Question

The density (in g mL\(^{-1}\)) of a \(3.60 \text{ M}\) sulphuric acid solution that is \(29\%\) \(\text{H}_2\text{SO}_4\) (molar mass \(= 98 \text{ g mol}^{-1}\)) by mass will be:

• A. \(1.64\)
• B. \(1.88\)
• C. \(1.22\)
• D. \(1.45\)

Hint:

Deriving concentration relation short-cuts can save immense time during competitive examinations. Memorize the identity \(M = \frac{10 \cdot \% \cdot d}{M_{\text{solute}}}\) to transform multi-step mass-to-volume logic tracks into a single computational step.

Answer 1

The Correct Option is C

Concept: Molarity (\(M\)) is defined as the number of moles of solute dissolved per liter (\(1000 \text{ mL}\)) of solution. Mass percentage (% w/w) represents the mass of solute present in \(100 \text{ g}\) of solution. These two concentration terms can be interconnected using the density (\(d\)) of the solution through fundamental conversion variables or a direct relational formula: \[ M = \frac{10 \times (\% \text{ by mass}) \times d}{\text{Molar Mass of Solute}} \] Step 1: Setting up the values from the given problem statement.
We are given the following values for the sulphuric acid (\(\text{H}_2\text{SO}_4\)) solution:
• Molarity (\(M\)) \(= 3.60 \text{ mol/L}\)
• Mass percentage (%\( \text{ w/w}\)) \(= 29\%\)
• Molar mass of solute (\(M_B\)) \(= 98 \text{ g/mol}\)
• Density of solution \(= d \text{ g/mL}\)

Step 2: Substituting the parameters into the relationship formula.
Using our conversion identity to isolate the variable \(d\): \[ 3.60 = \frac{10 \times 29 \times d}{98} \] \[ 3.60 = \frac{290 \times d}{98} \]

Step 3: Solving for the density \(d\).
Rearranging the equation to isolate \(d\): \[ d = \frac{3.60 \times 98}{290} \] \[ d = \frac{352.8}{290} \approx 1.2165 \text{ g/mL} \] Rounding to two decimal places to match the given choices, we get: \[ d \approx 1.22 \text{ g mL}^{-1} \]