Problem 99 - A Complete Resource Book in Chemistry for JEE Main

Question

The dissociation constant of water \(\left(\mathrm{H}_{2} \mathrm{O}\right)\) is \(1.0 \times 10-{ }^{14}\) and that of heavy water \(\left(\mathrm{D}_{2} \mathrm{O}\right)\) is \(3.0 \times 10^{-15}\). This is because the extent of dissociation of \(\mathrm{H}_{2} \mathrm{O}\) is about (a) \(1 / 3\) times that of \(\mathrm{D}_{2} \mathrm{O}\) (b) 3 times that of \(\mathrm{D}_{2} \mathrm{O}\) (c) 3000 times that of \(\mathrm{D}_{2} \mathrm{O}\) (d) 30 times that of \(\mathrm{D}_{2} \mathrm{O}\)

Step-by-Step Solution

Answer

The extent of dissociation of light water is approximately 3 times that of heavy water.

Step 1: Understand the Dissociation Constants

The dissociation constant (K_w) indicates the extent to which water molecules dissociate into ions. For light water, K_w= 1.0  imes 10^{-14}, and for heavy water, K_w= 3.0  imes 10^{-15}. A higher dissociation constant implies greater dissociation.

Step 2: Set Up the Ratio

Determine the ratio of the dissociation constants of light water to heavy water: \\[\text{Ratio} = \frac{1.0 \times 10^{-14}}{3.0 \times 10^{-15}}\]

Step 3: Simplify the Ratio

Simplify the expression to find the factor by which the dissociation of light water is greater than that of heavy water: \\[\text{Ratio} = \frac{1.0}{3.0} \cdot 10^{1} = \frac{10}{3} \approx 3.33\]

Step 4: Interpret the Result

The ratio 3.33 suggests that the extent of dissociation of light water is approximately 3.33 times that of heavy water. Therefore, the closest answer choice is option (b), which states it is 3 times the dissociation extent of D_2O.

Key Concepts

Light Water vs Heavy Water

Light water, commonly known as regular water, is composed of two hydrogen atoms and one oxygen atom, chemically represented as \( \text{H}_2\text{O} \). On the other hand, heavy water, represented as \( \text{D}_2\text{O} \), consists of two deuterium atoms, which are heavier isotopes of hydrogen, and one oxygen atom. The difference in the atomic composition between light water and heavy water leads to different physical and chemical properties. One of the most notable differences is in the rate at which they dissociate or break apart into ions. This difference arises because deuterium atoms in heavy water have more mass compared to regular hydrogen atoms, affecting the vibrations and the strength of the molecular bonds.

Understanding the distinction between light and heavy water is crucial in various scientific fields, including chemistry and nuclear physics, where heavy water plays a significant role as a moderator in nuclear reactors because of its neutron-absorbing properties.

Dissociation Extent

The term 'dissociation extent' refers to the degree or level to which molecules dissociate into ions in a solution. For example, when water dissociates, it forms hydronium ions \( (\text{H}^+) \) and hydroxide ions \( (\text{OH}^-) \). In the context of the exercise, dissociation extent is used to compare how much more light water dissociates compared to heavy water.

When comparing light water and heavy water, light water has a dissociation constant of \( 1.0 \times 10^{-14} \), whereas heavy water has a dissociation constant of \( 3.0 \times 10^{-15} \). The higher dissociation constant indicates that light water has a greater extent of dissociation compared to heavy water. Simplifying the ratio of their dissociation constants helps in quantifying this difference and shows that light water dissociates approximately 3.33 times more than heavy water.

Comparison of Dissociation Constants

The dissociation constant, often denoted by \( K_w \), is a numerical value that represents the equilibrium concentration of ions formed when a molecule dissociates. For water, \( K_w \) is specifically related to how water dissociates into hydronium and hydroxide ions.

To compare the dissociation constants of light and heavy water, we use their respective \( K_w \) values given in the exercise: \( K_w \) for light water is \( 1.0 \times 10^{-14} \) and for heavy water, it is \( 3.0 \times 10^{-15} \). By calculating the ratio \( \frac{1.0 \times 10^{-14}}{3.0 \times 10^{-15}} \), we determine that this ratio simplifies to approximately 3.33. This means light water dissociates about 3.33 times more extensively than heavy water, validating the step-by-step solution which concludes that the dissociation extent of light water relative to heavy water is close to 3 times as much.