From the following: pure water solution of \(\mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11}(m=0.01)\) in water solution of \(\mathrm{NaCl}(m=0.01)\) in water solution of \(\mathrm{CaCl}_{2}(m=0.01)\) in water Choose the one with the a. highest freezing point. b. lowest freezing point. c. highest boiling point. d. lowest boiling point. e. highest osmotic pressure.

When a solute is dissolved in a solvent, the freezing point of the solution is lower than that of the pure solvent. This phenomenon is known as freezing point depression. It occurs because the solute particles disrupt the formation of the ordered crystal structure of the solid solvent, which is necessary for freezing. The extent of freezing point depression depends on the number of solute particles in the solution; thus, it's a colligative property, which means it's related to the number of particles rather than their identity.

The formula to calculate freezing point depression is \( \Delta T_f = i \cdot K_f \cdot m \) where \( \Delta T_f \) is the freezing point depression, \( i \) is the van 't Hoff factor (which is the number of particles the compound dissociates into), \( K_f \) is the freezing point depression constant for the solvent, and \( m \) is the molality of the solution. To illustrate, the solution with the lowest number of solute particles, which is the \( \mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11} \) solution in our exercise, will have the smallest freezing point depression and therefore the highest freezing point.

Similar to freezing point depression, the addition of a solute to a solvent also causes the boiling point to increase, which is known as boiling point elevation. Solute particles in the solution result in a lower vapor pressure, requiring more heat to bring the solvent to boiling, which elevates the boiling point. This effect is also a colligative property and depends solely on the quantity of solute particles.

The boiling point elevation is given by the formula \( \Delta T_b = i \cdot K_b \cdot m \), where \( \Delta T_b \) is the increase in boiling point, \( i \) represents the van 't Hoff factor, \( K_b \) is the boiling point elevation constant, and \( m \) is the molality of the solution. In our exercise, the \( \mathrm{CaCl}_{2} \) solution has the greatest number of particles due to dissociation and therefore, has the highest boiling point elevation and the highest boiling point.

Osmotic pressure is a critical concept that explains the tendency of water to move across a semi-permeable membrane from the side with lower solute concentration to the side with higher concentration. It is a colligative property that is directly proportional to the molarity of the solute particles and the absolute temperature.

The equation for osmotic pressure \( \Pi \) is \( \Pi = i \cdot M \cdot R \cdot T \) where \( \Pi \) is the osmotic pressure, \( i \) is the van 't Hoff factor, \( M \) is the molarity of the solution, \( R \) is the gas constant, and \( T \) is the absolute temperature. In context, the \( \mathrm{CaCl}_{2} \) solution has more particles post-dissociation, resulting in a higher osmotic pressure relative to the other solutions.

The dissociation of ionic compounds into their respective ions is a fundamental process that occurs when these compounds dissolve in water. The number of ions produced, which is also referred to as the van 't Hoff factor \( i \) in colligative properties equations, significantly affects the properties of the solution. For instance, \( \mathrm{NaCl} \) dissociates into \( \mathrm{Na}^{+} \) and \( \mathrm{Cl}^{-} \) ions, resulting in two particles per formula unit.

Dissociation and Colligative Properties

Understanding this process is important for predicting how the solution will behave with regard to freezing point depression, boiling point elevation, and osmotic pressure. A compound like \( \mathrm{CaCl}_{2} \) will dissociate into one \( \mathrm{Ca}^{2+} \) and two \( \mathrm{Cl}^{-} \) ions, creating three particles and thus having a larger impact on the solution's colligative properties compared to a compound that does not dissociate or dissolves as molecules, such as \( \mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11} \).