If an amount of energy $Q_{o}^{\prime \prime}\left(\mathrm{J} / \mathrm{m}^{2}\right)$ 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
$$
T(x, t)-T_{i}=\frac{Q_{o}^{\prime \prime}}{\rho c(\pi \alpha t)^{1 / 2}} \exp \left(-x^{2} / 4 \alpha t\right)
$$
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 $\mathrm{P}$ into Si is significantly enhanced by the high-temperature environment.
A Si wafer with $1-\mu \mathrm{m}$-thick P film is suddenly placed in a furnace at $1000^{\circ} \mathrm{C}$, and the resulting distribution of $P$ is characterized by an expression of the form
$$
C_{\mathrm{P}}(x, t)=\frac{M_{\mathrm{P}, o}^{\prime \prime}}{\left(\pi D_{\mathrm{P}-\mathrm{Si}} t\right)^{1 / 2}} \exp \left(-x^{2} / 4 D_{\mathrm{P}-\mathrm{Si}} t\right)
$$
where $M_{\mathrm{P}, o}^{\prime \prime}$ is the molar area density $\left(\mathrm{kmol} / \mathrm{m}^{2}\right)$ of $\mathrm{P}$ associated with the film of concentration $C_{\mathrm{P}}$ and thickness $d_{o}$.
(a) Explain the correspondence between variables in the analogous temperature and concentration distributions.
(b) Determine the mole fraction of $P$ at a depth of $0.1 \mu \mathrm{m}$ in the Si after $30 \mathrm{~s}$. The diffusion coefficient is $D_{\mathrm{P}-\mathrm{Si}}=1.2 \times 10^{-17} \mathrm{~m}^{2} / \mathrm{s}$. The mass densities of $\mathrm{P}$ and $\mathrm{Si}$ are 2000 and $2300 \mathrm{~kg} / \mathrm{m}^{3}$, respectively, and their molecular weights are $30.97$ and $28.09 \mathrm{~kg} / \mathrm{kmol}$.