Quote:
Originally Posted by pigglet
2. due to the smaller volume of liquid in the opened bottle, the temperature gradient is more extreme within the liquid. thus, less time is required to lower the temperature of the smaller volume of liquid, and phase change occurs more rapidly.
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It would be an issue of thermal mass, not temperature gradient. Assuming the bottle is a tall cylinder, even if they were both closed, the temperature gradients would be very similar. This is because the smaller dimension is the radius, not the height, and since decreasing the amount of water in the bottle does not affect the radius and only affects the height (assuming it was standing relatively upright), the direction containing the more drastic temperature gradient would not change. The bottle would only cool faster because there is actually less to cool.
The temperature gradients would look something like this... Imagine taking a rainbow and shaping it into a two-dimentional rectangle (with the longer dimension being height, such that it somewhat resembles a vertical bottle) with rounded corners. The colors would represent the temperature profile. The full bottle would be that rainbow, the other would be one that was 1/4 shorter.