At certain temperature, a 5.12% solution of cane sugar is isotonic with a 0.9% solution of an unknown solute. The molar mass of solute is:

60

To determine the molar mass of the unknown solute, we can use the concept of isotonic solutions.

Isotonic solutions have the same osmotic pressure, which means that they exert the same pressure on a semi-permeable membrane. In this case, the \(5.12\)% solution of cane sugar and the 0.9\% solution of the unknown solute are isotonic.

The percentage of a solution can be expressed as $\frac{\text{mass solute}}{\text{mass solution}} \times 100\%$.

Let's assume that we have 100 g of each solution.

For the \(5.12\)% solution of cane sugar:
Mass of cane sugar \(= 5.12 g\)
Mass of solvent (water) \(= 100 g - 5.12 g = 94.88 g\)

For the 0.9\% solution of the unknown solute:
Mass of unknown solute \(= 0.9 g\)
Mass of solvent (water) \(= 100 g - 0.9 g = 99.1 g\)

Now, we can calculate the number of moles for each solute.

For cane sugar ($C_{12}H_{22}O_{11}$):
Molar mass of cane sugar \(= 12 \times 12.01 g/mol + 22 \times 1.01 g/mol + 11 \times 16.00 g/mol = 342.34 g/mol\)

Moles of cane sugar \(= \frac{\text{Mass of cane sugar}}{\text{Molar mass of cane sugar}}\)
                                   \(= \frac{5.12 g}{342.34 g/mol}\)

For the unknown solute:
The molar mass of the unknown solute = x g/mol (to be determined)

Moles of the unknown solute \(= \frac{\text{Mass of unknown solute}}{\text{Molar mass of the unknown solute}}\)
                                            \(= \frac{0.9 g}{x g/mol}\)

Since the solutions are isotonic, the number of moles of cane sugar and the number of moles of the unknown solute should be equal.

Therefore, we can set up the equation:

Moles of cane sugar = Moles of the unknown solute
\(\frac{5.12 g}{342.34 g/mol} = \frac{0.9 g}{x g/mol}\)

Solving for x, the molar mass of the unknown solute:

\(x = \frac{0.9 g \times 342.34 g/mol}{5.12 g}\)

\(x \approx 60 g/mol\)

Therefore, the molar mass of the unknown solute is approximately 60 g/mol.

Hence, the correct answer is (1) 60.