Problem 38 - Chemistry and Chemical Reactivity

Question

Calculate the osmotic pressure of a \(0.0120 \mathrm{M}\) solution of NaCl in water at \(0^{\circ} \mathrm{C}\). Assume the van't Hoff factor, \(i\), is 1.94 for this solution.

Step-by-Step Solution

Answer

The osmotic pressure is approximately 0.521 atm.

Step 1: Understand the formula

The formula for calculating osmotic pressure is \( \Pi = iMRT \), where \( \Pi \) is the osmotic pressure, \( i \) is the van't Hoff factor, \( M \) is the molarity of the solution, \( R \) is the ideal gas constant, and \( T \) is the temperature in Kelvin.

Step 2: Convert temperature to Kelvin

The given temperature is \(0^{\circ} C\). To convert this to Kelvin, use the formula: \( T(K) = T(^{\circ}C) + 273.15 \). So, \( T = 0 + 273.15 = 273.15 \) K.

Step 3: Identify known values

The molarity \( M = 0.0120 \) M, \( i = 1.94 \), and \( R = 0.0821 \text{ L atm K}^{-1} \text{ mol}^{-1} \). We have already calculated \( T = 273.15 \) K.

Step 4: Substitute known values into the formula

Substitute the known values into the osmotic pressure formula: \( \Pi = (1.94)(0.0120 \text{ M})(0.0821 \text{ L atm K}^{-1} \text{ mol}^{-1})(273.15 \text{ K}) \).

Step 5: Calculate the osmotic pressure

Perform the calculation: \( \Pi = 1.94 \times 0.0120 \times 0.0821 \times 273.15 \approx 0.521 \text{ atm} \).

Key Concepts

Van 't Hoff Factor

The van't Hoff factor, denoted as \( i \), is a key element in the calculation of colligative properties such as osmotic pressure. It reflects the degree of dissociation of a solute in a solution. For ionic compounds like NaCl, the factor indicates how many moles of ions are produced from a single mole of solute. In an ideal scenario, NaCl dissociates into two ions: Na⁺ and Cl⁻, giving an ideal van't Hoff factor of 2. However, due to non-ideal behavior in real solutions, the observed factor can be less.

This concept helps account for ion pairing, where some ions in solution are not fully dissociated. In this exercise, the van't Hoff factor is given as 1.94, indicating slight deviation from complete dissociation. Understanding and using the correct \( i \) value is crucial to accurately determining colligative properties, such as osmotic pressure.

Molarity

Molarity \((M)\) is a measure of the concentration of a solute in a solution, expressed in the number of moles of solute per liter of solution. It is a vital parameter in the formula for calculating osmotic pressure and is frequently used in chemistry to express solution concentration.

To calculate molarity, you would use the formula:

• Molarity \( (M) = \frac{\text{moles of solute}}{\text{liters of solution}} \)

In this exercise, the molarity of the sodium chloride solution is given as 0.0120 M, signifying that there are 0.0120 moles of NaCl dissolved in every liter of solution. This measurement enables the calculation of colligative properties like osmotic pressure when combined with other factors like the van't Hoff factor and temperature.

Temperature Conversion

Temperature conversion is a necessary step when dealing with gas laws and colligative properties, mainly because these formulas require temperature to be in Kelvin rather than Celsius or Fahrenheit. This is because Kelvin is an absolute temperature scale, keeping all values positive and consistent for calculations.

To convert Celsius to Kelvin, you apply this quick formula:

• \(T(K) = T(°C) + 273.15\)

For example, a temperature of \(0^{\circ}C\) becomes \(273.15\) K when converted. In colligative property calculations like osmotic pressure, using the correct Kelvin value ensures precision in your results. It standardizes the impact of temperature on the movement and pressure of molecules, a fundamental aspect of understanding the behavior of solutions.

Ideal Gas Constant

The ideal gas constant \((R)\) plays a significant role in calculating osmotic pressure as it is part of the formula \( \Pi = iMRT \). This constant relates to the energy scale of a mole of particles at a given temperature, connecting pressure, volume, and temperature in gas calculations.

The value of \(R\) used here is \(0.0821 \text{ L atm K}^{-1} \text{ mol}^{-1}\), which is suited for calculations involving pressure in atmospheres and volume in liters.

Choosing the correct version of \(R\) depends on the units used in the problem. If other units are used, \(R\) values such as \(8.314\ \text{J K}^{-1} \text{mol}^{-1}\) for energy calculations would be appropriate. Ensuring that the units of \(R\) match those in the rest of the equation is crucial to obtaining an accurate result when calculating properties like osmotic pressure.