Question

A solid mixture contains \(\mathrm{MgCl}_{2}\) and \(\mathrm{NaCl}\) . When 0.5000 \(\mathrm{g}\) of this solid is dissolved in enough water to form 1.000 \(\mathrm{L}\) of solution, the osmotic pressure at \(25.0^{\circ} \mathrm{C}\) is observed to be 0.3950 \(\mathrm{atm} .\) What is the mass percent of \(\mathrm{MgCl}_{2}\) in the solid? (Assume ideal behavior for the solution.)

Short answer

The mass percent of MgCl₂ in the solid mixture is approximately 88.72%.

Step-by-step solution

Convert the given temperature to Kelvin

: To convert the given temperature from Celsius to Kelvin, use the equation: K = °C + 273.15 Given temperature T = 25.0 °C, T(K) = 25 + 273.15 T(K) = 298.15 K

Determine the total molarity

: Use the osmotic pressure formula (Π = iMRT) to find the total molarity of the solution. Π = 0.3950 atm, R = 0.08206 L*atm/mol*K, T = 298.15 K, Rearrange the formula for M: M = Π / (iRT) Since the moles of MgCl2 and NaCl are not equal, we use the average van 't Hoff factor, so i = 2.5 (since i = 3 for MgCl₂ and i = 2 for NaCl): M_total = Π / (2.5 * R * T) M_total = 0.3950 / (2.5 * 0.08206 * 298.15) M_total = 0.002660 mol/L

Create two equations to find mole fractions of MgCl2 and NaCl

: Let x be the mole fraction of MgCl₂ and (1 - x) be the mole fraction of NaCl. Then, we can create two equations: 1) Molality_MgCl2 * 3 + Molality_NaCl * 2 = Molality_total (Using the van 't Hoff factor for each solute) 2) Molality_MgCl2 + Molality_NaCl = M_total (Total molality) Substitute (1 - x) for Molality_NaCl: Molality_MgCl2 * 3 + (1 - x) * 2 = Molality_total Molality_MgCl2 + (1 - x) = M_total

Solve the equations for the mole fractions

: We can solve the system of equations to find the mole fraction of each solute, x and (1 - x). To simplify the equations, replace Molality_total with M_total found in Step 2: Molality_MgCl2 * 3 + (1 - x) * 2 = 0.002660 Molality_MgCl2 + (1 - x) = 0.002660 By solving the linear system of equations, we find: x = 0.8286 1 - x = 0.1714

Find the moles of MgCl₂ and NaCl and their masses

: Using the mole fractions we found, now we can find the moles of MgCl₂ and NaCl in the solid mixture: Moles_MgCl2 = 0.8286 * 0.002660 mol/L = 0.002203 moles Moles_NaCl = 0.1714 * 0.002660 mol/L = 0.0004567 moles Using the molar masses for MgCl₂ and NaCl, we can find their masses: m_MgCl2 = moles_MgCl2 * molar_mass_MgCl2 m_NaCl = moles_NaCl * molar_mass_NaCl m_MgCl2 = 0.002203 * 95.211 g/mol = 0.2097 g m_NaCl = 0.0004567 * 58.442 g/mol = 0.02668 g

Calculate the mass percent of MgCl₂

: Now we can calculate the mass percentage of MgCl₂ in the solid mixture: Mass_percent_MgCl2 = (m_MgCl2 / (m_MgCl2 + m_NaCl)) * 100 Mass_percent_MgCl2 = (0.2097 / (0.2097 + 0.02668)) * 100 Mass_percent_MgCl2 ≈ 88.72 % The mass percent of MgCl₂ in the solid mixture is approximately 88.72%.

Key concepts

Mole Fraction

The mole fraction is a way to express the concentration of a component in a mixture. It is the ratio of moles of one component to the total moles of all the components in the mixture. In the context of our exercise with a solution containing \(\text{MgCl}_{2}\) and \(\text{NaCl}\), the mole fraction \(x\) for \(\text{MgCl}_{2}\) can be expressed as the moles of \(\text{MgCl}_{2}\) divided by the total moles of both \(\text{MgCl}_{2}\) and \(\text{NaCl}\). Similarly, the mole fraction for \(\text{NaCl}\) is \(1 - x\).
This ratio is dimensionless and always falls between 0 and 1. Calculating mole fractions helps in understanding the proportion of each solute in the solution, which is proportional to their contributions to properties like osmotic pressure.

Van 't Hoff Factor

The van 't Hoff factor \(i\) is crucial in determining how solute particles affect the colligative properties of solutions. It measures the effect of a solute on properties that depend on the number of particles in a solution rather than their nature.
For example, when \(\text{MgCl}_{2}\) dissolves, it dissociates into three ions: one \(\text{Mg}^{2+}\) and two \(\text{Cl}^{-}\), giving it an \(i\) value of 3. In contrast, NaCl dissociates into two ions (one \(\text{Na}^{+}\) and one \(\text{Cl}^{-}\)), having an \(i\) value of 2. In mixed solutions, an average van 't Hoff factor is used to reflect the weighted average impact of both solutes on the solution's properties.

Molarity

Molarity is a widely used measure of concentration, specifically indicating the moles of solute per liter of solution. The formula is expressed as \(M = \frac{\text{moles of solute}}{\text{liters of solution}}\). In the given exercise, the osmotic pressure formula \(\Pi = iMRT\) involves molarity \(M\), indicating that molarity helps calculate properties that depend on solute concentration, such as osmotic pressure.
Knowing the change in molarity is essential since it directly influences equation-based solutions and the understanding of solution behavior. In our exercise, determining the total molarity is key to calculating individual contribution to the osmotic pressure of a mixed solution, revealing how each solute component affects overall concentration.