The mass of urea \((NH_2CONH_2)\) required to make \(2.5\, \ kg\) of \(0.25\) molal aqueous solution will be:

37.5 g

The correct answer is option 3. 37.5 g.

Molality is a concentration unit that is defined as the number of moles of solute (in this case, urea) per kilogram of solvent (in this case, water).

\(\text{Molality} (m) = \frac{\text{moles of solute}}{\text{mass of solvent (in kg)}}\)

Given:

Molality (\(m\)) = 0.25 molal

Mass of solvent (water) = 2.5 kg

To find the number of moles of urea needed, rearrange the molality formula:

\(\text{moles of urea} = m \times \text{mass of solvent (in kg)}\)

Substitute the values:

\(\text{moles of urea} = 0.25 \times 2.5 = 0.625 \text{ moles}\)

So, we need 0.625 moles of urea to prepare the solution.

The molar mass is the mass of one mole of a substance. For urea \((\text{NH}_2\text{CONH}_2)\):

Nitrogen (\(N\)): 14 g/mol (and there are 2 nitrogen atoms)

Hydrogen (\(H\)): 1 g/mol (and there are 4 hydrogen atoms)

Carbon (\(C\)): 12 g/mol

Oxygen (\(O\)): 16 g/mol

Calculate the total molar mass:

\(\text{Molar mass of urea} = (2 \times 14) + (4 \times 1) + 12 + 16 = 28 + 4 + 12 + 16 = 60 \text{ g/mol}\)

So, the molar mass of urea is 60 g/mol.

Now that you know the number of moles of urea (0.625 moles) and the molar mass of urea (60 g/mol), you can calculate the mass of urea needed:

\(\text{Mass of urea} = \text{moles of urea} \times \text{molar mass of urea}\)

Substitute the values:

\(\text{Mass of urea} = 0.625 \times 60 = 37.5 \text{ g}\)

Conclusion

To make \(2.5 \, \text{kg}\) of a \(0.25\) molal aqueous solution, you need 37.5 g of urea.

Thus, the correct answer is Option 3 (37.5 g).