# How does the temperature in a bucket of water with ice change over time?

I'm curious what happens to the temperature of the water in a bucket of water with ice in it. Let's start with a bucket with just ice in it, and we add water, in a room which is room temperature. It seems like there are two possibilities. This is intuitively what I figure happens:

• If the water is too hot, then all the ice will rapidly melt. The temperature at that moment will depend on how much ice there was and how hot the water us. From then on, the temperature will slowly reach equilibrium with the temperature of the room.
• If the water is not too hot, then the ice won't melt all at once. The temperature of the water will keep getting colder, and the ice will keep melting, up until some intermediate equilibrium point. At that point, the water doesn't keep getting colder, but the ice does continue to melt. Once all the ice has melted, then the temperature starts reaching equilibrium with the temperature of the room.

My specific question is: first of all, is the above correct, and if so, what's the temperature of the water at that "intermediate equilibrium point"? Is it zero degrees? If so why doesn't some of the existing water turn into ice?