Fermi estimate: ΔS of the human body

Problem

Estimate the net change in environmental entropy produced by your body in one day, knowing that you dissipate a metabolic power $P\approx 100\;\text{W}$ at a body temperature $T_\text{body}\approx 310\;\text{K}$, releasing heat to the air at $T_a\approx 295\;\text{K}$.

Solution

Setup: in one day ($t=86400\;\text{s}$) the body transfers to the environment the heat $Q=P\,t=100\cdot 86400\approx 8{,}6\cdot 10^6\;\text{J}$

The body releases $Q$ at $T_\text{body}$ (losing entropy $Q/T_\text{body}$), the air absorbs it at $T_a$ (gaining $Q/T_a$). The net balance is: $\Delta S_\text{univ}=Q\left(\frac{1}{T_a}-\frac{1}{T_\text{body}}\right)=8{,}6\cdot 10^6\left(\frac{1}{295}-\frac{1}{310}\right)\approx 1{,}4\cdot 10^3\;\text{J/K}$

Every day your body thus produces about a thousand J/K of entropy in the environment: it is the thermodynamic price of staying alive and organised. $\ev{\Delta S_\text{univ}\approx 10^3\;\text{J/K per day}}$

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Topics: Entropia e secondo principio Concepts: Entropia Skills: Stime di Fermi · Bilancio entropico