Need help with latent heat involving ice and steam

[lilmul123]

Homework Statement

A well-insulated bucket of negligible heat capacity contains 177 g of ice at 0°C.

(a) If 20 g of steam at 100°C is injected into the bucket, what is the final equilibrium temperature of the system? I've already solved this part to be 0 degrees Celsius.

(b) What mass of ice remains?

Homework Equations

Q = mc(deltaT)

m = Q/Lf

The Attempt at a Solution

From the example in the textbook, it seems I only have to find the heat necessary to cool the steam from 100 degrees to 0 degrees, and then divide this number by the latent heat of ice (333.5 kJ/kg). 2.02 is the specific heat of steam. So, I did (.02)(2.02)(100). This gave me a heat of 4.04 kJ. I then divided this number by the latent heat of ice, 333.5, and came out with an answer of .01211 kg. I converted this to g (12.11 g), and subtract this from 177. My final answer was 164.886457 g, but this is incorrect. Help?

 

[lilmul123]

Does nobody know?

 

[denverdoc]

Phase change for water is different going from steam to liquid and then to water:

Total energy change= that going from steam to water-that going to from 100 to zero degrees-and then that from going liquid to freezing.

There are three separate constants at work.