Animated Solution for Chemistry - Basic Concepts in Chemistry: The density (in ${\text{g mL}}^{-1}$) 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

Imagine you are working in a chemistry lab, and you are handed a beaker containing a sulphuric acid ($H_{2}S{O}_4$) solution. The label on the beaker tells you two things: the solution has a molarity of $3.60\text{ M}$ and a mass percentage of $29\%\text{ w/w}$. Your task is to find the density of this solution.

At first glance, it might seem like you need to assume a certain volume, calculate the mass of the solute, find the mass of the solvent, and then somehow piece it all together to find the density. While that fundamental approach works perfectly, competitive exams like JEE demand speed and precision. This is where mastering direct interconversion formulas becomes your superpower.

There is a beautiful, time-saving relationship that directly connects Molarity ($M$), mass percentage ($x\%$), and density ($d$). The formula is:

Here, $M_w$ represents the molar mass of the solute. But where does this formula come from? Let's break it down intuitively.

If you have $1\text{ L}$ ($1000\text{ mL}$) of the solution, its total mass would be $1000\times d$ grams. Since the mass percentage of the solute is $x\%$, the mass of the solute in this $1\text{ L}$ solution is $\frac{x}{100}\times 1000\times d=10\times d\times x$ grams.

To find the molarity (which is the number of moles of solute per liter of solution), you simply divide this mass by the molar mass of the solute ($M_w$). And just like that, the formula $M=\frac{10\times d\times x}{M_w}$ is born!

Now, let's bring our specific values into the equation. We know: - Molarity, $M=3.60\text{ M}$ - Mass percentage, $x=29$ - Molar mass of $H_{2}S{O}_4$, $M_w=98{\text{ g mol}}^{-1}$

Substituting these into our master equation, we get:

Our only unknown here is the density, $d$. Let's isolate it by rearranging the terms:

Don't rush the arithmetic here; this is where silly mistakes often happen. Multiplying the numerator gives us $352.8$, and the denominator is simply $290$.

Looking at our options, we need to round this off to two decimal places. The digit after $1$ is $6$, so we round up to get:

This perfectly matches option (c). By trusting the formula and executing the math carefully, we bypassed a lengthy derivation and arrived at the correct answer efficiently.