The Cooling Coffee Paradox · AI Generator #GEN-250EC643

Physics · thermodynamics

Problem

A barista notices something curious about coffee cooling. She prepares two identical cups of coffee at T0=85°CT_0 = 85°\text{C} in a room at Troom=22°CT_{\text{room}} = 22°\text{C}. To the first cup, she immediately adds cold milk at Tmilk=5°CT_{\text{milk}} = 5°\text{C}. To the second cup, she adds the same amount of milk after waiting exactly t=5t = 5 minutes. The coffee cups have mass mc=200  gm_c = 200\;\text{g} with specific heat cc=4200  J/(kg⋅K)c_c = 4200\;\text{J/(kg·K)}. The milk has mass mm=50  gm_m = 50\;\text{g} with specific heat cm=3900  J/(kg⋅K)c_m = 3900\;\text{J/(kg·K)}. Coffee follows Newton's law of cooling with time constant τ=12\tau = 12 minutes (meaning T(t)=Troom+(T0−Troom)e−t/τT(t) = T_{\text{room}} + (T_0 - T_{\text{room}})e^{-t/\tau} for pure coffee). When milk is added, assume instantaneous thermal equilibrium between coffee and milk, but the cooling time constant remains τ=12\tau = 12 minutes for the mixture. (a) What is the temperature of the first cup (milk added immediately) after t=5t = 5 minutes? (b) What is the temperature of the second cup (milk added after 5 minutes of coffee cooling) immediately after the milk is added? (c) Which cup is hotter after the milk has been added to both, and by how much?

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