Blackbody radiation at temperature has energy density and entropy density ; the…

PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Earth, Climate & Energy Physics · The laws of thermodynamics

Energy from Physics, Ch The solar resource resolved: geometry, spectrum and the limits of concentration

Problem

Blackbody radiation at temperature TT has energy density u=aT4u=aT^4 and entropy density s=43aT3s=\tfrac43aT^3; the surroundings' own radiation at T0T_0 has u0=aT04u_0=aT_0^4 and s0=43aT03s_0=\tfrac43aT_0^3. ((a)) Starting from the exergy per unit volume x=(u−u0)−T0(s−s0)x=(u-u_0)-T_0(s-s_0), show that x/u=1−43(T0/T)+13(T0/T)4x/u=1-\tfrac43(T_0/T)+\tfrac13(T_0/T)^4. ((b)) Show that this fraction is zero when T=T0T=T_0, and explain why it must be. ((c)) Show that the Carnot factor exceeds it by 13(T0/T)[1−(T0/T)3]\tfrac13(T_0/T)\left[1-(T_0/T)^3\right], positive for every T>T0T>T_0.

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