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Why does heat flow spontaneously from a hotter object to a colder object?

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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.

Heat flows spontaneously from a hotter object to a colder one because this process results in a net increase in the entropy of the system, which is a fundamental requirement of the second law of thermodynamics.12

When an amount of heat $Q$ moves from a reservoir at temperature $T_{1}$ to one at a lower temperature $T_{2}$ , the net change in entropy ( $Δ S$ ) is calculated as:

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

Because $T_{1} > T_{2}$ , the term $\frac{Q}{T_{2}}$ is larger than $\frac{Q}{T_{1}}$ , ensuring that $Δ S$ is positive. While the laws of physics at the microscopic level—involving individual molecular collisions—are time-reversible, the spontaneous flow of heat from cold to hot is never observed in macroscopic systems because it is statistically improbable. In a system containing an enormous number of molecules, the probability of random thermal motions resulting in a decrease in entropy is so infinitesimal that it is effectively impossible. Thus, the second law of thermodynamics dictates that spontaneous processes in an isolated system always proceed in the direction of increasing entropy, leading the system toward thermal equilibrium.123