If an amount of energy Q''_o (J/m²) is released instantaneously, as, for example, from a pulsed laser, and it is absorbed by the surface of a semi-infinite medium, with no attendant losses to the surroundings, the subsequent temperature distribution in the medium is where T_i is the initial, uniform temperature of the medium. Consider an analogous mass transfer process involving deposition of a thin layer of phosphorous (P) on a silicon (Si) wafer at room temperature. If the wafer is placed in a furnace, the diffusion of Pinto Si is significantly enhanced by the high-temperature environment, A Si wafer with 1-μm-thick P film is suddenly placed in a furnace at 1000^oC, and the resulting distribution of P is characterized by an expression of the form where M''_p,0, is the molar area density (k mol/m²) of P associated with the film of concentration C p and thickness do.

(a) Explain the correspondence between variables in the analogous temperature and concentration distributions.

(b) Determine the mole fraction of P at a depth of 0.1μm in the Si after 30s. The diffusion coefficient is D_{p – Si} = 1.2 x 10^-17 m²/s. The mass densities of P and Si are 2000 and 2300 kg/m³, respectively, and their molecular weights are 30.97 and 28.09 kg/k mol.

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Tix, 1) — т, — pe(Tat)2 exp(-x³14a1) 1/2 Tx,0) = T; P, c, k Tx,) Limit qAr = Q" (At - 0) Mp, Cp(x, t) = (TDp-st)2 exp( -x/4Dp-s)