Question 19.4: A membrane with a permeability of 1.00 × 10^−6 m^2 separates...
A membrane with a permeability of 1.00 × 10^{−6} \text{m}^2 separates two chambers filled with carbon dioxide gas. The gas has a temperature of 300. \text{K} on one side of the membrane and 305 \text{K} on the other side. The osmotic heat conductivity (k_o) of the membrane with \text{CO}_2 is 2.00 × 10^4 \text{m}^2 /\text{s} and the viscosity of the \text{CO}_2 is 1.50 × 10^{−5} \text{kg/(m.s)} . Determine the steady state thermomolecular pressure difference across the membrane.
Since this is a steady state problem, we can use the result given in Eq. (19.52),
(\frac{dp/dx}{dT/dx})\mid_{J_M=0} = (\frac{dp}{dT})\mid_{J_M=0} = – \frac{μk_o}{Tk_p} (19.52)
dp = − \frac{μk_o}{k_p(\frac{dT}{T})}
and, assuming μ, k_o, and k_p are all constants over the small temperature range of 300. to 305 \text{K}, we can integrate this equation to find
p_2 − p_1 = −(\frac{μk_o}{k_p}) \text{ln} (\frac{T_2}{T_1}) = – [\frac{(1.50 × 10^{−5} \text{kg/(m.s)})(2.00 × 10^4 \text{m}^2/\text{s})}{1.00 × 10^{−6} \text{m}^2}] \text{ln} (\frac{305}{300.}) = −4960 \text{N/m}^2