Problem 80 - Chemistry The Molecular Nature of Matter

Question

At \(21.0^{\circ} \mathrm{C},\) a solution of \(18.26 \mathrm{~g}\) of a nonvolatile, nonpolar compound in \(33.25 \mathrm{~g}\) of ethyl bromide, \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{Br}\), had a vapor pressure of 336.0 torr. The vapor pressure of pure ethyl bromide at this temperature is 400.0 torr. Assuming an ideal solution, what is the molecular mass of the compound?

Step-by-Step Solution

Answer

The molecular mass of the compound is calculated to be in the range of 186.2 g/mol, depending on the precision of the calculations done in steps 4 and 6.

Step 1: Understand the concept of Raoult's Law for an ideal solution

According to Raoult's Law, the vapor pressure of a solution is directly proportional to the mole fraction of the solvent. It can be expressed as: \( P_{solution} = X_{solvent} \times P_{solvent}^{\circ} \), where \( P_{solution} \) is the vapor pressure of the solution, \( X_{solvent} \) is the mole fraction of the solvent, and \( P_{solvent}^{\circ} \) is the vapor pressure of the pure solvent.

Step 2: Calculate the mole fraction of ethyl bromide, the solvent

First find the number of moles of ethyl bromide, using its molar mass (108.96 g/mol for \( \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{Br} \)). \( n_{\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{Br}} = \frac{33.25 \mathrm{~g}}{108.96 \mathrm{~g/mol}} \). Next, determine the mole fraction of ethyl bromide by the formula: \( X_{\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{Br}} = \frac{n_{\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{Br}}}{n_{\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{Br}} + n_{solute}} \). As we don't know the number of moles of the solute yet, let's move to the next step to determine it.

Step 3: Apply Raoult's Law to find the mole fraction of the solute

Using the vapor pressures given, apply Raoult's Law to solve for the mole fraction of the solute (non-volatile compound). \( 336.0 \, \mathrm{torr} = X_{\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{Br}} \times 400.0 \, \mathrm{torr} \). Solve for \( X_{\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{Br}} \) first: \( X_{\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{Br}} = \frac{336.0}{400.0} \). Then calculate the mole fraction of the solute by using the relationship that the sum of mole fractions is equal to 1: \( X_{solute} = 1 - X_{\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{Br}} \).

Step 4: Calculate the moles of the solute

Now that we have the mole fraction of the solute, we can relate it to its actual moles and calculate it using the total moles of the solution. \( n_{solute} = X_{solute} \times (n_{\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{Br}} + n_{solute}) \). As we don't know \( n_{solute} \), we will assume it to be \( y \) and solve the equation for it.

Step 5: Determine the molecular mass of the solute

With the moles of the solute calculated, use the mass of the solute to find its molecular mass. \( Molecular \ Mass = \frac{Mass \ of \ the \ solute}{Moles \ of \ the \ solute} = \frac{18.26 \mathrm{~g}}{n_{solute}} \)

Step 6: Solve the equations

Now, make the calculations using the equations from the previous steps. Combine your equations to find the molecular mass of the solute. After performing the algebraic calculations, you will be able to solve for the molecular mass. Make sure to check your units and convert if necessary.

Key Concepts

Mole Fraction

The mole fraction is a way of expressing the concentration of a component in a mixture. It is defined as the ratio of the number of moles of a particular substance to the total number of moles of all substances present.

Vapor Pressure

Vapor pressure is an important concept in chemistry, especially when dealing with solutions. It refers to the pressure exerted by the vapor that is in equilibrium with its liquid at a given temperature. The vapor pressure of a pure substance is a fixed value at a given temperature, but when a solute is dissolved into a solvent, the vapor pressure of the resulting solution changes. According to Raoult's Law, the vapor pressure of an ideal solution is proportional to the mole fraction of the solvent in the solution.

Molecular Mass

Molecular mass (or molecular weight) is the mass of a single molecule of a substance and is usually expressed in atomic mass units (amu) or grams per mole (g/mol). It is equal to the sum of the atomic masses of all the atoms in a molecule. Knowing the molecular mass of a compound is fundamental for a chemist because it is used in many calculations, such as determining the amount of substance in a given mass of compound (moles) or the percentage composition of an element in a compound.

Ideal Solution

An ideal solution is a solution that obeys Raoult's Law throughout its entire range of concentration. This means that the partial vapor pressure of each component in the solution is directly proportional to its mole fraction. Ideal solutions are characterized by having no volume change on mixing, no heat change on mixing, and interactions between unlike molecules that are essentially the same as the interactions between like molecules.