Question 508: Thermodynamics & KTG

Question

Consider a spherical shell of radius $$R$$ at temperature $$T$$. The black body radiation inside it can be considered as an ideal gas of photons with internal energy per unit volume $$u = {U \over V}\, \propto \,{T^4}$$ and pressure $$p = {1 \over 3}\left( {{U \over V}} \right)$$ . If the shell now undergoes an adiabatic expansion the relation between $$T$$ and $$R$$ is:

Accepted Answer

As, $$P = {1 \over 3}\left( {{U \over V}} ight)$$
But $$\,\,\,\,$$ $${U \over V} = KT{}^4$$
So, $$\,\,\,\,\,P = {1 \over 3}K{T^4}$$
or $$\,\,\,\,{{uRT} \over V} = {1 \over 3}K{T^4}\,\,\,\,$$
$$\left[ \, ight.$$ As $$PV = uRT$$ $$\left. \, ight]$$
$${4 \over 3}\pi {R^3}{T^3} = $$ $$constant$$
$$	herefore$$ $$\,\,\,\,$$ $$T \propto {1 \over R}$$

Answer

$$T\, \propto {1 \over R}$$

Answer

$$T\, \propto {1 \over {{R^3}}}$$

Answer

$$T\, \propto \,{e^{ - R}}$$

Answer

$$T\, \propto \,{e^{ - 3R}}$$

Thermodynamics & KTG

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