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Chapter 1 Solutions
Types Of Solutions
Solutions are homogeneous mixtures where components have uniform composition and properties throughout. A solution consists of a solvent (present in largest quantity, determining the physical state) and one or more solutes (present in smaller quantities). Solutions can exist in gaseous, liquid, or solid states, with the solute and solvent being in any of these states. Examples include gas mixtures (oxygen and nitrogen), gases dissolved in liquids (oxygen in water), solids dissolved in liquids (glucose in water), and solids dissolved in solids (copper in gold).
Concentration, the quantitative description of a solution's composition, can be expressed in several ways:
- Mass Percentage (w/w): Mass of component / Total mass of solution × 100.
- Volume Percentage (V/V): Volume of component / Total volume of solution × 100.
- Mass by Volume Percentage (w/V): Mass of solute / Volume of solution (in mL) × 100.
- Parts Per Million (ppm): Number of parts of component / Total parts of all components × 10⁶. Useful for trace amounts.
- Mole Fraction (x): Moles of component / Total moles of all components. The sum of mole fractions for all components in a solution is always 1.
- Molarity (M): Moles of solute / Volume of solution (in Litres). It is temperature-dependent.
- Molality (m): Moles of solute / Mass of solvent (in kg). It is independent of temperature.
Expressing Concentration Of Solutions
Concentration is expressed quantitatively to describe the composition of solutions:
- Mass Percentage (w/w): (Mass of component / Total mass of solution) × 100.
- Volume Percentage (V/V): (Volume of component / Total volume of solution) × 100.
- Mass by Volume Percentage (w/V): Mass of solute (g) / Volume of solution (mL) × 100. Commonly used in medicine.
- Parts Per Million (ppm): (Parts of component / Total parts of all components) × 10⁶. Useful for expressing concentrations of trace substances.
- Mole Fraction (x): (Moles of component / Total moles of all components). The sum of mole fractions of all components in a solution is 1.
- Molarity (M): Moles of solute / Volume of solution in litres. It is temperature-dependent as volume changes with temperature.
- Molality (m): Moles of solute / Mass of solvent in kg. It is independent of temperature as mass does not change with temperature.
Solubility
Solubility is the maximum amount of a substance (solute) that can dissolve in a specified amount of solvent at a given temperature and pressure to form a saturated solution. It depends on the nature of the solute and solvent ("like dissolves like": polar solutes dissolve in polar solvents, nonpolar solutes in nonpolar solvents), temperature, and pressure.
Solubility Of A Solid In A Liquid
The solubility of solids in liquids is primarily influenced by temperature. Generally, if the dissolution process is endothermic ($Δ_{sol}H > 0$), solubility increases with temperature. If it's exothermic ($Δ_{sol}H < 0$), solubility decreases with increasing temperature. Pressure has a negligible effect on the solubility of solids in liquids due to their low compressibility.
Solubility Of A Gas In A Liquid
The solubility of gases in liquids is significantly affected by both temperature and pressure.
- Effect of Pressure: According to Henry's Law, the solubility of a gas in a liquid at a constant temperature is directly proportional to the partial pressure of the gas above the liquid. Mathematically, $p = K_H x$, where $p$ is the partial pressure, $x$ is the mole fraction (measure of solubility), and $K_H$ is the Henry's law constant. Higher $K_H$ means lower solubility. Solubility of gases increases with increasing pressure. Applications include carbonating beverages and the risks faced by scuba divers (bends) due to increased gas solubility at high pressure.
- Effect of Temperature: The solubility of gases in liquids generally decreases with an increase in temperature, as the dissolution process is typically exothermic. This is why aquatic organisms are more comfortable in cold waters.
Vapour Pressure Of Liquid Solutions
When a non-volatile solute is added to a volatile solvent, the vapour pressure of the solution decreases. This reduction is because the solute molecules occupy some of the surface area, decreasing the rate at which solvent molecules escape into the vapour phase.
Vapour Pressure Of Liquid-Liquid Solutions
In solutions of two volatile liquids, both components contribute to the vapour pressure. Raoult's Law states that the partial vapour pressure of each volatile component in a solution is directly proportional to its mole fraction in the solution ($p_i = x_i p^0_i$, where $p^0_i$ is the vapour pressure of the pure component). The total vapour pressure is the sum of the partial vapour pressures (Dalton's Law). Vapour phase composition depends on the partial pressures; the vapour phase is richer in the more volatile component.
Raoult’s Law As A Special Case Of Henry’s Law
Raoult's Law can be seen as a special case of Henry's Law where the Henry's law constant ($K_H$) is equal to the vapour pressure of the pure volatile component ($p^0_i$). Both laws describe the relationship between partial pressure and mole fraction but apply to different scenarios (Raoult's for volatile solutes/solvents, Henry's for gases in liquids).
Vapour Pressure Of Solutions Of Solids In Liquids
When a non-volatile solute is dissolved in a volatile solvent, the solvent's vapour pressure decreases. Raoult's Law for such solutions states that the partial vapour pressure of the solvent ($p_1$) is equal to its mole fraction ($x_1$) multiplied by its vapour pressure in the pure state ($p^0_1$), i.e., $p_1 = x_1 p^0_1$. The relative lowering of vapour pressure ($ (p^0_1 - p_1) / p^0_1 $) is equal to the mole fraction of the solute ($x_2$).
Ideal And Non-Ideal Solutions
Liquid-liquid solutions can be classified based on their adherence to Raoult's Law:
Ideal Solutions
Ideal solutions obey Raoult's Law over the entire range of concentrations. They exhibit zero enthalpy of mixing ($Δ_{mix}H = 0$) and zero volume of mixing ($Δ_{mix}V = 0$), meaning no heat is absorbed or released, and volumes are additive upon mixing. This behaviour occurs when intermolecular forces between solute-solute (A-A) and solvent-solvent (B-B) molecules are similar to those between solute-solvent molecules (A-B). Examples include mixtures of benzene and toluene, or n-hexane and n-heptane.
Non-Ideal Solutions
Non-ideal solutions deviate from Raoult's Law. They exhibit either positive or negative deviations:
- Positive Deviation: Vapour pressure is higher than predicted by Raoult's Law. This occurs when solute-solvent interactions (A-B) are weaker than solute-solute (A-A) or solvent-solvent (B-B) interactions, making molecules escape more easily. Examples include mixtures of ethanol-water or chloroform-acetone. These solutions often form minimum boiling azeotropes.
- Negative Deviation: Vapour pressure is lower than predicted. This happens when solute-solvent interactions (A-B) are stronger than A-A or B-B interactions, reducing the tendency of molecules to escape. Examples include mixtures of phenol-aniline or acetone-chloroform. These solutions can form maximum boiling azeotropes.
Azeotropes: Binary mixtures with the same composition in liquid and vapour phases that boil at a constant temperature. They cannot be separated by fractional distillation. Ethanol-water forms a minimum boiling azeotrope (~95% ethanol), while nitric acid-water forms a maximum boiling azeotrope.
Colligative Properties And Determination Of Molar Mass
Colligative properties depend on the number of solute particles in a solution, not their chemical identity. They are observed when a non-volatile solute is added to a volatile solvent. These properties are connected to the lowering of vapour pressure:
Relative Lowering Of Vapour Pressure
The relative lowering of vapour pressure ($ (p^0_1 - p_1) / p^0_1 $) is equal to the mole fraction of the solute ($x_2$). For dilute solutions, this is approximately equal to the ratio of moles of solute to moles of solvent ($ n_2 / n_1 $). This property can be used to determine the molar mass of the solute ($M_2$) using the formula: $ (p^0_1 - p_1) / p^0_1 = (w_2 \times M_1) / (M_2 \times w_1) $.
Elevation Of Boiling Point
The boiling point of a solvent increases when a non-volatile solute is added because the vapour pressure is lowered. The elevation in boiling point ($ΔT_b$) is directly proportional to the molality ($m$) of the solute: $ΔT_b = K_b m$, where $K_b$ is the molal elevation constant. This allows calculation of the solute's molar mass ($M_2$).
Depression Of Freezing Point
The freezing point of a solvent decreases when a non-volatile solute is added. The depression in freezing point ($ΔT_f$) is directly proportional to the molality ($m$) of the solute: $ΔT_f = K_f m$, where $K_f$ is the molal freezing point depression constant. This property is also used to determine the molar mass of the solute.
Osmosis And Osmotic Pressure
Osmosis is the spontaneous net movement of solvent molecules from a region of higher solvent concentration (lower solute concentration) to a region of lower solvent concentration (higher solute concentration) through a semipermeable membrane (SPM). Osmotic pressure (P) is the excess pressure that must be applied to the solution to prevent this inward flow of solvent. It's a colligative property and can be calculated using the formula $P = iCRT$ or $PV = i n_2 R T$, where $i$ is the van't Hoff factor, $C$ is molarity, $R$ is the gas constant, $T$ is temperature, $n_2$ is moles of solute, and $V$ is the volume of solution. Osmotic pressure is particularly useful for determining the molar masses of macromolecules.
Isotonic solutions have the same osmotic pressure. Hypertonic solutions have higher osmotic pressure, causing solvent to leave cells (shrinking), while hypotonic solutions have lower osmotic pressure, causing solvent to enter cells (swelling).
Reverse Osmosis And Water Purification
Reverse Osmosis (RO) occurs when pressure exceeding the osmotic pressure is applied to the solution side, forcing the solvent to move from the solution to the pure solvent across an SPM. This process is used for desalination of seawater and water purification, utilizing semipermeable membranes like cellulose acetate.
Abnormal Molar Masses
Colligative properties are used to determine molar masses. However, for electrolytes that dissociate in solution (e.g., NaCl, KCl) or solutes that associate (e.g., acetic acid in benzene), the observed colligative properties deviate from values calculated assuming no change in the number of particles. This leads to abnormally high (due to dissociation) or low (due to association) molar masses.
The van't Hoff factor (i) quantifies this deviation: $i = (\text{Observed colligative property}) / (\text{Calculated colligative property})$ or $i = (\text{Total moles of particles after dissociation/association}) / (\text{Number of moles of particles before dissociation/association})$.
- For non-electrolytes or non-associating/dissociating solutes, $i = 1$.
- For electrolytes that completely dissociate into $n$ ions, $i = n$ (e.g., $i \approx 2$ for NaCl, $i \approx 3$ for $K_2SO_4$).
- For solutes that associate into $n$ particles, $i = 1/n$.
The colligative property equations are modified by multiplying with the van't Hoff factor ($i$) to account for these phenomena.
Intext Questions
Question 1.1. Calculate the mass percentage of benzene ($C_6H_6$) and carbon tetrachloride ($CCl_4$) if 22 g of benzene is dissolved in 122 g of carbon tetrachloride.
Answer:
Question 1.2. Calculate the mole fraction of benzene in solution containing 30% by mass in carbon tetrachloride.
Answer:
Question 1.3. Calculate the molarity of each of the following solutions: (a) 30 g of $Co(NO_3)_2 \cdot 6H_2O$ in 4.3 L of solution (b) 30 mL of 0.5 M $H_2SO_4$ diluted to 500 mL.
Answer:
Question 1.4. Calculate the mass of urea ($NH_2CONH_2$) required in making 2.5 kg of 0.25 molal aqueous solution.
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Question 1.5. Calculate (a) molality (b) molarity and (c) mole fraction of KI if the density of 20% (mass/mass) aqueous KI is 1.202 g $mL^{-1}$.
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Question 1.6. $H_2S$, a toxic gas with rotten egg like smell, is used for the qualitative analysis. If the solubility of $H_2S$ in water at STP is 0.195 m, calculate Henry’s law constant.
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Question 1.7. Henry’s law constant for $CO_2$ in water is $1.67 \times 10^8$ Pa at 298 K. Calculate the quantity of $CO_2$ in 500 mL of soda water when packed under 2.5 atm $CO_2$ pressure at 298 K.
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Question 1.8. The vapour pressure of pure liquids A and B are 450 and 700 mm Hg respectively, at 350 K . Find out the composition of the liquid mixture if total vapour pressure is 600 mm Hg. Also find the composition of the vapour phase.
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Question 1.9. Vapour pressure of pure water at 298 K is 23.8 mm Hg. 50 g of urea ($NH_2CONH_2$) is dissolved in 850 g of water. Calculate the vapour pressure of water for this solution and its relative lowering.
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Question 1.10. Boiling point of water at 750 mm Hg is 99.63°C. How much sucrose is to be added to 500 g of water such that it boils at 100°C.
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Question 1.11. Calculate the mass of ascorbic acid (Vitamin C, $C_6H_8O_6$) to be dissolved in 75 g of acetic acid to lower its melting point by 1.5°C. $K_f$ = 3.9 K kg $mol^{-1}$.
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Question 1.12. Calculate the osmotic pressure in pascals exerted by a solution prepared by dissolving 1.0 g of polymer of molar mass 185,000 in 450 mL of water at 37°C.
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Exercises
Question 1.1. Define the term solution. How many types of solutions are formed? Write briefly about each type with an example.
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Question 1.2. Give an example of a solid solution in which the solute is a gas.
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Question 1.3. Define the following terms:
(i) Mole fraction
(ii) Molality
(iii) Molarity
(iv) Mass percentage.
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Question 1.4. Concentrated nitric acid used in laboratory work is 68% nitric acid by mass in aqueous solution. What should be the molarity of such a sample of the acid if the density of the solution is 1.504 g mL$^{–1}$?
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Question 1.5. A solution of glucose in water is labelled as 10% w/w, what would be the molality and mole fraction of each component in the solution? If the density of solution is 1.2 g mL$^{–1}$, then what shall be the molarity of the solution?
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Question 1.6. How many mL of 0.1 M HCl are required to react completely with 1 g mixture of $Na_2CO_3$ and $NaHCO_3$ containing equimolar amounts of both?
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Question 1.7. A solution is obtained by mixing 300 g of 25% solution and 400 g of 40% solution by mass. Calculate the mass percentage of the resulting solution.
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Question 1.8. An antifreeze solution is prepared from 222.6 g of ethylene glycol ($C_2H_6O_2$) and 200 g of water. Calculate the molality of the solution. If the density of the solution is 1.072 g mL$^{–1}$, then what shall be the molarity of the solution?
Answer:
Question 1.9. A sample of drinking water was found to be severely contaminated with chloroform ($CHCl_3$) supposed to be a carcinogen. The level of contamination was 15 ppm (by mass):
(i) express this in percent by mass
(ii) determine the molality of chloroform in the water sample.
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Question 1.10. What role does the molecular interaction play in a solution of alcohol and water?
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Question 1.11. Why do gases always tend to be less soluble in liquids as the temperature is raised?
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Question 1.12. State Henry’s law and mention some important applications.
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Question 1.13. The partial pressure of ethane over a solution containing $6.56 \times 10^{–3}$ g of ethane is 1 bar. If the solution contains $5.00 \times 10^{–2}$ g of ethane, then what shall be the partial pressure of the gas?
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Question 1.14. What is meant by positive and negative deviations from Raoult's law and how is the sign of $\Delta_{mix}H$ related to positive and negative deviations from Raoult's law?
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Question 1.15. An aqueous solution of 2% non-volatile solute exerts a pressure of 1.004 bar at the normal boiling point of the solvent. What is the molar mass of the solute?
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Question 1.16. Heptane and octane form an ideal solution. At 373 K, the vapour pressures of the two liquid components are 105.2 kPa and 46.8 kPa respectively. What will be the vapour pressure of a mixture of 26.0 g of heptane and 35 g of octane?
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Question 1.17. The vapour pressure of water is 12.3 kPa at 300 K. Calculate vapour pressure of 1 molal solution of a non-volatile solute in it.
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Question 1.18. Calculate the mass of a non-volatile solute (molar mass 40 g $mol^{–1}$) which should be dissolved in 114 g octane to reduce its vapour pressure to 80%.
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Question 1.19. A solution containing 30 g of non-volatile solute exactly in 90 g of water has a vapour pressure of 2.8 kPa at 298 K. Further, 18 g of water is then added to the solution and the new vapour pressure becomes 2.9 kPa at 298 K. Calculate:
(i) molar mass of the solute
(ii) vapour pressure of water at 298 K.
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Question 1.20. A 5% solution (by mass) of cane sugar in water has freezing point of 271K. Calculate the freezing point of 5% glucose in water if freezing point of pure water is 273.15 K.
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Question 1.21. Two elements A and B form compounds having formula $AB_2$ and $AB_4$. When dissolved in 20 g of benzene ($C_6H_6$), 1 g of $AB_2$ lowers the freezing point by 2.3 K whereas 1.0 g of $AB_4$ lowers it by 1.3 K. The molar depression constant for benzene is 5.1 K kg $mol^{–1}$. Calculate atomic masses of A and B.
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Question 1.22. At 300 K, 36 g of glucose present in a litre of its solution has an osmotic pressure of 4.98 bar. If the osmotic pressure of the solution is 1.52 bars at the same temperature, what would be its concentration?
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Question 1.23. Suggest the most important type of intermolecular attractive interaction in the following pairs.
(i) n-hexane and n-octane
(ii) $I_2$ and $CCl_4$
(iii) $NaClO_4$ and water
(iv) methanol and acetone
(v) acetonitrile ($CH_3CN$) and acetone ($C_3H_6O$).
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Question 1.24. Based on solute-solvent interactions, arrange the following in order of increasing solubility in n-octane and explain. Cyclohexane, KCl, $CH_3OH$, $CH_3CN$.
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Question 1.25. Amongst the following compounds, identify which are insoluble, partially soluble and highly soluble in water?
(i) phenol
(ii) toluene
(iii) formic acid
(iv) ethylene glycol
(v) chloroform
(vi) pentanol.
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Question 1.26. If the density of some lake water is 1.25g mL$^{–1}$ and contains 92 g of $Na^+$ ions per kg of water, calculate the molarity of $Na^+$ ions in the lake.
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Question 1.27. If the solubility product of CuS is $6 \times 10^{–16}$, calculate the maximum molarity of CuS in aqueous solution.
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Question 1.28. Calculate the mass percentage of aspirin ($C_9H_8O_4$) in acetonitrile ($CH_3CN$) when 6.5 g of $C_9H_8O_4$ is dissolved in 450 g of $CH_3CN$.
Answer:
Question 1.29. Nalorphene ($C_{19}H_{21}NO_3$), similar to morphine, is used to combat withdrawal symptoms in narcotic users. Dose of nalorphene generally given is 1.5 mg. Calculate the mass of $1.5 \times 10^{–3}$ m aqueous solution required for the above dose.
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Question 1.30. Calculate the amount of benzoic acid ($C_6H_5COOH$) required for preparing 250 mL of 0.15 M solution in methanol.
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Question 1.31. The depression in freezing point of water observed for the same amount of acetic acid, trichloroacetic acid and trifluoroacetic acid increases in the order given above. Explain briefly.
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Question 1.32. Calculate the depression in the freezing point of water when 10 g of $CH_3CH_2CHClCOOH$ is added to 250 g of water. $K_a = 1.4 \times 10^{–3}$, $K_f = 1.86$ K kg $mol^{–1}$.
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Question 1.33. 19.5 g of $CH_2FCOOH$ is dissolved in 500 g of water. The depression in the freezing point of water observed is $1.0^\circ C$. Calculate the van’t Hoff factor and dissociation constant of fluoroacetic acid.
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Question 1.34. Vapour pressure of water at 293 K is 17.535 mm Hg. Calculate the vapour pressure of water at 293 K when 25 g of glucose is dissolved in 450 g of water.
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Question 1.35. Henry’s law constant for the molality of methane in benzene at 298 K is $4.27 \times 10^5$ mm Hg. Calculate the solubility of methane in benzene at 298 K under 760 mm Hg.
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Question 1.36. 100 g of liquid A (molar mass 140 g $mol^{–1}$) was dissolved in 1000 g of liquid B (molar mass 180 g $mol^{–1}$). The vapour pressure of pure liquid B was found to be 500 torr. Calculate the vapour pressure of pure liquid A and its vapour pressure in the solution if the total vapour pressure of the solution is 475 Torr.
Answer:
Question 1.37. Vapour pressures of pure acetone and chloroform at 328 K are 741.8 mm Hg and 632.8 mm Hg respectively. Assuming that they form ideal solution over the entire range of composition, plot $p_{total}$, $p_{chloroform}$, and $p_{acetone}$ as a function of $x_{acetone}$. The experimental data observed for different compositions of mixture is:
| $100 \times x_{acetone}$ | 0 | 11.8 | 23.4 | 36.0 | 50.8 | 58.2 | 64.5 | 72.1 |
|---|---|---|---|---|---|---|---|---|
| $p_{acetone}$ /mm Hg | 0 | 54.9 | 110.1 | 202.4 | 322.7 | 405.9 | 454.1 | 521.1 |
| $p_{chloroform}$ /mm Hg | 632.8 | 548.1 | 469.4 | 359.7 | 257.7 | 193.6 | 161.2 | 120.7 |
Plot this data also on the same graph paper. Indicate whether it has positive deviation or negative deviation from the ideal solution.
Answer:
Question 1.38. Benzene and toluene form ideal solution over the entire range of composition. The vapour pressure of pure benzene and toluene at 300 K are 50.71 mm Hg and 32.06 mm Hg respectively. Calculate the mole fraction of benzene in vapour phase if 80 g of benzene is mixed with 100 g of toluene.
Answer:
Question 1.39. The air is a mixture of a number of gases. The major components are oxygen and nitrogen with approximate proportion of 20% is to 79% by volume at 298 K. The water is in equilibrium with air at a pressure of 10 atm. At 298 K if the Henry’s law constants for oxygen and nitrogen at 298 K are $3.30 \times 10^7$ mm and $6.51 \times 10^7$ mm respectively, calculate the composition of these gases in water.
Answer:
Question 1.40. Determine the amount of $CaCl_2$ (i = 2.47) dissolved in 2.5 litre of water such that its osmotic pressure is 0.75 atm at $27^\circ$ C.
Answer:
Question 1.41. Determine the osmotic pressure of a solution prepared by dissolving 25 mg of $K_2SO_4$ in 2 litre of water at $25^\circ$ C, assuming that it is completely dissociated.
Answer: