Question: Calculate the concentration gradient for a $4.3$ mm-thick sheet of palladium at $500^\circ$C. Assume that the concentrations at the high- and low-pressure sides of the plates are $2.3$ and $0.35$ kg/m$^3$ of hydrogen per cubic meter of palladium, and that steady-state conditions have been attained.
1. **State the problem:** We need to calculate the concentration gradient $\frac{\Delta C}{\Delta x}$ for hydrogen diffusing through a palladium sheet of thickness $4.3$ mm, with given concentrations at each side.
2. **Given data:**
- Thickness of sheet, $\Delta x = 4.3$ mm $= 4.3 \times 10^{-3}$ m
- Concentration at high-pressure side, $C_1 = 2.3$ kg/m$^3$
- Concentration at low-pressure side, $C_2 = 0.35$ kg/m$^3$
3. **Formula:** The concentration gradient under steady-state conditions is given by:
$$\frac{\Delta C}{\Delta x} = \frac{C_1 - C_2}{\Delta x}$$
4. **Calculate the concentration difference:**
$$C_1 - C_2 = 2.3 - 0.35 = 1.95 \text{ kg/m}^3$$
5. **Calculate the concentration gradient:**
$$\frac{\Delta C}{\Delta x} = \frac{1.95}{4.3 \times 10^{-3}}$$
6. **Simplify the fraction:**
$$\frac{\Delta C}{\Delta x} = \frac{1.95}{\cancel{4.3 \times 10^{-3}}} \times \frac{\cancel{1}}{1} = 453.49 \text{ kg/m}^4$$
7. **Interpretation:** The concentration gradient is approximately $453.49$ kg/m$^4$, which means the concentration changes by this amount per meter thickness of the palladium sheet.
**Final answer:**
$$\boxed{\frac{\Delta C}{\Delta x} = 453.49 \text{ kg/m}^4}$$