This problem handles boiling-point elevation, which means we will use the formula:
ΔT = Kb * m
Where ΔT is the difference of Temperature between boiling points of the solution and the pure solvent (Tsolution - Tsolvent). Kb is the ebullioscopic constant of the solvent (2.64 for benzene), and m is the molality of the solution.
Knowing that benzene's boiling point is 80.1°C, we solve for m:
Tsolution - Tsolvent = Kb * m
80.23 - 80.1 = 2.64 * m
m = 0.049 m
We use the definition of molality to calculate the moles of azulene:
0.049 m = Xmoles azulene / 0.099 kgBenzene
Xmoles azulene = 4.87 x10⁻³ moles azulene
We use the mass and the moles of azulene to calculate its molecular weight:
0.640 g / 4.875 x10⁻³ mol = 130.28 g/mol
A molecular formula that would fulfill that molecular weight is C₁₀H₁₀. So that's the result of solving this problem.
The actual molecular formula of azulene is C₁₀H₈.