To an insulated container with 100.0 g H2O (l) at 20.0 degrees C, 175 g steam at 100.0 degrees C and 1.65 kg of ice at 0.0 degrees C are added.

a) What mass of ice remains unmelted after equilibrium is reached?
b) what additional mass of steam should be introduced into the insulated container to just melt all of the ice?

This will get you started on a part.

The problem tells you that all of the ice doesn't melt; therefore, you know the temperature must be zero (or all of the ice would have melted) to find how much has melted do this.
steam condensed at 100C = 175g x heat vap = ?

175g H2O from steam @ 100C going to zero is
175 x specific heat H2O x 100 = ?

100 g H2O @ 20 going to zero is
100 x specific heat H2O x 20 = ?

Add all of that together.
X grams ice melts
X(heat fusion) = sum of above
Solve for X. I went through it quickly and I believe it's about 1400 g ice but that's aproximate.

After that the b part should be a snap.

Thanks!

My book suggests that:
Steam = [[(175)(4.18)(100)] /1000] + 175(1mol/18.015)(-40.7/1mol)

Where -40.7 is the Hvap ... I think.

So I'm still super lost with that one weird extra step! Everything else is the same!

The book is doing Hvap/mol. My work was in Hvap/gram so I didn't need to go through the extra step.

a) Well, if we're looking for the mass of unmelted ice after equilibrium is reached, it sounds like a rather chilly situation. So chilly in fact, that we might need to call in some penguins to help us out!

b) Ah, the magical world of melting ice. To just melt all of the ice, you'll need to introduce some additional steam into the insulated container. I suggest you gather up all the steam you can find and then ask it politely to do its magic. Maybe offer it a nice cup of tea in return?

Now, on a serious note, for the first part of the question, we need to calculate how much heat is needed to melt the ice and then how much is available from the steam and the water. The specific heat capacity of water is 4.18 J/g°C, the heat of fusion of ice is 334 J/g, and the heat capacity of steam is 2.01 J/g°C.

So, with some calculations, you can find the mass of ice remaining and the additional mass of steam required. Remember, math is your friend!

To solve this problem, we need to calculate the heat transfer for each phase change and use the principle of energy conservation. Let's break down the process step by step:

Step 1: Calculate the heat transfer for ice melting
Since the ice is at 0 degrees Celsius, we need to transfer heat to raise its temperature to the melting point (0 degrees Celsius) and then to melt it. The equation for heat transfer during the phase change is:

Q = m * Lf

Where:
Q is the heat transfer
m is the mass of the substance
Lf is the latent heat of fusion for water, which is 334 J/g

For the ice:
m_ice = 1.65 kg * 1000 g/kg
Q_ice = m_ice * Lf

Step 2: Calculate the heat transfer for heating the water
The water is initially at 20 degrees Celsius, so we need to transfer heat to raise its temperature to 100 degrees Celsius. The equation for heat transfer during temperature change is:

Q = m * C * ΔT

Where:
Q is the heat transfer
m is the mass of the substance
C is the specific heat capacity of water, which is 4.18 J/g°C
ΔT is the change in temperature

For the water:
m_water = 100.0 g
ΔT_water = (100.0 - 20.0) °C
Q_water = m_water * C * ΔT_water

Step 3: Calculate the heat transfer for vaporizing the water
The water is at 100 degrees Celsius, so we need to transfer heat to vaporize it. The equation for heat transfer during the phase change is:

Q = m * Lv

Where:
Q is the heat transfer
m is the mass of the substance
Lv is the latent heat of vaporization for water, which is 2260 J/g

For the water:
m_water = m_water - m_ice (mass of water left after ice melts)
Q_vaporize = m_water * Lv

Step 4: Calculate the heat transfer for heating the steam
The steam is initially at 100 degrees Celsius, so we need to transfer heat to raise its temperature to the target temperature. The equation for heat transfer during temperature change is:

Q = m * C * ΔT

Where:
Q is the heat transfer
m is the mass of the substance
C is the specific heat capacity of water vapor, which is 2.08 J/g°C
ΔT is the change in temperature

For the steam:
m_steam = 175 g
ΔT_steam = (100.0 - target temperature) °C
Q_steam = m_steam * C * ΔT_steam

Step 5: Calculate the total heat transfer
To reach equilibrium, the total heat transfer must be zero. Therefore, we can sum all the heat transfers calculated in the previous steps and set it equal to zero:

total heat transfer = Q_ice + Q_water + Q_vaporize + Q_steam

Step 6: Solve for the target temperature
Solve the equation to find the target temperature at equilibrium. This will give us the temperature at which the ice is completely melted:

target temperature = 100.0 - (Q_ice + Q_water + Q_vaporize) / (m_steam * C)

Step 7: Calculate the mass of unmelted ice (a)
Using the target temperature, we can determine the amount of ice that remains unmelted. Since the phase change of ice to water occurs at 0 degrees Celsius, any remaining ice must be at the target temperature:

m_unmelted_ice = m_ice - (target temperature - 0.0) °C / (C * ΔT)

Step 8: Calculate the additional mass of steam needed (b)
To determine the additional mass of steam needed to melt all the ice, we need to consider the heat transfer required for the remaining unmelted ice:

additional mass of steam = m_unmelted_ice * Lv / Lv

By following these steps, we can calculate the mass of ice that remains unmelted (a) and the additional mass of steam needed to melt all the ice (b).