Problem 83 - Chemistry: The Central Science in SI Units, Global Edition

Question

The osmotic pressure of a \(0.010 \mathrm{M}\) aqueous solution of \(\mathrm{CaCl}_{2}\) is found to be \(68.3 \mathrm{kPa}\) at \(25^{\circ} \mathrm{C}\). Calculate the van't Hoff factor, \(i\), for the solution.

Step-by-Step Solution

Answer

The van't Hoff factor, \(i\), for the \(0.010 \; M\) aqueous solution of calcium chloride is approximately \(2.79\).

Step 1: Convert temperature to Kelvin

The temperature given is in Celsius, we need to convert it to Kelvin for the osmotic pressure formula. \(T_{\text{Kelvin}}=T_{^\circ\text{C}}+273.15\) \[ T_{\text{Kelvin}}=25^\circ\mathrm{C}+273.15=298.15\; \mathrm{K} \]

Step 2: Calculate theoretical osmotic pressure

The theoretical osmotic pressure is calculated using the given molarity and assuming that the solution does not dissociate, i.e., \(i=1\). Use the osmotic pressure equation and the ideal gas constant, \(R=8.314 \frac{\mathrm{J}}{\mathrm{mol} \cdot \mathrm{K}}\) (converted to \(kPa\)). \[ \Pi_{\text{theoretical}}=i_{\text{assumed}}MRT \] \[ \Pi_{\text{theoretical}}=(1)(0.010 \; \mathrm{M})(0.0821 \; \mathrm{\frac{L \cdot kPa}{mol \cdot K}})(298.15 \; \mathrm{K}) \] \[ \Pi_{\text{theoretical}}=24.5 \; \mathrm{kPa} \]

Step 3: Calculate van't Hoff factor

Now that we have both the experimental and theoretical osmotic pressure, we can calculate the van't Hoff factor. Rearrange the osmotic pressure equation to solve for \(i\). \[ i= \frac{\Pi}{MRT} \] Substitute the experimental osmotic pressure (\(68.3 \; \mathrm{kPa}\)) and the previously calculated values for \(M\) and \(T\). \[ i= \frac{68.3 \; \mathrm{kPa}}{(0.010 \; \mathrm{M})(0.0821 \; \mathrm{\frac{L \cdot kPa}{mol \cdot K}})(298.15 \; \mathrm{K})} \] \[ i \approx 2.79 \] The van't Hoff factor, \(i\), for the \(0.010 \; M\) aqueous solution of calcium chloride is approximately \(2.79\). This indicates that calcium chloride dissociates into ions in the solution, leading to higher osmotic pressure than theoretically expected for a non-dissociating solute.

Key Concepts

van't Hoff factor

The van't Hoff factor, denoted as \(i\), is a measure of how many particles a solute breaks into when dissolved in a solvent. It is crucial for understanding colligative properties, like osmotic pressure, which depend on the number of solute particles rather than their identity. For non-electrolytes, or solutes that don't dissociate, \(i\) is 1. However, electrolytes, which break into ions, usually have \(i\) greater than 1.
Given in the example, calcium chloride in solution has a van't Hoff factor of roughly 2.79. This number indicates that the solute dissociates into multiple particles, producing an osmotic pressure higher than predicted by nonelectrolytic behavior. This calculated \(i\) helps in determining the extent of dissociation, which is greater for calcium chloride than a neutral solute.

• The importance of the van't Hoff factor lies in its ability to reflect the dissociative nature of the substance in the solution.
• A greater \(i\) implies a larger effect on the colligative properties.

calcium chloride dissociation

Calcium chloride, \(\text{CaCl}_2\), dissociates in water, breaking down into its constituent ions. In an aqueous solution, it separates into one calcium ion, \(\text{Ca}^{2+}\), and two chloride ions, \(\text{Cl}^-\), as described by the dissociation equation:
\[ \text{CaCl}_2(s) \rightarrow \text{Ca}^{2+}(aq) + 2\text{Cl}^{-}(aq) \]
This dissociation process results in three ions per formula unit of calcium chloride, predicting a theoretical van't Hoff factor of 3 if dissociation were complete. However, real-world interactions could lead to a slightly different value due to ion pairing or incomplete dissociation.

• Dissociation affects the solution's osmotic pressure, as observed in the example with a higher than expected pressure.
• The van’t Hoff factor gives insight into the actual number of particles in the solution compared to the initial mole concentration.

temperature conversion to Kelvin

Converting temperature from Celsius to Kelvin is crucial when calculating osmotic pressure and other thermodynamic properties. The Kelvin scale is an absolute temperature scale where 0 K represents absolute zero, the lowest possible temperature. To convert Celsius to Kelvin, add 273.15 to the Celsius temperature:
\[ T_{\text{Kelvin}} = T_{^\circ\text{C}} + 273.15 \]
In the provided exercise, a temperature of 25 °C is converted to 298.15 K. Using Kelvin ensures consistency in scientific calculations, particularly when applying equations involving the ideal gas constant, \(R\).

• Kelvin eliminates negative values, simplifying calculations involving temperature ratios.
• Critical for calculations involving gas laws and colligative properties.

ideal gas constant

The ideal gas constant, symbolized as \(R\), is a key component in the calculation of osmotic pressure via the formula \( \Pi = iMRT \). This constant relates energy to temperature and amount of substance for ideal gases and solutions.
In consistent units with pressure and temperature, \(R\) is typically given as 0.0821 L·kPa/mol·K for osmotic pressure calculations, allowing us to predict how a solution behaves under different conditions. In the practical scenario, \(R\)'s significance lies in adjusting the ideal gas law concept to solutions, analyzing solute-particle effects on properties like osmotic pressure.

• It is a crucial link between temperature and pressure in gas laws and solution behavior theories.
• Uniform units for \(R\) ensure accurate and consistent results across related calculations.