The Cooling Coffee Paradox · AI Generator #GEN-6233B63B

Physics · thermodynamics

Problem

A barista notices something puzzling about two identical coffee cups. Cup A contains

200\;\text{mL}

of coffee at

85°\text{C}

, while Cup B contains

100\;\text{mL}

of the same coffee at

85°\text{C}

plus

100\;\text{mL}

of room temperature milk at

20°\text{C}

(which was mixed in immediately). Both cups are then left to cool in a room at

20°\text{C}

. After exactly

10

minutes, the barista measures the temperatures and finds that Cup A has cooled to

65°\text{C}

. Surprisingly, Cup B (which started cooler after mixing) is now at exactly the same temperature as Cup A! Assuming Newton's law of cooling applies and that coffee and milk have identical thermal properties (specific heat

c = 4200\;\text{J/(kg·K)}

and density

\rho = 1000\;\text{kg/m}^3

), what is the thermal time constant

\tau

for these cups?

Related theory

thermodynamics

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