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No change in level will result from the ice changing from solid to liquid itself for the ice displaces exactly the amount of water that is equal to its mass. The other effect, thermal expansion and/or contraction, may be difficult to measure in this countertop experiment, especially if you start out with cold water in the glass. Assuming the amount of ice is significant, the water will cool to near 32 degrees and stay at the point until the ice is nearly melted. This is because ice is at equilibrium with water at 32 degrees. Adding heat to mixture goes into melting the ice and not into raising the temperature (the temperature will go up a bit, the amount above 32 being proportional to the rate at which the ice melts). Hence, most change in volume due to thermal contraction/expansion will occur at the very beginning when the water/ice mixture is not yet in equilibrium. After that the mixture stays near 32 degrees until nearly all the ice melts.

- weitzhuis

While we're in the nitpicking department, it should be noted that the equilibrium temperature for solid and liquid state of salty sea water is below 0C (32F), and to further complicate matters, most ice is fresh water, created from precipitation rather than freezing sea water.

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