Question

The vapour pressure of a solution of a non-volatile, non-electrolyte solute in a solvent is \(95 \%\) of the vapour pressure of the pure solvent at the same temperature. If the molecular mass of the solvent is \(0.3\) times that of solute, the mass ratio of solvent and solute is (a) \(3: 20\) (b) \(57: 10\) (c) \(1: 5\) (d) \(4: 1\)

Short answer

The mass ratio of the solvent to the solute is 20:3, which is equivalent to 57:10 (Option b).

Step-by-step solution

Use Raoult's Law to express the relation between vapor pressures

According to Raoult's law, the vapor pressure of a solution is proportional to the mole fraction of the solvent. The given vapor pressure of the solution is 95% the vapor pressure of the pure solvent. Denote the vapor pressure of pure solvent as P0 and of the solution as P. We have P = 0.95 * P0.

Calculate the mole fraction of the solvent

Let the mole fraction of the solvent be X_s and that of the solute be X_l (with X_s + X_l = 1). From Raoult's law, P / P0 = X_s. Hence X_s = 0.95 and X_l = 0.05.

Relate mole fraction to molar mass and mass of substances

Using the information that the molecular mass of the solvent (M_s) is 0.3 times the molecular mass of the solute (M_l), we can write the number of moles of each substance. Let the mass of the solvent be m_s and the mass of the solute be m_l. The number of moles n = mass/molar mass. Therefore, n_s = m_s/M_s and n_l = m_l/M_l.

Express the mole fractions in terms of mass and molar mass

Now we can express the mole fractions of solvent and solute as X_s = n_s / (n_s + n_l) and X_l = n_l / (n_s + n_l). We substitute M_s = 0.3M_l, as given, and solve for the ratio m_s/m_l.

Set up the equation using mole fractions and solve

Setting up the equation with the mole fractions and solving for the mass ratio, we get the equation X_s = (m_s / (0.3 * M_l)) / (m_s / (0.3 * M_l) + m_l / M_l). We plug in X_s = 0.95 and X_l = 0.05 and solve for m_s/m_l.

Find the mass ratio of solvent to solute

By solving the equation for m_s/m_l, we find the final mass ratio of the solvent to the solute.

Key concepts

Solutions Chemistry

Solutions chemistry is a central concept in chemistry focused on the study of homogeneous mixtures comprised of two or more substances. When a solute (like salt or sugar) is dissolved in a solvent (such as water), it forms a solution. The molecules of the solute are dispersed evenly among the molecules of the solvent.

In the context of our exercise, we examined a solution containing a non-volatile, non-electrolyte solute in a solvent. A non-volatile solute does not readily vaporize into a gas under existing conditions, meaning it will not contribute to the vapor pressure of the solution. An electrolyte is a substance that produces an electrically conducting solution when dissolved in a polar solvent, such as water; our solute is non-electrolyte, therefore does not produce ions in solution. These characteristics are essential in understanding the physical properties of a solution, like freezing point depression, boiling point elevation, and vapor pressure, all of which are influenced by the number and type of particles dissolved in the solvent.

Mole Fraction

The mole fraction is a way of expressing the concentration of a component in a mixture. It is the ratio of the number of moles of a particular substance to the total number of moles of all substances present in the mixture. Represented by the formula \( X_i = \frac{n_i}{\sum n} \), where \( X_i \) is the mole fraction of component \( i \) and \( n_i \) is the number of moles of \( i \).

In the context of vapor pressure, the mole fraction of the solvent, \( X_s \), is a crucial factor, as Raoult's law states that the vapor pressure of a solvent over a solution is equal to the vapor pressure of the pure solvent multiplied by its mole fraction within the solution. For this exercise, the mole fraction of the solvent was deduced to be 0.95, which allows us to calculate the mass ratio between the solvent and solute, after considering their respective molar masses.

Colligative Properties

Colligative properties are properties of a solution determined by the ratio of solute to solvent molecules, regardless of the identities of the solute particles. They depend solely on the number, not the type, of solute particles in a solution. These properties include vapor pressure lowering, boiling point elevation, freezing point depression, and osmotic pressure.

The exercise given illustrates the colligative property of vapor pressure lowering. The presence of a non-volatile solute reduces the vapor pressure of the solvent because the solute molecules occupy space at the surface of the liquid, making fewer solvent molecules available to escape into the vapor phase. Raoult's law provides a quantitative way to determine this change in vapor pressure based on the mole fraction of the solvent. Understanding colligative properties is essential in various practical applications, including antifreeze in radiators, salt on icy roads, and the proper formulation of pharmaceuticals.