Problem 59 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

A solution containing \(2.60 \mathrm{~g}\) of a nonvolatile, non-electrolyte solute in \(200 \mathrm{~g}\) of water boils at \(100.130^{\circ} \mathrm{C}\) at \(1 \mathrm{~atm}\). What is the molar mass of the solute? \(\left[K_{\mathrm{b}}\left(\mathrm{H}_{2} \mathrm{O}\right)\right.\) \(=0.52 \mathrm{~K}-\mathrm{kg} / \mathrm{mol}]\) (a) \(52.0 \mathrm{~g} \mathrm{~mol}^{-1}\) (b) \(152.0 \mathrm{~g} \mathrm{~mol}^{-1}\) (c) \(104 \mathrm{~g} \mathrm{~mol}^{-1}\) (d) \(204 \mathrm{~g} \mathrm{~mol}^{-1}\)

Step-by-Step Solution

Answer

The molar mass of the solute is (b) 152.0 g/mol.

Step 1: Calculate the Boiling Point Elevation

The boiling point elevation \(\Delta T_b\) is the difference between the boiling point of the solvent with the solute present and the pure solvent. In this case, the boiling point of the solution is \(100.130^\circ C\) while pure water boils at \(100^\circ C\). Thus, \(\Delta T_b = 100.130^\circ C - 100^\circ C = 0.130^\circ C\).

Step 2: Use the Boiling Point Elevation Formula

The boiling point elevation can be calculated using the formula \(\Delta T_b = iK_b \cdot m\), where \(i\) is the van 't Hoff factor (which is 1 for a non-electrolyte), \(K_b\) is the ebullioscopic constant for water, and \(m\) is the molality of the solution. Since the solute is non-electrolyte \(i = 1\). Plug in the values to get the molality \(m = \frac{\Delta T_b}{K_b}\).

Step 3: Calculate the Molality

Using the values \(\Delta T_b = 0.130^\circ C\) and \(K_b = 0.52 \frac{K\cdot kg}{mol}\), the molality \(m\) can be found as \(m = \frac{0.130}{0.52}\) mol/kg.

Step 4: Determine the Molar Mass

The molality is also defined as the number of moles of solute per kilogram of solvent. We can rearrange the definition of molality to find the molar mass \(M\) of the solute: \(m = \frac{n}{mass\_of\_solvent\_in\_kg}\). So, \(M = \frac{2.60\_g}{m \cdot 0.2\_kg}\), where \(2.60\_g\) is the mass of the solute and \(0.2\_kg\) is the mass of the solvent in kilograms.

Step 5: Compute the Molar Mass

Substitute the values into the rearranged molality equation: \(M = \frac{2.60\_g}{0.250 \cdot 0.2\_kg}\) to find the molar mass of the solute.

Step 6: Final Answer

After simplifying the equation, you get the molar mass of the solute, which should correspond to one of the choices given.

Key Concepts

Boiling Point Elevation

When a nonvolatile solute is dissolved in a solvent, the boiling point of the resulting solution is higher than that of the pure solvent. This phenomenon is known as boiling point elevation. It occurs because the addition of a solute lowers the vapor pressure of the solvent, which means it requires a higher temperature to reach the vapor pressure necessary for boiling.

Boiling point elevation is directly proportional to the molality of the solution, which is the concentration of the solute in the solution. For every substance, there is a specific constant known as the ebullioscopic constant, which must be used to calculate the degree of boiling point elevation experienced.

Ebullioscopic Constant

The ebullioscopic constant (\(K_b\)) is a property of the solvent that provides a relationship between the molality of a solution and the extent to which the boiling point is elevated. It is defined as the boiling point elevation experienced for each mole of a solute added to one kilogram of solvent.

The value of the ebullioscopic constant varies with the solvent, reflecting how susceptible the solvent's boiling point is to the presence of a solute. Water, for example, has an ebullioscopic constant of 0.52 K·kg/mol, which is applied in our exercise to determine the boiling point elevation. By understanding the relationship these constants have with boiling point changes, students can reliably calculate the impact of solutes within various solvents.

Molality

Molality (\(m\)) is a measure of the concentration of a solution, expressed as the number of moles of solute present in one kilogram of solvent. Unlike molarity, which is dependent on the volume of the solution, molality is only affected by the mass of the solvent, which makes it particularly useful under conditions where temperature changes can cause volume fluctuations.

To calculate molality, you can use the formula: \(m = \frac{n}{mass\_of\_solvent\_in\_kg}\), where \(n\) is the number of moles of solute. In the context of boiling point elevation, once you've derived the molality using the ebullioscopic constant and the observed boiling point elevation, it can then be employed to find the molar mass of the solute by rearranging the formula to solve for the number of moles.