A student claims that "doubling the temperature doubles everything thermal"

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Condensed Matter, Materials & Soft Matter · Origins of quantum theory

Thermal Physics, Ch Photons and phonons: blackbody radiation and the heat capacity of solids

Problem

A student claims that "doubling the temperature doubles everything thermal". Test the claim on two quantities of this chapter, using a blackbody furnace wall and a nickel sample. The furnace wall is a black surface, radiating freely into cold surroundings whose return radiation may be neglected. The nickel sample is

1.00\,\mathrm{mol}

of a Debye solid with Debye temperature

\theta_D = 450\,\mathrm{K}

; at low temperature its heat capacity is

C_V = (12\pi^4/5)Nk_B(T/\theta_D)^3

, which per mole is

1944\,(T/\theta_D)^3\,\mathrm{J\,K^{-1}\,mol^{-1}}

, and at high temperature it tends to

3Nk_B

. ((a)) Find the flux radiated by the wall at

900\,\mathrm{K}

, and the factor by which the flux changes when the wall is taken to

1800\,\mathrm{K}

. ((b)) Find

C_V

of the nickel at

9.0\,\mathrm{K}

, and the factor by which it changes on warming to

18\,\mathrm{K}

. ((c)) Treating both

900\,\mathrm{K}

and

1800\,\mathrm{K}

as high enough above

\theta_D

for the high-temperature limit to apply, find the factor by which

C_V

of the nickel changes on going from one to the other. Take

\sigma_{\rm SB} = 5.67\times10^{-8}\,\mathrm{W\,m^{-2}\,K^{-4}}

and

R = 8.314\,\mathrm{J\,K^{-1}\,mol^{-1}}

. Ignore melting, and ignore the electronic contribution to the heat capacity.

Related theory

Quantum Mechanics

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