Estimate: entropy of a cup of coffee

Problem

Estimate the increase in entropy of the universe produced by a single cup of coffee ($m \approx 0{,}2$ kg, which cools from $80\ ^\circ$C to $20\ ^\circ$C) left to cool in a room at $20\ ^\circ$C. Comment: if all students in Italy drank a coffee in the morning, what order of magnitude of entropy increase would they collectively produce?

Solution

We treat coffee as water, $c_s \approx 4180\ \text{J/(kg}\cdot\text{K)}$. The temperatures in kelvin are $T_\text{caldo}=353$ K and $T_\text{amb}=293$ K.

Entropy lost by the cooling coffee: $\Delta S_\text{caffè} = m c_s\ln\frac{T_\text{amb}}{T_\text{caldo}} = 0{,}2\cdot 4180\cdot\ln\frac{293}{353} \approx -156\;\text{J/K}$

Heat given up to the environment: $Q = m c_s (T_\text{caldo} - T_\text{amb}) = 0{,}2\cdot 4180\cdot 60 = 50\,160\;\text{J}$

Entropy gained by the environment (thermostat at $T_\text{amb}$): $\Delta S_\text{amb} = \frac{Q}{T_\text{amb}} = \frac{50\,160}{293} \approx +171\;\text{J/K}$

Total balance: $\Delta S_\text{tot} = -156 + 171 \approx +15\;\text{J/K}$ $\ev{\Delta S_\text{tot} \sim 10^1\;\text{J/K}}$

Comment: a single cup produces an entropy increase of the order of tens of J/K. With a number of students of order $10^6$–$10^7$, the overall morning increase is of order $10^7$–$10^8$ J/K: a colossal contribution, yet still just a drop compared with the entropy produced every instant by the entire planet.

Collegamenti

Argomenti: Entropy and the second law Concetti: Entropy · Second law of thermodynamics · Specific heat and heat capacity Competenze: Fermi estimates · Entropy balance