Fall 2007 10 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.




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

\begin{align} c n R^3 (t) [T(t) - T_0] \sim Q \end{align}

where $c \sim k$ is the specific heat per molecule. Therefore,

\begin{align} T(t) - T_0 \sim \frac{Q}{k n} \left ( \frac{\sigma^2 n^2 m}{k T_0} \right )^{3/4} \frac{1}{t^{3/2}} \end{align}
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