Write down the relations for steady one-dimensional heat conduction and mass diffusion through a plane wall, and identify the quantities in the two equations that correspond to each other.

For one-dimensional steady heat conduction through a plane wall, we use Fourier's law, which is the basic governing equation of heat conduction. The equation is given as: \begin{equation} q = -k\dfrac{dT}{dx} \end{equation} where, \(q\): Heat flux (W/m²), \(k\): Thermal conductivity of the material (W/mK), \(\frac{dT}{dx}\): Temperature gradient in the direction of heat flow (K/m).

For one-dimensional steady mass diffusion through a plane wall, we use Fick's first law, which is the basic relation for mass diffusion. The equation is given as: \begin{equation} J = -D\dfrac{dC}{dx} \end{equation} where, \(J\): Mass flux (kg/m²s), \(D\): Diffusivity of the material (m²/s), \(\frac{dC}{dx}\): Concentration gradient in the direction of mass flow (kg/m³).

Now we will compare the two equations and identify the parallel quantities in both relations: 1. Heat flux \(q\) and mass flux \(J\) - these parameters represent the flow rate of heat and mass, respectively. 2. Temperature gradient \(\frac{dT}{dx}\) and concentration gradient \(\frac{dC}{dx}\) - these parameters relate to the driving forces for heat flow and mass flow in the system. 3. Thermal conductivity \(k\) and diffusivity \(D\) - these parameters define the capacity of the medium to conduct heat and allow mass diffusion, respectively. In conclusion, Fourier's law and Fick's law both describe the flow of heat and mass through a plane wall under steady-state conditions. The key quantities in both equations that correspond to each other are heat flux (q) and mass flux (J), temperature (T) gradient and concentration (C) gradient, and thermal conductivity (k) and diffusivity (D).