A 0.0500 kg ice cube at -30.0◦C is placed in 0.400 kg of 35.0◦C water in a very well insulated container.

What is the final temperature? (Note: how can you check if all the ice melts at equilibrium temperature?)

To determine the final temperature, we need to use the principle of conservation of energy.

Step 1: Calculate the heat gained by the water:
The heat gained by the water can be calculated using the formula:

Qwater = mcΔT

Where:
Qwater = heat gained by water
m = mass of water (0.400 kg)
c = specific heat capacity of water (4.18 kJ/kg°C)
ΔT = change in temperature

ΔT = final temperature - initial temperature

Let's calculate ΔT:

ΔT = final temperature - 35.0°C

Step 2: Calculate the heat lost by the ice:
The heat lost by the ice can be calculated using the formula:

Qice = mcΔT

Where:
Qice = heat lost by ice
m = mass of ice (0.0500 kg)
c = specific heat capacity of ice (2.09 kJ/kg°C)
ΔT = change in temperature

ΔT = final temperature - (-30.0°C)

Step 3: Calculate the heat gained by the ice (during phase change):
When the ice melts, it absorbs heat without changing its temperature. The heat gained during the phase change can be calculated using the formula:

Qphase_change = mLf

Where:
Qphase_change = heat gained during phase change
m = mass of ice (0.0500 kg)
Lf = latent heat of fusion for ice (334 kJ/kg)

Step 4: Calculate the total heat gained:
The total heat gained by the system is equal to the heat gained by the water plus the heat gained during the phase change:

Qtotal = Qwater + Qphase_change

Step 5: Set up the equation for conservation of energy:
Since the system is well insulated, the total heat gained by the system is equal to the total heat lost by the system:

Qtotal = Qice

Step 6: Solve for the final temperature:
Setting up the equation, we have:

Qwater + Qphase_change = Qice

Substituting the formulas for Qwater, Qphase_change, and Qice, we get:

mcΔT + mLf = mcΔT

The mcΔT terms cancel out:

Lf = 0

Therefore, there is no phase change occurring and the ice does not melt completely. The final temperature will be below 0°C, but we cannot determine the exact value without additional information.

To check if all the ice melts at equilibrium temperature, we need to compare the final temperature with the melting point of ice (0°C). If the final temperature is below 0°C, it means that some ice remains. If the final temperature is 0°C or above, it means all the ice has melted.