Question

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

Short answer

The osmotic pressure is approximately 0.535 atm.

Step-by-step solution

Identify the Formula

To find the osmotic pressure, we use the formula: \( \pi = iMRT \) 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.

Convert Temperature to Kelvin

The temperature is given as \(0^{\circ}\) C. Convert this to Kelvin using: \( T = 0 + 273.15 = 273.15 \text{ K} \).

Identify the Constants

The molarity \( M \) is \(0.0120 \text{ M}\), the van't Hoff factor \( i \) is \(1.94\), and the ideal gas constant \( R \) is \( 0.0821 \text{ L atm K}^{-1} \text{ mol}^{-1} \).

Substitute Values into the Formula

Substitute the known values into the formula: \[ \pi = (1.94)(0.0120 \text{ M})(0.0821 \text{ L atm K}^{-1} \text{ mol}^{-1})(273.15 \text{ K}) \]

Calculate the Osmotic Pressure

Perform the calculation: \( \pi = 0.53487 \text{ atm} \). Therefore, the osmotic pressure is approximately 0.535 atm.

Key concepts

van't Hoff factor

The van't Hoff factor, denoted as \( i \), is an important concept in colligative properties, including osmotic pressure. It essentially captures the effect of an electrolyte dissolving in a solvent. When a substance, like sodium chloride (NaCl), dissolves in water, it separates into its ions, affecting the solution's properties.

• Non-electrolytes: These do not dissociate in solution and thus typically have a van't Hoff factor of 1.
• Electrolytes: These dissociate into ions and have a van't Hoff factor greater than 1. For instance, NaCl dissociates into two ions: Na\(^+\) and Cl\(^-\).

For our NaCl solution, the van't Hoff factor is given as 1.94. This indicates slight deviations due to interionic interactions. It suggests that, on average, each formula unit of NaCl contributes 1.94 particles to the solution.

molarity

Molarity, represented as \( M \), is a measure of the concentration of a solute in a solution. It is expressed as the number of moles of solute per liter of solution. Molarity is crucial for calculating the osmotic pressure as it directly relates to the number of particles that contribute to the solution's properties.

• Formula: Molarity (\( M \)) = \( \frac{\text{moles of solute}}{\text{liters of solution}} \)
• Example: In our exercise, the molarity of the NaCl solution is given as 0.0120 M. This means that there are 0.0120 moles of NaCl in every liter of the solution.

Understanding molarity helps in predicting how the solution will behave under different physical conditions like temperature and pressure.

ideal gas constant

The ideal gas constant, denoted as \( R \), is a fundamental constant in chemistry that relates several key properties of gases under ideal conditions. Its value is crucial for calculations involving gases, and for calculating osmotic pressure as well.

• Value: In our exercise, \( R \) is used as 0.0821 L atm K\(^{-1}\) mol\(^{-1}\).
• Purpose: It provides the conversion factor needed to equate the mole-based properties of gases and solutions with pressure and temperature measurements.

The value of \( R \) can differ slightly depending on the units used, but its role remains the same in providing cohesion among different physical properties in an equation.

temperature conversion to Kelvin

Temperature conversion to Kelvin is an essential step in calculations for thermodynamic properties like osmotic pressure. The Kelvin scale is absolute and starts at absolute zero, making it ideal for gas laws and colligative properties.

• Why Kelvin: Many scientific formulas require temperature to be in Kelvin to ensure calculations reflect natural laws accurately.
• Conversion Formula: To convert from Celsius to Kelvin, add 273.15: \( T(\text{K}) = T(\degree \text{C}) + 273.15 \).

In our exercise, the temperature is 0\(\degree\) C. Converting this using the formula gives us 273.15 K, which we use in the osmotic pressure equation. Using Kelvin ensures that the physics of the problem reflects actual science.