Problem 92 - Chemistry: An Atoms-Focused Approach

Question

Suppose \(100.0 \mathrm{mL}\) of \(2.50 \mathrm{m} M \mathrm{NaCl}\) is mixed with \(80.0 \mathrm{mL}\) of \(3.60 \mathrm{mM} \mathrm{MgCl}_{2}\) at \(20^{\circ} \mathrm{C} .\) Calculate the osmotic pressure of each starting solution and that of the mixture, assuming that the volumes are additive and that both salts dissociate completely into their component ions.

Step-by-Step Solution

Answer

Answer: The osmotic pressures of the starting solutions are 121.03 atm for NaCl and 318.37 atm for MgCl2. The osmotic pressure of the final mixture is 440.53 atm.

Step 1: Calculate moles of each solute

Using the given concentrations and volumes, we can calculate the moles of each solute. For NaCl: n = concentration × volume = 2.50 mol/L × 0.100 L = 0.250 moles For MgCl2: n = concentration × volume = 3.60 mmol/L × 0.080 L = 0.288 moles (note that 1 mmol = 0.001 mol)

Step 2: Calculate the osmotic pressure of each starting solution

We'll use the osmotic pressure formula, Π = n * i * R * T / V, with the ideal gas constant R = 0.0821 L atm / (mol K), and temperature T = 20°C = (20 + 273.15) K = 293.15 K For NaCl: Π = 0.250 moles * 2 * 0.0821 L atm / (mol K) * 293.15 K / 0.100 L = 121.03 atm For MgCl2: Π = 0.288 moles * 3 * 0.0821 L atm / (mol K) * 293.15 K / 0.080 L = 318.37 atm

Step 3: Calculate the moles of each ion and total moles in the mixture

When both solutions are mixed, the total volume becomes 0.100 L + 0.080 L = 0.180 L. For NaCl: Na+ ions = 0.250 moles; Cl- ions from NaCl = 0.250 moles For MgCl2: Mg2+ ions = 0.288 moles; Cl- ions from MgCl2 = 0.288 moles × 2 = 0.576 moles Total Cl- ions = 0.250 + 0.576 = 0.826 moles

Step 4: Calculate the osmotic pressure of the mixture

We'll use the osmotic pressure formula again, Π = n * i * R * T / V. Since both salts dissociate completely into their component ions, we'll consider the total moles of Cl- ions and the sum of moles of Na+ and Mg2+ ions when calculating osmotic pressure. For the mixture: Π = (0.250 + 0.288 + 0.826) moles * 0.0821 L atm / (mol K) * 293.15 K / 0.180 L = 440.53 atm The osmotic pressure of the starting solutions are 121.03 atm for NaCl and 318.37 atm for MgCl2. The osmotic pressure of the mixture is 440.53 atm.

Key Concepts

Molarity

Molarity is a common way of expressing the concentration of a solute in a solution. It is defined as the number of moles of solute per liter of solution. In the original exercise, molarity is used to calculate the amount of solute in both the NaCl and MgCl2 solutions.

To determine molarity, you can use the formula:

• \[ ext{Molarity (M)} = \frac{n}{V} \]

Where \( n \) is the number of moles of solute and \( V \) is the volume of the solution in liters.

It's crucial to always convert any given volume into liters before using this formula since molarity uses volume in liters as the standard unit. This conversion ensures consistency and accuracy across calculations.

Dissociation

Dissociation involves the separation of a compound into smaller components, usually ions, when dissolved in water. The exercise assumes complete dissociation of both NaCl and MgCl2.

For NaCl, dissociation occurs as follows:

• \[ ext{NaCl} \rightarrow ext{Na}^+ + ext{Cl}^- \]

This means that for every mole of NaCl, one mole of Na⁺ and one mole of Cl^- ions are produced.

Similarly, MgCl2 dissociates into:

• \[ ext{MgCl}_2 \rightarrow ext{Mg}^{2+} + 2 ext{Cl}^- \]

Here, each mole of MgCl2 yields one mole of Mg²⁺ and two moles of Cl^- ions. Recognizing these dissociation patterns is key for accurately calculating the number of ions in solution, which is essential for determining osmotic pressure.

Ideal Gas Constant

The Ideal Gas Constant, denoted as \( R \), is a fundamental constant in the equation for calculating osmotic pressure and other gas-related calculations. Its value is approximately 0.0821 L atm / (mol K).

In the context of the original exercise, the Ideal Gas Constant facilitates the calculation of osmotic pressure using the formula:

• \[ ext{Π} = n imes i imes R imes T / V \]

Where \( i \) is the van 't Hoff factor, \( T \) is temperature in Kelvin, and \( V \) is the volume of the solution.

Using the proper value and units for \( R \) is crucial, as it ensures the consistency and correctness of the calculated quantities in any thermodynamic equation, including osmotic pressure.

Temperature Conversion

Temperature conversion is important for consistency in calculations, particularly in equations involving thermodynamic properties, like osmotic pressure.

In most scientific formulas, temperature must be expressed in Kelvin. To convert from Celsius to Kelvin, you use the formula:

• \[ T( ext{K}) = T( ext{°C}) + 273.15 \]

In the original problem, the temperature is provided as 20°C. By converting to Kelvin, we achieve the accurate value for the calculations:

• \[ 20°C + 273.15 = 293.15 ext{ K} \]

Without this conversion, using Celsius in the formulas would lead to incorrect results, demonstrating why it's vital to check and adjust the temperature unit in thermodynamic equations.