Verified answer

Part of the heat from the mercury will raise the temp of the water to 100oC. Th reat will go to vaporizing the water

Lowering the temp of Hg to 100oC will put m*v*deltaT energy into the system

Q in = 3.86kg*138J/kg-K*(204-100) = 55400J Available energy from the Hg

Some of this goes into the water to raise its temp to 100. That amt = m*c*deltaT

0.332kg*4190J/kg-K(100-82.2) = 24761J

This means there are 55400-24761 = 30639J available to vaporize the water

So now we need to calculate how much water this would vaporize using Q = m*Lv

so m = Q/Lv = 30639J/2.256x10^6J/kg = 0.0136kg turns to steam