# Understanding calculations about phase changes

I am given the following example calculation about phase transitions, however multiple things are not clear to me during the calculations. Could someone explain them to me?

Problem:

Suppose mixture of 500g of water and 100g of ice that both are in thermic equilibrium. We then add 200g of steam which has a temperature of 100°C. Calculate the equilibrium temperature and the composition of the final mixture.

Used formula's:

$Q = mL$ where Q = heat, m = mass, L = latent heat

$Q = mc\Delta t$ where m=mass, c=specific heat, $\Delta t$ = change in temperature

Given:

• $m_{water} = 500g$

• $m_{ice} = 100g$

• $m_{steam} = 200g$

• $T_{water} = 0°C$

• $T_{ice} = 0°C$

• $T_{solid} = 100°C$

• $L_{1,2} = 334J/g$ (latent het from solid to liquid state)

• $L_{2,3} = 2249J/g$ (latent heat from liquid to gas state)

1) Suppose $T_{final}$ = 0°C

Heat balance: $Q_{Given By Steam} = Q_{Absorbed By Ice}$

$m_{solid} L_{2,3} + m_{solid} C_{water}(T_{solid} - T_{water}) = m"_{ice}L_{1,2}$

after some calculations: $m"_{ice} = 1597,4g$

We conclude this supposition is wrong. We only have 200g of ice so there is no way 1597,4g which can melt.

2) Suppose $T_{final}$=100°C

Heat balance: $Q_{Given By Steam} = Q_{Absorbed By Ice AND Water}$

$m"_{solid}L_{2,3} = m_{ice}L_{1,2} + (m_{ice} + m_{water})c_{water}(T_{solid}-T_{water})$

after some calculations: $m"_{solid} = 126,5g$

This is not in contradiction with our supposition.

Solution:

We can now tell that the final composition is 73,5g of steam, 726,5g of water and no ice.

My questions

1) At the first supposition we suppose (and that is what is being used for the calculations):

$Q_{given by steam} = Q_{absorbed by ice}$

Why isn't it: $Q_{Given By Steam} = Q_{Absorbed By Ice AND Water}$ (like in the second supposition) as the ice and the water are both in thermic equilibrium?

2) Why is at the first supposition the formula $Q=mc\Delta t$ associated with the heat given by the steam and at the second supposition with the heat absorbed by the ice and water?

Thank you very much, 