5: Colligative Properties

The van’t Hoff factor \(i\) compares the actual number of particles in solution with the number expected from the formula units dissolved. A non-dissociating solute does not split into ions, and no association is stated either. Thus each dissolved formula unit remains one effective particle, so the initial value is \(i=1\). Volatility does not change this particle-count factor.

For a solution with a non-volatile solute, Raoult’s law states that the partial vapour pressure of the solvent is \(p=x_1p_1^0\). Substituting the given values gives \(p=0.75\times80=60\) units. The pure-solvent pressure is not used unchanged because the solute lowers the solvent’s mole fraction and therefore lowers its vapour pressure.

Depression in freezing point is a colligative property. For dilute solutions it is expressed as \(\Delta T_f=iK_fm\), so it depends on the effective number of dissolved particles, represented by \(i\), rather than on the chemical identity of those particles. Surface tension, viscosity, and refractive index generally depend on the nature and interactions of the substances as well.

Molality is defined as moles of solute divided by the mass of solvent in kilograms: \(m=n_{solute}/mass_{solvent}\). Convert 600 grams to 0.600 kilograms, then calculate \(m=0.3/0.600=0.5\,\mathrm{mol\,kg^{-1}}\). The mass of the solution is not used; only the mass of the solvent belongs in the denominator.

The elevation in boiling point for a solution is calculated using \(\Delta T_b=iK_bm\). Substituting the values gives \(\Delta T_b=2\times0.52\times1=1.04\,\mathrm{K}\). A temperature interval in kelvin has the same numerical size as an interval in degrees Celsius, so 1.04 K is also a rise of 1.04 °C. Therefore, option B is correct.

Depression in freezing point is calculated from \(\Delta T_f=iK_fm\). Substitution gives \(\Delta T_f=3\times1.86\times0.2=1.116\,\mathrm{K}\). The factor \(i=3\) indicates that the solute produces three effective particles per formula unit under the stated conditions, so omitting it would give the smaller and incorrect value 0.372 K.

The symbol \(\Delta T_b\) represents the elevation, or increase, in the boiling point caused by the solute. Therefore, the boiling point of the solution is found by adding the elevation to the pure-solvent boiling point: \(T_b(solution)=90+1.8=91.8^{\circ}\mathrm{C}\). Subtraction would describe a decrease and is inappropriate here.

Isotonic solutions are solutions that have equal osmotic pressure at the same temperature. Osmotic pressure depends on the number of dissolved solute particles and, for dilute solutions, is expressed as \(\pi=CRT\). Isotonicity does not require equal boiling points, molar masses, colours, or chemical identities. Therefore, the property common to isotonic solutions is osmotic pressure, so A is correct.

For dilute solutions, osmotic pressure follows \(\pi=CRT\). Comparing two solutions at the same temperature means that both \(R\) and \(T\) are common constants. Consequently, \(\pi_1/\pi_2=C_1/C_2\). Since the osmotic-pressure ratio is \(1:4\), the molar-concentration ratio must also be \(1:4\). Therefore, option C is correct.

For dilute solutions at the same temperature, osmotic pressure is given by \(\pi=CRT\). Because \(R\) and \(T\) are fixed, a higher value of \(\pi\) means a higher molar concentration \(C\), or more dissolved solute particles per unit volume. Therefore, the first solution is more concentrated. It is not pure or particle-free, so option B is correct.

For a colligative property, the van’t Hoff factor multiplies the ideal particle-based effect. For example, ΔTf = iKf m, ΔTb = iKb m, and π = iCRT. If i = 2.5 while concentration and the solvent remain unchanged, the effective number of particles and therefore the observed effect are about 2.5 times those for a normal non-electrolyte. Thus A is correct.

Colligative properties depend primarily on the number of dissolved solute particles, not on their chemical identity. The four standard examples are relative lowering of vapour pressure, elevation of boiling point, depression of freezing point, and osmotic pressure. Viscosity depends on particle size, shape, interactions, and solvent structure, so it is not colligative. Hence D is correct.

The mole fraction of a component equals its number of moles divided by the total number of moles of all components. Here, total moles = 0.2 + 0.8 = 1.0 mol. Therefore, the solvent mole fraction is Xsolvent = 0.8/1.0 = 0.8. The mole fractions of solute and solvent together must add to one, confirming option C.

For a dilute solution of a non-volatile solute, Raoult’s law gives the relative lowering of vapour pressure as \(\frac{p^0-p}{p^0}=x_{solute}\), where \(p^0\) is the vapour pressure of the pure solvent and \(p\) is the vapour pressure of the solution. Since \(x_{solute}=0.12\), the relative lowering is 0.12. Thus option A is correct; it is not the solvent mole fraction, 0.88.

Osmotic pressure is a colligative property because, for a dilute solution, it is described by π = iCRT. Thus, at fixed temperature and concentration, the effective number of solute particles determines the osmotic pressure. Viscosity, surface tension, and refractive index depend substantially on the nature and interactions of the substances, so A is the only correct answer.

Molality is defined as the number of moles of solute divided by the mass of solvent in kilograms: m = moles of solute / kilograms of solvent. Convert 250 g of solvent to 0.250 kg. Therefore, m = 0.5/0.250 = 2.0 mol kg⁻¹. The correct answer is option C.

For a non-electrolyte, the freezing-point depression is ΔTf = Kf m because i = 1. Solving for molality gives m = ΔTf/Kf. Substituting the values, m = 0.8/2.0 = 0.4 mol kg⁻¹. The unit K cancels appropriately with the unit in Kf, leaving mol kg⁻¹. Therefore, option B is correct.

The depression in freezing point is defined as the difference between the freezing point of the pure solvent and that of the solution: ΔT_f = T_f° − T_f. Substituting the given values, ΔT_f = 0 − (−0.72) = 0.72 °C. The negative sign shows that the solution freezes at a lower temperature, while the depression itself is reported as a positive magnitude.

For a dilute solution, the osmotic-pressure equation is π = iCRT. If concentration C and the van’t Hoff factor i remain constant, osmotic pressure is directly proportional to the absolute temperature T. Thus increasing temperature increases π. Temperature must be expressed in kelvin in the equation, although the direction of change is the same when temperature rises in Celsius.

A colligative property depends primarily on the number of dissolved particles present in a solution, rather than on the chemical identity of those particles. Important examples are relative lowering of vapour pressure, elevation of boiling point, depression of freezing point, and osmotic pressure. Electrolytes may produce more particles because they dissociate into ions, so their effects can be larger at the same concentration.

Osmosis is the spontaneous movement of solvent molecules through a semipermeable membrane from the side containing a more dilute solution, or nearly pure solvent, to the side containing a more concentrated solution. The membrane permits solvent molecules to pass but prevents solute particles from crossing. This movement continues until the chemical potential difference is balanced or external pressure stops it.

The net movement of solvent across a semipermeable membrane is driven by a difference in osmotic pressure or effective solute-particle concentration. When two solutions have no net solvent flow under the stated conditions, their osmotic pressures are equal. Such solutions are called isotonic. Hypertonic and hypotonic describe unequal effective concentrations relative to one another, while unsaturated refers to solubility, not osmotic balance.

The word colligative means related to collecting or counting particles. Thus, these properties depend on the total number of dissolved solute particles relative to the amount of solvent, not mainly on the particles’ chemical identity. Vapour-pressure lowering, boiling-point elevation, freezing-point depression, and osmotic pressure are standard examples. Dissociation or association changes the effective particle count and therefore changes the magnitude of the effect.

The van’t Hoff factor compares the actual number of particles in solution with the number expected if no association or dissociation occurred. A value below 1 means that the number of independent particles has decreased. This commonly occurs when two solute molecules associate to form one dimer. Thus, association by dimer formation is the most appropriate conclusion.

Use the relation ΔTb = iKb m. Substituting the given values gives 0.052 = i × 0.52 × 0.1. The product 0.52 × 0.1 equals 0.052, so i = 0.052/0.052 = 1. A van’t Hoff factor of 1 indicates that the solute behaves as though it neither dissociates into additional particles nor associates significantly in the solution.