(a) No
When the water and the ice mix, the amount of heat released by the water is absorbed by the ice. The heat released by the water is "used" by the ice as follows:
1- partially to increase its temperature from -21.0 ∘C to its melting point, 0 ∘C
2- if there is still enough heat, then it is used to melt the ice
3- if there is still enough heat, then it is used to increase the temperature of the "new sample" of water, until the two samples of water are at same temperature (equilibrium)
Let's start with process 1). The heat required for this process is
where
is the mass of the ice
is the heat specific capacity of ice
is the final temperature of the ice
is the initial temperature
Substituting,
When the ice has reached this temperature, then it will continue absorb heat from the water to melt completely. In order to completely melt the ice, the water must have "enough heat" to give off. The maximum heat that the water can give to the ice is when it reaches thermal equilibrium with it, so
where
is the mass of the water
is the heat specific capacity of water
is the final temperature at equilibrium
is the initial temperature of the water
Substituting,
Part of this heat is used for process (1), while the rest is used for process (2) (and if there is still heat available, for process 3). However, the heat required to completely melt the ice (process 2) is
where
is the mass of the ice
is the latent heat of fusion of ice
Substituting,
We see that is larger than Q: this means that not all the ice melts.
(b) 0.266 kg,
Since we know that not all the ice has melted, we can consider processes 1) and 2) only, and we can write that the heat given off by the water is used partially to heat the ice, and the rest to melt part of the ice:
where is the mass of ice that has melted. Solving for this variable, we find:
So, the mass of ice remained is
and the final temperature at equilbrium is 0 degrees, since not all ice has melted.