Problem 9 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

What quantity (in mL) of a \(45 \%\) acid solution of a mono-protic strong acid must be mixed with a \(20 \%\) solution of the same acid to produce \(800 \mathrm{~mL}\) of a \(29.875 \%\) acid solution? (a) 320 (b) 325 (c) 316 (d) 330

Step-by-Step Solution

Answer

The quantity is 316 mL.

Step 1: Define Variables

Let \( x \) be the volume in mL of the 45% acid solution, and \( 800 - x \) be the volume in mL of the 20% acid solution. We are to find the value of \( x \).

Step 2: Set Up Equation for Acid Content

For the 45% solution, the acid content is \( 0.45x \). For the 20% solution, it is \( 0.20(800 - x) \). The total acid content in the final 29.875% solution should be \( 0.29875 \times 800 \).

Step 3: Create the Equation

Create the equation based on acid content: \[0.45x + 0.20(800 - x) = 0.29875 \times 800\]

Step 4: Simplify and Solve the Equation

Expand and simplify the equation:\[0.45x + 160 - 0.20x = 239\]Combine like terms:\[0.25x + 160 = 239\]Subtract 160 from both sides:\[0.25x = 79\]Divide by 0.25:\[x = \frac{79}{0.25} = 316\]

Step 5: Verify the Solution

Check that the volumes add up to 800 mL and acid content is correct:- Volume of 45% solution: 316 mL.- Volume of 20% solution: \(800 - 316 = 484\) mL.Calculate acid contents:- For 45%: \(0.45 \times 316 = 142.2\).- For 20%: \(0.20 \times 484 = 96.8\).Total acid: \(142.2 + 96.8 = 239\), matches required 239 mL of acid (29.875% of 800 mL).

Key Concepts

Acid-Base Mixtures

Acid-base mixtures are common in chemistry and involve combining different concentrations of acid solutions to achieve a desired concentration. A strong, mono-protic acid is one that donates one proton (or hydrogen ion) per molecule to a solution. Examples include hydrochloric acid (HCl) and nitric acid (HNO3).
Mixing solutions of different concentrations involves understanding how percent concentrations work. These are weight/volume or weight/weight measures detailing how much solute (acid) is in a given amount of solvent or solution.
When combining two solutions, the goal is to achieve a target concentration in the final mixture. This is done by balancing the amounts of acid contributed by each solution, ensuring their combined total meets the desired target. It's similar to averaging but involves measuring precisely based on mass or volume ratios.

• Solution 1 contributes a certain percentage of acid per mL.
• Solution 2 also contributes its distinct percentage.
• The final mixture must have an acid concentration matching the target percentage, expressed as a simple mathematical relationship.

Understanding this setup allows chemists to predict the outcomes of mixing and adjust concentrations accordingly.

Mathematical Modelling in Chemistry

Mathematical modelling in chemistry involves representing chemical processes through mathematical equations. In the case of acid-base mixtures, it helps to determine the precise quantities needed to achieve the desired outcomes.
The key step in this process is defining variables that represent the unknown quantities, such as the volume of each solution. By expressing these values with variables, such as "x" for one component and "800 - x" for the other, we translate a chemistry problem into a solvable mathematical task.
Next, we set equations to represent the conservation of mass. Each component contributes its acid content to the total, thus enabling the creation of an equation reflecting these contributions.

• Identify acid content contributed by each solution.
• Define the total acid amount as per the target concentration times the desired total volume.
• Solve the equation for the unknown variable to find the precise volumes needed.

This approach ensures accuracy and allows conceptual problems to be solved systematically using algebraic techniques.

Stoichiometry

Stoichiometry is the quantitative relationship between reactants and products in a chemical reaction. While typically used for reactions, it can also be applied to mixing problems by determining how much of each component is needed for a desired mixture.
In our exercise, stoichiometry helps to ensure precision as it applies to the concentrations and volumes of the acid solutions. By ensuring that all parts of the solution add up correctly, stoichiometry keeps the quantities aligned with the predicted outcome.
Using stoichiometry, we calculate the total amount of acid needed. This is represented in clear steps:

• Find the total acid in the desired final solution (using percentage and volume).
• Express each solution's contribution based on its concentration.
• Ensure the total equals the solution's required acid content by adjusting variables.

Understanding stoichiometry lays a strong foundation for tackling a variety of chemical calculations, whether in reaction settings or mixture problems, providing clarity and precision.