How does a vacuum flask keep a drink cold?

A vacuum is an excellent insulator. As there is nothing in the vacuum to transmit heat in or out, whatever is within the vacuum will hold its heat. Vacuum flasks don't have a perfect vacuum, so there is loss but quite slow.

The vacuum is held in the space between the outer case and the inner liner. More.

Lastly, conduction occurs when a hot object is placed next to a cold object. The heat is passed from the warmer to the colder object. This is why your hand gets burned when you touch a hot stove.

The heat requires a medium to move through. Heat moves through some materials very easily (conductors), and other materials resist the flow of heat (insulators). By having a double-walled glass flask with a vacuum between the layers of glass, there is no medium for the heat to flow through and the ice stays cold.

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