Using melting ice in a jug cools faster than using water at 0°C due to the energy absorption during the phase change.
Cooling is faster when using melting ice compared to water at 0°C because the process of melting ice requires energy to overcome the intermolecular forces, absorbing more heat from the surroundings and resulting in faster cooling. When water is at 0°C, it has already gained the heat necessary for its phase change, so the cooling process is slower.
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