = given mass /molecular weight

⇒ n = 1000/18 = 55.56 gmol^-1

Mole fraction of solute in solution,x₂ = moles of solute/(moles of solute + moles of water)

⇒ x₂ = 1/(1 + 55.56)

⇒ x₂ = 0.0177

Given vapour pressure of pure water at 300k is 12.3 kpa

Apply the formula:

⇒ P₁ = 12.0823kpa

which is the vapour pressure of the solution.

A 2.18 Given:

Molar mass of non-volatile solute = 40g

Let no. of moles of solute be n.

Mass of octane = 114g

Molar mass of octane(C₈H₁₈) = 12 × 8 + 1 × 18 = 114g/mol

Moles of octane = given mass/molar mass

⇒ n = 114/114 moles

⇒ n = 1 mole

Molar fraction of solute,

x₂ = moles of solute / moles of solute + moles of octane

⇒ x₂ = n/n + 1

Let the vapour pressure of original solvent(without solute) be p₁^⁰

Accordingly after addition of solute vapour pressure of solution reduces to 80% i.e.

0.8 p₁^⁰ = p₁

Applying the formula:

⇒ n/n + 1 = 0.2

⇒ 0.2n + 0.2 = n

⇒ n = 0.25 moles

Hence, mass of solute is:

moles = given mass/molar mass

⇒ 0.25moles = mass/40g

⇒ mass = 10g.

A 2.19 Given: mass of solute = 30g

Let the molar mass of solute be x g and vapour pressure of pure water at 298k be P₁ ^⁰

Mass of water(solvent) = 90g

Molar mass of water = H₂O = 1 × 2 + 16 = 18g

Moles of water = mass of water/molar mass

⇒ n = 90/18 moles

⇒ n = 5moles

Molar fraction of solute,

x₂ = moles of solute / moles of solute + moles of octane

x₂ = (30/x) / (30/x) + 5

x₂ = 30 / 30+5x

Vapour pressure of solution (p₁) = 2.8kpa

Applying the formula:

According to second condition when we add 18g of water to solution vapour pressure becomes 2.9kpa

Moles of water = mass/molar mass

⇒ n = 90 + 18/18

⇒ n = 6moles

Molar fraction of solute,

x₂ = moles of solute / moles of solute + moles of octane

x₂ = (30/x) / (30/x) + 6

x₂ = 30 / 30+6x

Applying the formula:

Solving 1 and 2:

Dividing (2) by (1) we get

Substituting value of x in 1 we get

(i) molar mass of the solute = 78g

(ii) vapour pressure of water at 298 K = 537kpa

A 2.20 5% solution means 5g of cane sugar is present in 100g of solution

Freezing point of solution = 271k

Freezing point of pure water = 273.15k

Molar mass of cane sugar(C₁₂H₂₂O₁₁) = 12 × 12 + 1 × 22 + 16 × 11 = 342g

Moles of cane sugar = mass/molar mass = 5/342

⇒ n = 0.0146mol

Molality of solution = moles of solute/mass of solvent(in kg)

⇒ M = 0.0146/0.095

⇒ Molality = 0.154M

Depression in freezing point = ΔT_f = 273.15-271 = 2.15k

Applying the formula: ΔT_f = K_f × M

Where

ΔT_f = depression in freezing point

K_f = molal depression constant

M = molality of solution

⇒ K_f = 2.15/0.154

⇒ K_f = 13.96k kg mol^-1

Second condition: mass of glucose = 5g

Molar mass of glucose(C₆H₁₂O₆) = 12 × 6 + 1 × 12 + 16 × 6 = 180g

Moles of glucose = mass/molar mass

⇒ n = 5/180moles

⇒ n = 0.0278mol

Molality of solution = moles of solute/mass of solvent(in kg)

⇒ molality = 0.0278/0.095

⇒ M = 0.2926M

Applying the formula: ΔT_f = K_f × M

Where

ΔT_f = depression in freezing point

K_f = molal depression constant

M = molality of solution

⇒ ΔT_f = 13.96 × 0.2926

⇒ ΔT_f = 4.08k

Freezing point of solution = 273.15 + 4.08 = 277.234k

A 2.21 let the molar masses of AB₂ and AB₄ be x and y respectively.

Molar mass of benzene(C₆H₆) = 12 × 6 + 1 × 6 = 78 g/mol

Moles of benzene = mass/molar mass = 20/78

n = 0.256mol

⇒ ΔT_f = 2.3 K

K_f = 5.1K kg mol^-1

For AB₂

Applying the formula: ΔT_f = K_f × M

Where

ΔT_f = depression in freezing point

K_f = molal depression constant

M = molality of solution

⇒ 2.3 = 5.1 × M₁

⇒ M₁ = 0.451mol/kg

For AB₄

Applying the formula: ΔT_f = K_f × M

ΔT_f = depression in freezing point

K_f = molal depression constant

M = molality of solution

⇒ 1.3 = 5.1 × M₂

⇒ M₂ = 0.255mol/kg

M₁ = moles of solute/mass of solvent(in kg)

M₁ = 1/x / 0.02 = 1 / 0.02x = 0.451

⇒ X = 110.86g

M₂ = moles of solute/mass of solvent(in kg)

M₂ = 1/y / 0.02 = 1 / 0.02y = 0.255

⇒ y = 196.1g

atomic mass of A be a and that of B be b g respectively.

So, AB₂ : a + 2b = 110.86

AB_4: a + 4b = 196.1

⇒ a = 25.59g

⇒ b = 42.64g

A 2.22 Here,

T = 300 K

π = 1.52 bar

R = 0.083 bar L

Applying the relation, π = CRT

where

π = osmotic pressure of solution

C = concentration of solution

R = universal gas constant

T = temperature

⇒C  = π / RT = 1.52 / 0.083 X 300

⇒ C = 0.061mol/L

Concentration of the solution is 0.061mol/L

• A 2.23 (i) Both the compounds are non-polar and they do not attract each other because they do not form any polar ions. Vanderwaals forces of attraction will be dominant in between them as vanderwaals forces of attraction are not a result of any chemical or electronic bond.
• (ii) now here both the compounds are non-polar because in I₂ both the atoms are same so they have same electronegativity and hence there will be no displacement of electron cloud, it will be in the centre. In case of CCl₄ molecule, it has tetrahedral shape so two Cl atoms will cancel the attraction effect from two opposite Cl atoms, hence molecule as a whole is non polar. Therefore they will also have vanderwaals forces of attraction.
• (iii) NaClO₄ is ionic in nature as Na, Cl and O all have different electronegativity and their shape is also not symmetric. So molecular will be ionic in nature and we know that water is polar because O will attract the electron cloud towards it(more electronegative) hence there will be formation of dipole(two oppositely charged ions separated by a short distance). Therefore there will be ion-dipole interaction between them.
• (iv) methanol and acetone are both polar molecules because of the presence of electron withdrawing O atom in methanol and ketone group in acetone. So they will have dipole-dipole interaction.
• (v) acetonitrile is polar compound due to presence of electronegative N atom and acetone due to ketone group. So, dipole-dipole interaction.

A 2.24 Now n-octane is non-polar solvent due to long chain saturated structure. We know that “like dissolves like” so a non-polar compound will be more soluble in non-polar solvent as compared to polar compound.

So cyclohexane is non-polar due to symmetric structure. KCl is ionic in nature as it will dissociate into K ⁺ and Cl^- ions. CH₃CN is polar as mentioned above and CH₃OH is also polar in nature.

The order of increasing polarity is:

Cyclohexane < CH₃CN < CH₃OH < KCl (O is more electronegative than N)

Therefore, the order of increasing solubility is:

KCl < CH₃OH < CH₃CN < Cyclohexane

1. A 2.25 Water is a polar compound (due to electronegativity difference between O and H) . We know that “like dissolves like”. So, a non-polar compound will be more soluble in non-polar solvent as compared to polar compound.
2. Phenol has the polar group -OH and non-polar group –C₆H₅ and it can not form H bonding with water(presence of bulky non-polar group) . Thus, phenol is partially soluble in water
3. Toluene has no polar Thus, toluene is insoluble in water.
4. Formic acid (HCOOH) has the polar group -OH and can form H-bond with water. Thus, formic acid is highly soluble in water
5. Ethylene glycol(OH-CH₂-CH_2-OH) has polar -OH group and can form H-bond with Thus, it is highly soluble in water.
6. Chloroform is partly soluble as CHCl₃ is polar in nature due to high electronegativity of Cl atoms, there will be production of partial + charge on H atom so it can form H bonding with water but it is also surrounded by 3 Cl atoms, so partly soluble
7. Pentanol(C₅H₁₁OH) has polar -OH group, but it also contains a very bulky non- polar group –C₅H₁₁. Thus, pentanol is partially soluble in water

A 2.26 Mass of ions = 92g

Molar mass of ions = Na⁺ = 23g(neglect the mass lost due to absence of a electron)

Moles of ions = mass of ions/molar mass

⇒ n = 92/23 moles

⇒ n = 4moles

Molality of solution = moles of solute/mass of solvent(in kg) Molality = 4/1 = 4M

A 2.27 The Solubility product of CuS (ksp) = 6 × 10^-16

CuS → Cu ⁺⁺ + S^2-

Let the s be solubility of CuS in mol/L

Ksp = [ Cu ⁺⁺ ][S²]

Ksp = solubility product

6 × 10^-16 = s × s = s²

⇒ S = 2.45 × 10^-8 mol/L

Hence, the maximum molarity of CuS in an aqueous solution is 2.45 × 10^-8 mol/L

A 2.28 Total mass of solution = 6.5g + 450g = 456.5g

Therefore mass percentage of aspirin in solution = (mass of aspirine/total mass of solution) = 6.5/456.5

⇒ mass % of aspirine = 1.424%

A 2.29 molar mass of Nalorphene = 311g/mol

Now 1000g of solution contains 1.5 × 10^-3 moles of Nalorphene(Molality of solution = moles of solute/mass of solvent (in kg)

⇒ 1.5 × 10^-3 moles of Nalorphene = 1.5 × 10^-3 × 311 = 0.4665g of Nalorphene

Therefore, total mass of the solution = (1000 + 0.4665) g

⇒ total mass = 1000.4665 g

This implies that the mass of the solution containing 0.4665 g of nalorphene is 1000.4665 g.

Therefore, mass of the solution containing 1.5 mg of nalorphene is:

Mass = 1000.4665 X 1.5 X 10^-3 / 4.665 g

⇒ mass of solution containing required ions = 3.22 g

Hence, the mass of aqueous solution required is 3.22 g.

A 2.30 Volume of the solution = 250mL = 0.25L

Let the no. of moles of solute be n

Molarity = No. of moles of solute/volume of solution

⇒ 0.15 = n/0.25

⇒ n = 0.0375moles

Molar mass of C₆H₅OH = 6×12 + 5×1 + 16 + 1 = 94g

Moles = mass/molar mass

⇒ 0.0375 = m/94

Mass of benzoic acid required = 3.525g.

A 2.31 The depression in freezing point of water observed for the same amount of acetic acid, trichloroacetic acid and trifluoroacetic acid increases in the pattern,

Acetic acid< trichloroacetic acid< trifluoroacetic acid.

This is because fluorine is more electronegative than chlorine. So, trifluoracetic acid is a stronger acid in comparison to trichloroacetic acid and acetic acid. And also, acetic acid is the weakest of all.

Explanation: Stronger acid produces more number of ions, therefore it has more ΔT_f(depression in freezing point), hence lower freezing point. As the acidic strength increases, the acid gets more and more ionised.

Trifluoracetic acid ionizes to the largest extent. Hence, in this case, trifluoracetic acid being the strongest acid produces more number of ions(extent of ionisation and concentration of ions are more), high ΔT_f(depression in freezing point)and lower freezing point and vice versa.

A 2.32 Mass of CH₃CH₂CHClCOOH = 10 g

Mass of water = 250g

K_a = 1.4 × 10^–3,

K_f = 1.86 K kg mol^–1

Molar mass of CH₃CH₂CHClCOOH = 12 + 3 + 12 + 2 + 12 + 1 + 35.5 + 2 + 16 + 16 + 1

= 122.5 g mol^–1

Number of moles of solute = Mass of Solute / Molar Mass

→ No. of moles = 10g / 122.5 g/mol

∴ No. of moles = 8.6 X 10^–2 mol

Now, Molality is given as,

M = Number of moles of solute / kg of solvent

M= 8.6 X 10^–2 X 1000 g/mol / 250 g

M = 0.3264 kg/mol

CH₃CH₂CHClCOOH = CH₃CH₂CHClCOO^- + H ⁺

Total moles at equilibrium = (1-α) + 2 α

= 1 + α

In order to find out the depression in freezing point, 

values of i(vant Hoff’s factor) and α(degree of dissociation) are to be found out.

To find out degree of dissociation, _α

∴ i = 1.0654

Now, to find out the depression in freezing point,

A 2.33 Given- w₁ = 500g

W₂ = 19.5g

K_f = 1.86 K kg mol^-1

Molar mass of CH₂FCOOH = 12 + 2 + 19 + 12 + 16 + 16 + 1

= 78 g mol^-1

The depression in freezing point is calculated by,

→ (where, m is the molality)

= 1.86 X 19.5 / 78 X 1000/500

= 1.86 X 19.5 / 78 X 2

=0.93

∴ Δt_f (calculated) = 0.93

To find out the vant Hoff’s factor, we use the formula,

i = observed Δt_f / calculated Δt_f

_{i = 1.0 (given) / 0.93}

∴ i= 1.07

CH₂FCOOH → CH₂FCOO^- + H ⁺

To find out the degree of dissociation α, we use

Thus, the vant Hoff’s factor is 1.07 an the dissociation constant is 2.634x10^-3

A 2.34 Given- Vapour pressure of water,

P_A⁰  = 17.535 mm Hg  

W_B= 25 g of glucose

W_A = 450g of water

Molar mass of water, H₂O = 1 + 1 + 16 = 18 g mol^-1

Molar mass of glucose, C₆H₁₂O₆ = (12×6) + (1×12) + (16×6) = 180 g mol^-1

Using Raoult’s law for solution of non-volatile solute,

P_A⁰ - P_A / P_A⁰ = x_B→ Equation 1

where x_B is the mole fraction of the solute

x_B = W_B/M_B X M_B/W_B

=25/180 X 18 / 450

x_B = 1/180

Substituting the value of x_B in equation 1, we get,

Thus, the vapour pressure of water at 293 K at the given conditions is 17.437 mm Hg

A 2.35 Given-

Henry’s law constant K_H = 4.27X 10⁵ mm Hg,

p = 760mm Hg,

Using Henry’s law,

Using the formula of lowering vapour pressure,

Thus, the solubility of methane in benzene is 0.023 moles

A 2.36 Given-

Mass of liquid A, W_A = 100g, Molar mass, M_A = 140 g mol^-1

Mass of liquid B, W_B = 1000 g, Molar mass, M_B = 180 g mol^-1

Using the formula below calculate the no. of moles in liquid A and B.

Number of moles = Mass / Molar Mass

Number of moles of liquid A, M_A = 100/140 = 0.714 mol^-1

Number of moles of liquid B, M_B = 1000/ 180 = 5.556 mol^-1

Using the formula,

mole fraction of a liquid = No. of moles of the liquid / total no of moles

we calculate the mole fraction of liquids A and B.

→ Mole fraction of A,

x_A = 0.714 / (0.714 + 5.556)

∴ x_A = 0.114

→ Mole fraction of B,

x_B = 1- x_{A = 1 - 0.114}

∴ x_B = 0.886

Vapour pressure of pure liquid B, P^o = 500 torr (given)

According to Henry`s law,

Having given the total vapour pressure of the solution, P_total = 475 torr,

Thus, the vapour pressure of pure liquid A = 280.7 torr and

vapour pressure of liquid A in the solution = 32 torr

A 2.37 The P_total for the values given in the graph is found out and plotted in the graph.

It can be observed from the graph that the plot for the p total of the solution curves downwards. Therefore, the solution shows negative deviation from the ideal behaviour.

A 2.38 Given-

Thus, mole fraction of benzene in vapour phase is 0.6744

A 2.39 Given-

K_H for O₂ = 3.30 × 10⁷ mm Hg,

K_H for N₂ = 6.51 × 10⁷ mm Hg

Percentage of oxygen (O₂) = 20 %

Percentage of nitrogen (N₂) = 79%

Total pressure = 10 atm

Using Henry’s law,

where, p is the partial pressure of gas in the solution and K_H is Henry’s constant.

Thus, the mole fraction of oxygen in solution, x_oxy = 4.61x10^-5

and the mole fraction of nitrogen in solution, x_nit is 9.22x10^-5

A 2.40 Given-

Vant Hoff’s factor, i = 2.47

osmotic pressure, π = 0.75 atm

Volume of solution = 2.5L.

To determine the amount of CaCl₂, we use vant Hoff’s equation for dilute solutions, given as,

πV = inRT

where, n is the number of moles of solute, R is solution constant which is equal to the gas constant and T is the absolute temperature.

Hence, the amount of CaCl₂ dissolved is 3.425g

A 2.41 Given-

Mass of K₂SO₄, w = 25 mg = 25 X 10^-3 g,

Molar mass of K₂SO₄ = (39×2) + (32×1) + (16×4) = 174 g mol^-1

Volume V = 2 liter

T = 25⁰C + 273 = 298 K (add 273 to convert in Kelvin)

The reaction of dissociation of K₂SO₄ is written as,

K₂SO₄ → 2K ⁺ + SO₄^2-

Number if ions produced = 2 + 1 = 3, hence vant Hoff’s factor, i = 3

Here, we use vant Hoff’s equation for dilute solutions, given as,

^{πV = inRT}

where, n is the number of moles of solute, R is solution constant which is equal to the gas constant(0.082) and T is the absolute temperature (298 K).

Hence, the osmotic pressure of a solution is 5.27x10^-3atm