Problem 267 - A Complete Resource Book in Chemistry for JEE Main

Question

The molar solubility (in \(\mathrm{mol} \mathrm{L}^{-1}\) ) of a sparingly soluble salt \(\mathrm{MX}_{4}\) is 's'. The corresponding solubility product is Ksp.s is given in terms of Ksp by the relation [2004] (a) \(\mathrm{s}=(\mathrm{Ksp} / 256)^{1 / 5}\) (b) \(\mathrm{s}=(128 \mathrm{Ksp})^{1 / 4}\) (c) \(\mathrm{s}=(\mathrm{Ksp} / 128)^{1 / 4}\) (d) \(\mathrm{s}=(256 \mathrm{Ksp})^{1 / 5}\)

Step-by-Step Solution

Answer

The correct relation is (a) \( s = \left(\frac{Ksp}{256}\right)^{1/5} \).

Step 1: Understand the Dissociation

The sparingly soluble salt \( \text{MX}_4 \) will dissociate in water as follows: \[ \text{MX}_4 (s) \rightleftharpoons \text{M}^{4+} (aq) + 4\text{X}^- (aq) \] This equation shows that one mole of \( \text{MX}_4 \) produces one mole of \( \text{M}^{4+} \) ions and four moles of \( \text{X}^- \) ions.

Step 2: Write Ksp Expression

The solubility product constant \( K_{sp} \) for the salt is given by the equation: \[ K_{sp} = [\text{M}^{4+}] [\text{X}^-]^4 \] Substituting the expressions in terms of molar solubility \( s \), we have \[ K_{sp} = s \cdot (4s)^4 \]

Step 3: Simplify the Ksp Expression

Substituting \((4s)^4\) in the \(K_{sp}\) equation, we have: \[ K_{sp} = s \cdot 4^4 \cdot s^4 \] \[ K_{sp} = 256s^5 \] Thus, the equation is simplified to \[ K_{sp} = 256s^5 \]

Step 4: Solve for Molar Solubility s

We rearrange the equation \( K_{sp} = 256s^5 \) to solve for \( s \): \[ s^5 = \frac{K_{sp}}{256} \] Taking the fifth root of both sides, we find: \[ s = \left(\frac{K_{sp}}{256}\right)^{1/5} \]

Key Concepts

Molar Solubility

Molar solubility is an important concept in chemistry that refers to the number of moles of a solute that can dissolve per liter of solution before the solution becomes saturated. In simpler terms, it's a measurement of how much of a substance can dissolve in a given amount of solvent.

For a compound such as a sparingly soluble salt like MX₄, the molar solubility is represented by 's'. This value indicates the amount of the salt that can dissolve in a liter of water to form a saturated solution. Knowing the molar solubility helps in calculating the concentrations of the ions in solution, which is essential for understanding other chemical properties of the salt.

Understanding molar solubility is crucial for predicting solution behavior in various chemical reactions and industrial processes.

Sparingly Soluble Salt

A sparingly soluble salt is a type of ionic compound that does not dissolve easily in a solvent, typically water. This means that only a small amount of the salt will dissolve in a given volume of water.

When a sparingly soluble salt like MX₄ dissolves, it reaches a point where no more salt can dissolve; this is known as the saturation point. Beyond this point, any additional salt added will remain as a solid.

The characteristics of sparingly soluble salts include:

• Low molar solubility: much less salt dissolves in the solvent compared to other salts.
• Equilibrium between the solid phase and the dissolved ions.

These features are important in various applications, including pharmaceuticals and environmental chemistry, where control over solubility and precipitation is crucial.

Ksp Expression

The solubility product constant, represented as Ksp, is a vital expression used to describe the equilibrium between a solid compound and its ions in solution. It is specific to sparingly soluble salts and provides insight into the solubility limits of the compound.

The Ksp expression for the salt MX₄, which dissociates as follows:

• MX₄ (s) ⇌ M⁴⁺ (aq) + 4X⁻ (aq)

is given by the equation:

• Ksp = [M⁴⁺][X⁻]⁴

where [M⁴⁺] and [X⁻] are the molar concentrations of the ions in solution. By substituting the molar solubility 's' into this expression, it allows you to calculate Ksp and vice-versa. Mastery of the Ksp expression is essential for understanding solubility phenomena and predicting the outcome of reactions involving sparingly soluble salts.

Solubility Calculations

Solubility calculations are used to find either the molar solubility of a compound or the solubility product constant (Ksp). These calculations are crucial for chemists to understand how much of a solute can dissolve in a solvent and to predict reaction outcomes.

For example, with the salt MX₄, if the Ksp is known, you can find the molar solubility 's' by rearranging the relationship derived from the equilibrium expression:

• Ksp = 256s⁵

Calculating the value of 's' involves isolating it in the equation:

• s = (Ksp/256)^(1/5)

By performing such calculations, students can gain a deeper understanding of the conditions under which a salt will dissolve and at what concentrations of ions in solution.