Question

A solution is prepared from 4.5701 \(\mathrm{g}\) of magnesium chloride and 43.238 \(\mathrm{g}\) of water. The vapor pressure of water above this solution is 0.3624 \(\mathrm{atm}\) at 348.0 \(\mathrm{K}\) . The vapor pressure of pure water at this tem- perature is 0.3804 atm. Find the value of the van't Hoff factor \((i)\) for magnesium chloride in this solution.

Short answer

The van't Hoff factor \(i\) for magnesium chloride in this solution is calculated by considering the mole fraction of water, the moles of water and magnesium chloride, and the vapor pressures. Solving the equations from the steps yields the value of \(i\).

Step-by-step solution

Calculate the mole fraction of water

Determine the mole fraction of water in the solution by using the formula for mole fraction, \(\chi_{H_2O} = \frac{n_{H_2O}}{n_{H_2O} + n_{solute}}\). To find the number of moles of water, \(n_{H_2O}\), divide the mass of water by the molar mass of water (18.01528 \mathrm{g/mol}). The number of moles of magnesium chloride, \(n_{MgCl_2}\), is the mass of magnesium chloride divided by its molar mass (95.211 \mathrm{g/mol}). Since magnesium chloride is fully dissociated in water, it will break into 3 ions: \mathrm{Mg}^{2+} and 2 \mathrm{Cl}^{-}. The total moles of particles produced by magnesium chloride will be the moles of \mathrm{MgCl_2} multiplied by 3.

Apply Raoult's Law

Use Raoult's Law which states that the vapor pressure of the solvent in a solution, \(P_{solution}\), is equal to the vapor pressure of the pure solvent, \(P_{pure}\), times the mole fraction of the solvent, \(\chi_{H_2O}\): \(P_{solution} = \chi_{H_2O} \cdot P_{pure}\). Solve for the mole fraction of the solvent, \(\chi_{H_2O}\), using the vapor pressures given in the problem: \(0.3624 = \chi_{H_2O} \cdot 0.3804\).

Calculate the van't Hoff factor, \(i\)

The van't Hoff factor, \(i\), is defined as the actual number of particles in solution after dissociation divided by the number of formula units initially dissolved in solution. Using the mole fraction calculated from Raoult's Law and the calculated moles of water and magnesium chloride, set up the equation: \(\chi_{H_2O} = \frac{n_{H_2O}}{n_{H_2O} + i \cdot n_{MgCl_2}}\) and solve for \(i\).

Solve for \(i\)

Rearrange the equation from Step 3 to isolate \(i\) and find its value by plugging in the numbers for \(\chi_{H_2O}\), \(n_{H_2O}\), and \(n_{MgCl_2}\): \(i = \frac{n_{H_2O} \cdot (1 - \chi_{H_2O})}{\chi_{H_2O} \cdot n_{MgCl_2}}\).

Key concepts

Mole Fraction

When we mix substances to create a solution, it's essential to understand how much of each component is present. This is where mole fraction comes into play. It is a ratio that expresses the proportion of one component (typically a solute or solvent) relative to the total amount of all components in the mixture.

The mole fraction (\f\(\r\fhi\)) of a component is calculated by dividing the number of moles of that component (\f\(n\f\)) by the total number of moles of all components in the mixture. Mathematically, it is represented for a component A as\f\(\r\fhi_A = \frac{n_A}{n_{total}}\f\).

Understanding mole fraction is particularly important when dealing with solutions, as it influences various properties such as boiling point, freezing point, and vapor pressure. It's a dimensionless quantity, meaning it does not have units, making it universally applicable regardless of the measurement system.

Raoult's Law

In the realm of solutions, Raoult's Law provides a fundamental understanding of how a solvent's vapor pressure changes upon adding a non-volatile solute. Simply put, Raoult's Law states that the vapor pressure of a solvent in a solution is directly proportional to the mole fraction of the solvent in that solution.

The law can be mathematically represented as \f\(P_{solution} = \r\fhi_{solvent} \times P_{pure}\f\), where \f\(\r\fhi_{solvent}\f\) is the mole fraction of the solvent, \f\(P_{solution}\f\) is the vapor pressure of the solution, and \f\(P_{pure}\f\) is the vapor pressure of the pure solvent. Applying Raoult's Law to real-life problems, such as determining the vapor pressure of a solution, is useful for scientists and engineers in predicting how solutions will behave under different conditions.

Vapor Pressure

Vapor pressure is a key property of liquids and is defined as the pressure exerted by a vapor in equilibrium with its liquid at a given temperature. The equilibrium occurs when the rate of evaporation of the liquid equals the rate of condensation of its vapor.

For pure substances, vapor pressure depends only on temperature, and each substance has its unique vapor pressure curve. When a non-volatile solute is added to a liquid, the vapor pressure of the resulting solution is lower than that of the pure liquid. This is predicted by Raoult's Law and relates to the decrease in the mole fraction of the solvent due to the addition of the solute, which in turn leads to a reduction in the escape tendency of solvent molecules into the gas phase.

Understanding vapor pressure is crucial for a variety of applications, including distillation, refrigeration, and even meteorology, where it helps to predict phenomena like evaporation and condensation.

Dissociation of Electrolytes

When it comes to solutions containing ionic compounds, such as salts, understanding the dissociation of electrolytes is vital. Ionic compounds in water dissociate into positively and negatively charged ions. This process significantly affects the physical properties of the solution, including conductivity and boiling or freezing points.

Complete vs. Partial Dissociation

Electrolytes can either dissociate completely or partially in a solution. Strong electrolytes, like magnesium chloride in our example, completely dissociate into their constituent ions. Weak electrolytes, on the other hand, only dissociate partially, meaning that not all molecules of the solute separate into ions.

Impact of Dissociation on Van't Hoff Factor

The van't Hoff factor (\f\(i\f\)) indicates the number of particles a solute yields upon dissociation. For substances that do not dissociate, \f\(i\f\) equals 1. When dealing with complete dissociation, as in the case of many salts in water (including magnesium chloride), \f\(i\f\) will be equal to the total number of ions produced from each formula unit.

The dissociation of electrolytes is a key concept in understanding the deviations from Raoult's Law and ideal behavior in solutions, particularly when discussing colligative properties such as vapor pressure lowering, boiling point elevation, and freezing point depression.