Fall 2007
Fall 2007 1 Mech
Fall 2007 2 Mech
Fall 2007 3 EM
Fall 2007 4 EM
Fall 2007 5 QM
Fall 2007 6 QM
Fall 2007 7 SM
Fall 2007 8 SM
Fall 2007 9 Misc
Fall 2007 10 Misc
Fall 2007 11 Mech
Fall 2007 12 Mech
Fall 2007 13 EM
Fall 2007 14 EM
Fall 2007 15 QM
Fall 2007 16 QM
Fall 2007 17 SM
Fall 2007 18 SM
Fall 2007 19 Misc
Fall 2007 20 Misc
Ideal gas has density n, molecular mass m, initial temperature T0, and collisional cross-section $\sigma$. At time t = 0 a small amount of heat Q is released in a neighbourhood of a point inside the gas. Determine how the temperature difference T - T0 at that point decays at large time t. No detailed calculations are necessary; rather, use estimates and dimensional analysis to derive the scaling trend.
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Answer
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Small heat disturbances spread by thermal conductivity. At large time the size of the heated region is $R(t) \sim \sqrt{D_T t}$, where D is the thermal diffusion coefficient. In an ideal gas DT is of the same order as the regular diffusion coefficient $D = v_T l/3$, where vT is the thermal velocity and $l = 1/\sigma n$ is the mean-free path. The conservation of energy requires
(1)where $c \sim k$ is the specific heat per molecule. Therefore,
(2)