Problem 25 - Chemistry in 30 days

Question

What is the degree of dissociation for \(\mathrm{Ca}\left(\mathrm{NO}_{3}\right)_{2}\) solution if van't Hoff factor \(=2.4\) ? (a) \(0.8\) (b) \(0.7\) (c) \(0.5\) (d) \(0.9\)

Step-by-Step Solution

Answer

The degree of dissociation is \(0.7\).

Step 1: Understand the Chemical Dissociation

Recognize that \(\mathrm{Ca}\left(\mathrm{NO}_{3}\right)_{2}\) dissociates in water as follows:\[\mathrm{Ca}\left(\mathrm{NO}_{3}\right)_{2} \rightarrow \mathrm{Ca}^{2+} + 2\, \mathrm{NO}_{3}^{-}\]This shows that 1 mole of \(\mathrm{Ca}\left(\mathrm{NO}_{3}\right)_{2}\) produces 1 mole of \(\mathrm{Ca}^{2+}\) and 2 moles of \(\mathrm{NO}_{3}^{-}\), leading to a total of 3 ions.

Step 2: Define the van't Hoff Factor and Degree of Dissociation

The van't Hoff factor (\(i\)) accounts for the effect of ionization on colligative properties. For complete dissociation of 1 mole of \(\mathrm{Ca}\left(\mathrm{NO}_{3}\right)_{2}\), \(i = 3\) because there are 3 ions formed.

Step 3: Set up the Equation Relating van't Hoff Factor

The actual degree of dissociation affects \(i\). The relationship is \[i = 1 + (n - 1) \alpha\]where \(n\) is the number of particles produced (=3 for \(\mathrm{Ca}\left(\mathrm{NO}_{3}\right)_{2}\)) and \(\alpha\) is the degree of dissociation. Substitute \(i = 2.4\) and \(n = 3\) into the equation.

Step 4: Solve for the Degree of Dissociation (\(\alpha\))

Substitute the given values into the equation:\[2.4 = 1 + (3 - 1)\alpha\]\[2.4 = 1 + 2\alpha\]Rearrange to solve for \(\alpha\):\[2.4 - 1 = 2\alpha\]\[1.4 = 2\alpha\]\[\alpha = \frac{1.4}{2} = 0.7\]

Step 5: Verify and Conclude the Degree of Dissociation

Confirm the calculations and note that the degree of dissociation \(\alpha\) is 0.7. This means that 70% of \(\mathrm{Ca}\left(\mathrm{NO}_{3}\right)_{2}\) dissociates into ions.

Key Concepts

van't Hoff factor

The van't Hoff factor, represented as \( i \), is a crucial concept in physical chemistry, especially when discussing solutions and their behaviors. It measures the effect that solute particles have on the colligative properties of a solution. Colligative properties depend on the number of solute particles rather than their identity. The van't Hoff factor helps us understand how dissolving a substance affects properties such as boiling point elevation, freezing point depression, and osmotic pressure.

• Theoretical value: If a compound dissociates completely, the van't Hoff factor equals the number of particles resulting from the dissociation. For example, complete dissociation of \( \mathrm{Ca}\left(\mathrm{NO}_{3}\right)_{2} \) would produce three ions: one \( \mathrm{Ca}^{2+} \) ion and two \( \mathrm{NO}_{3}^{-} \) ions, resulting in \( i = 3 \).
• Experimental value: In practice, not all solutes dissociate completely, leading to a van't Hoff factor that is less than the theoretical value. This is where calculations for the degree of dissociation are beneficial.

Understanding the van't Hoff factor allows chemists to predict the changes in colligative properties more accurately.

Chemical Dissociation

Chemical dissociation refers to the process whereby compounds split into their constituent ions when they dissolve in a solvent, such as water. This process is common with ionic compounds, like salts. For example, when \( \mathrm{Ca}\left(\mathrm{NO}_{3}\right)_{2} \) dissolves, it dissociates into one \( \mathrm{Ca}^{2+} \) ion and two \( \mathrm{NO}_{3}^{-} \) ions.This breakdown is critical in understanding the behavior of solutions because it impacts the resultant concentration of ions, which influences the colligative properties of the solution.

• The dissociation results in an increase in the number of solute particles in the solution.
• This increase directly affects properties like osmotic pressure and boiling point.

The degree to which dissociation occurs can be altered by factors such as temperature, the nature of the solute, and the solvent's properties.

Colligative Properties

Colligative properties are those properties of solutions that depend on the ratio of the number of solute particles to the number of solvent molecules, and not on the identity of the solute. These include boiling point elevation, freezing point depression, vapor pressure lowering, and osmotic pressure. The extent of these changes is directly influenced by the van't Hoff factor, which takes into account the dissociation of solute particles. For instance:

• Increased ion concentration leads to an increased lowering of vapor pressure.
• More particles result in a higher boiling point and lower freezing point.

Understanding colligative properties and their dependence on the van't Hoff factor is essential for practical applications in areas such as chemistry, biology, and engineering, where accurate predictions of solute-solvent interactions in a solution are required.

Ionization

Ionization refers to the process by which neutral molecules are converted into charged ions. This process is distinct from dissociation and often involves the loss or gain of electrons, resulting in the formation of cations and anions from neutral atoms or molecules.When discussing electrolytes, ionization is a key factor because it facilitates the conduction of electricity through the solution. It also plays a role in the determination of the van't Hoff factor, especially for substances that do not originally exist in an ionic state, such as acids that ionize to release \( \text{H}^+ \) ions.

• The degree of ionization influences the number of particles in the solution.
• A higher degree of ionization leads to a higher van't Hoff factor.

Grasping the concept of ionization is crucial for understanding how certain molecular compounds enhance the conductivity of solutions and influence other related chemical processes.

Calculation of Degree of Dissociation

Calculating the degree of dissociation is a vital process to determine how much a compound dissolves into its constituent ions. The degree of dissociation, \( \alpha \), is the fraction of the initial solute molecules that dissociate into ions in solution.To find \( \alpha \), we use the relationship between the van't Hoff factor \( i \), and the number of particles following dissociation:\[ i = 1 + (n - 1) \alpha \]Where:

• \( i \) is the van't Hoff factor
• \( n \) is the total number of ions produced per formula unit of the compound (3 for \( \mathrm{Ca}\left(\mathrm{NO}_{3}\right)_{2} \))
• \( \alpha \) is the degree of dissociation

Using this formula, you can rearrange to solve for \( \alpha \), allowing you to quantify the extent of dissociation in a given solution. This understanding helps in making precise calculations needed to assess changes in colligative properties.