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Why does melting ice in hot water increase the total entropy of the system?

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Engraved portrait of Rudolf Julius Emanuel Clausius, a bearded man in formal attire with sideburns, facing forward. — thermodynamics Rudolf Clausius (1822 - 1888) is considered one of the central founders of the science of thermodynamics.

When ice melts in hot water, the total entropy of the system increases because the process is spontaneous and irreversible.12

From a thermodynamic perspective, entropy is a measure of the energy unavailable for work and is linked to the second law of thermodynamics, which dictates that spontaneous processes in an isolated system always proceed toward increasing entropy. When heat $Q$ flows from a hot reservoir at temperature $T_{1}$ to a colder one at $T_{2}$ , the net entropy change is calculated as:12

$Δ S = \frac{Q}{T_{2}} - \frac{Q}{T_{1}}$

Because $T_{1} > T_{2}$ , the term $\frac{Q}{T_{2}}$ is larger than $\frac{Q}{T_{1}}$ , resulting in a positive net entropy change ( $Δ S > 0$ ).12

Statistically, this increase reflects a transition from a state of lower probability to one of higher probability. Initially, thermal energy is partitioned, with slow-moving molecules in the ice and fast-moving molecules in the water. As the ice melts and the system reaches thermal equilibrium, this energy becomes uniformly distributed, representing a state of greater randomness and disorder.13