Problem 58 - A Complete Resource Book in Chemistry for JEE Main

Question

Solubility product \(\left(\mathrm{K}_{\mathrm{sp}}\right)\) of salts \(\mathrm{mx}, \mathrm{m} \mathrm{x}_{2}\) and \(\mathrm{mx}_{3}\) at T temperature are \(4 \times 10^{-8}, 3.2 \times 10^{-14}\) and \(2.7 \times 10^{-15}\) respectively. The solubilities of these salts are in the decreasing order? (a) \(\mathrm{mx}>\mathrm{mx}_{2}>\mathrm{m}_{3} \mathrm{x}\) (b) \(\mathrm{m}_{3} \mathrm{x}>\mathrm{m} \mathrm{x}_{2}>\mathrm{mx}\) (c) \(\mathrm{mx}_{2}>\mathrm{m}_{3} \mathrm{x}>\mathrm{mx}\) (d) \(m x>m_{3} x>m x_{2}\)

Step-by-Step Solution

Answer

(a) MX > MX2 > MX3

Step 1: Understanding the solubility product relation

The solubility product, or \( K_{sp} \), of a salt is related to the solubility \( S \) of the salt in solution. For a salt of the type \( MX \), the relationship is \( K_{sp} = S^2 \). For \( MX_2 \), \( K_{sp} = 4S^3 \), and for \( MX_3 \), \( K_{sp} = 27S^4 \). Understanding these relations is essential to finding the solubility.

Step 2: Calculate solubility of MX

For the salt \( MX \), where \( K_{sp} = 4 \times 10^{-8} \), use the formula \( K_{sp} = S^2 \). Thus, \( S = \sqrt{4 \times 10^{-8}} = 2 \times 10^{-4} \).

Step 3: Calculate solubility of MX2

For the salt \( MX_2 \), where \( K_{sp} = 3.2 \times 10^{-14} \), use the formula \( K_{sp} = 4S^3 \). Thus, \( S = \sqrt[3]{\frac{3.2 \times 10^{-14}}{4}} \). Calculating yields \( S \approx 2 \times 10^{-5} \).

Step 4: Calculate solubility of MX3

For the salt \( MX_3 \), where \( K_{sp} = 2.7 \times 10^{-15} \), use the formula \( K_{sp} = 27S^4 \). Thus, \( S = \sqrt[4]{\frac{2.7 \times 10^{-15}}{27}} \). Calculating yields \( S \approx 3 \times 10^{-6} \).

Step 5: Compare the solubilities

Now compare the solubilities from the calculations: \( S(MX) = 2 \times 10^{-4} \), \( S(MX_2) = 2 \times 10^{-5} \), \( S(MX_3) = 3 \times 10^{-6} \). The order of decreasing solubility is \( MX > MX_2 > MX_3 \).

Key Concepts

Ksp Calculation

To calculate the solubility product constant, denoted as \(K_{sp}\), you need to understand its relationship with salt solubility, \(S\). The solubility product \(K_{sp}\) is the equilibrium constant for a solid substance dissolving in an aqueous solution. It helps predict how much of the solid will dissolve.
For different types of salts, \(K_{sp}\) is expressed differently:

• For salts like \(MX\), \(K_{sp} = S^2\)
• For salts like \(MX_{2}\), \(K_{sp} = 4S^3\)
• For salts like \(MX_{3}\), \(K_{sp} = 27S^4\)

This information allows us to solve problems where you are asked to rank the solubilities of different salts based on their \(K_{sp}\) values. Each type of salt will have a different equation relating \(K_{sp}\) and \(S\) due to the number of ions it dissociates into upon dissolving in water.

Salt Solubility

Salt solubility relates to how much salt can be dissolved in a given amount of solvent at a specific temperature to form a saturated solution. Each salt has its own solubility that is influenced by factors like temperature, pressure, and the specific ions involved. Here, let's look at why \(MX\), \(MX_2\), and \(MX_3\) have different solubilities when their \(K_{sp}\) values vary.
In general, the solubility \(S\) of a salt increases with an increase in \(K_{sp}\) under constant conditions. However, the relationship depends on the formula used to extract \(S\) from \(K_{sp}\):

• For \(MX\), a higher \(K_{sp}\) directly results in higher \(S\), as they are related through a square root \(S = \sqrt{K_{sp}}\).
• For \(MX_{2}\), it’s a cubic root \(S = \sqrt[3]{\frac{K_{sp}}{4}}\).
• For \(MX_{3}\), it’s more complex, involving a fourth root \(S = \sqrt[4]{\frac{K_{sp}}{27}}\).

By comparing solutions for different salts, you can determine which one is more soluble by solving for \(S\) with these root calculations.

Chemistry Problem Solving

Solving chemistry problems involving \(K_{sp}\) and salt solubility requires a systematic approach. First, identify the type of salt and use the corresponding formula to relate \(K_{sp}\) to \(S\). Here’s a simple step-by-step method you could follow:
1. **Identify the salt type**: Determine if the salt is \(MX\), \(MX_{2}\), or \(MX_{3}\). This is essential because it dictates how the \(K_{sp}\) equation is structured.
2. **Use the appropriate formula**: Once identified, utilize the formula that links \(K_{sp}\) and \(S\) correctly. For instance, use \(S^2\) for \(MX\) salts.
3. **Do the math**: Solve the equation for \(S\). This is often the step where you calculate roots, which may require a calculator.
4. **Compare calculated solubilities**: Rank the solubility of each salt based on your findings. It helps determine which salt dissolves more or less in a solution.
This problem-solving process is universal across different salts and increases your understanding of the concept of solubility, making chemistry exercises more approachable and less daunting.