Diffusion of Methane Through Helium. A gas of CH₄ and He is contained in a tube at 101.32 kPa pressure and 298 K. At one point the partial pressure of methane is p_A1 = 60.79 kPa, and at a point 0.02 m distance away, p_A2 = 20.26 kPa. If the total pressure is constant throughout the tube, calculate the flux of CH₄ (methane) at steady state for equimolar counterdiffusion.

Diffusion of CO₂ in a Binary Gas Mixture. The gas CO₂ is diffusing at steady state through a tube 0.20 m long having a diameter of 0.01 m and containing N₂ at 298 K. The total pressure is constant at 101.32 kPa. The partial pressure of CO₂ is 456 mm Hg at one end and 76 mm Hg at the other end. The diffusivity D_AB is 1.67 × 10⁻⁵ m²/s at 298 K. Calculate the flux of CO₂ in cgs and SI units for equimolar counterdiffusion.

Equimolar Counterdiffusion of a Binary Gas Mixture. Helium and nitrogen gas are contained in a conduit 5 mm in diameter and 0.1 m long at 298 K and a uniform constant pressure of 1.0 atm abs. The partial pressure of He at one end of the tube is 0.060 atm and at the other end is 0.020 atm. The diffusivity can be obtained from Table 6.2-1. Calculate the following for steady-state equimolar counterdiffusion:

Equimolar Counterdiffusion of NH₃ and N₂ at Steady State. Ammonia gas (A) and nitrogen gas (B) are diffusing in counterdiffusion through a straight glass tube 2.0 ft (0.610 m) long with an inside diameter of 0.080 ft (24.4 mm) at 298 K and 101.32 kPa. Both ends of the tube are connected to large mixed chambers at 101.32 kPa. The partial pressure of NH₃ is constant at 20.0 kPa in one chamber and 6.666 kPa in the other. The diffusivity at 298 K and 101.32 kPa is 2.30 × 10⁻⁵ m²/s.

Diffusion of A Through Stagnant B and Effect of Type of Boundary on Flux. Ammonia gas is diffusing through N₂ under steady-state conditions with N₂ nondiffusing since it is insoluble in one boundary. The total pressure is 1.013 × 10⁵ Pa and the temperature is 298 K. The partial pressure of NH₃ at one point is 1.333 × 10⁴ Pa, and at the other point 20 mm away it is 6.666 × 10³ Pa. The D_AB for the mixture at 1.013 × 10⁵ Pa and 298 K is 2.30 × 10⁻⁵ m²/s.

Diffusion of Methane Through Nondiffusing Helium. Methane gas is diffusing in a straight tube 0.1 m long containing helium at 298 K and a total pressure of 1.01325 × 10⁵ Pa. The partial pressure of CH₄ is 1.400 × 10⁴ Pa at one end and 1.333 × 10³ Pa at the other end. Helium is insoluble in one boundary, and hence is nondiffusing or stagnant. The diffusivity is given in Table 6.2-1. Calculate the flux of methane in kg mol/sm² at steady state.

Mass Transfer from a Naphthalene Sphere to Air. Mass transfer is occurring from a sphere of naphthalene having a radius of 10 mm. The sphere is in a large volume of still air at 52.6°C and 1 atm abs pressure. The vapor pressure of naphthalene at 52.6°C is 1.0 mm Hg. The diffusivity of naphthalene in air at 0°C is 5.16 × 10⁻⁶ m²/s. Calculate the rate of evaporation of naphthalene from the surface in kg mol/s · m². [Note: The diffusivity can be corrected for temperature using the temperature-correction factor from the Fuller et al. equation (6.2-45).]

Estimation of Diffusivity of a Binary Gas. For a mixture of ethanol (CH₃CH₂OH) vapor and methane (CH₄), predict the diffusivity using the method of Fuller et al.

Diffusion Flux and Effect of Temperature and Pressure. Equimolar counterdiffusion is occurring at steady state in a tube 0.11 m long containing N₂ and CO gases at a total pressure of 1.0 atm abs. The partial pressure of N₂ is 80 mm Hg at one end and 10 mm at the other end. Predict the D_AB by the method of Fuller et al.

Evaporation Losses of Water in Irrigation Ditch. Water at 25°C is flowing in a covered irrigation ditch below ground. Every 100 ft there is a vent line 1.0 in. inside diameter and 1.0 ft long to the outside atmosphere at 25°C. There are 10 vents in the 1000-ft ditch. The outside air can be assumed to be dry. Calculate the total evaporation loss of water in lb_m/d. Assume that the partial pressure of water vapor at the surface of the water is the vapor pressure, 23.76 mm Hg at 25°C. Use the diffusivity from Table 6.2-1.

Time to Completely Evaporate a Sphere. A drop of liquid toluene is kept at a uniform temperature of 25.9°C and is suspended in air by a fine wire. The initial radius r₁ = 2.00 mm. The vapor pressure of toluene at 25.9°C is P_A1 = 3.84 kPa and the density of liquid toluene is 866 kg/m³.

Diffusion in a Nonuniform Cross-Sectional Area. The gas ammonia (A) is diffusing at steady state through N₂ (B) by equimolar counterdiffusion in a conduit 1.22 m long at 25°C and a total pressure of 101.32 kPa abs. The partial pressure of ammonia at the left end is 25.33 kPa and at the other end 5.066 kPa. The cross section of the conduit is in the shape of an equilateral triangle, the length of each side of the triangle being 0.0610 m at the left end and tapering uniformally to 0.0305 m at the right end. Calculate the molar flux of ammonia. The diffusivity is D_AB = 0.230 × 10⁻⁴ m²/s.

Diffusion of A Through Stagnant B in a Liquid. The solute HCl (A) is diffusing through a thin film of water (B) 2.0 mm thick at 283 K. The concentration of HCl at point 1 at one boundary of the film is 12.0 wt % HCl (density ρ₁ = 1060.7 kg/m³), and at the other boundary at point 2 it is 6.0 wt % HCl (ρ₂ = 1030.3 kg/m³). The diffusion coefficient of HCl in water is 2.5 × 10⁻⁹ m²/s. Assuming steady state and one boundary impermeable to water, calculate the flux of HCl in kg mol/s · m².

Diffusion of Ammonia in an Aqueous Solution. An ammonia (A)–water (B) solution at 278 K and 4.0 mm thick is in contact at one surface with an organic liquid at this interface. The concentration of ammonia in the organic phase is held constant and is such that the equilibrium concentration of ammonia in the water at this surface is 2.0 wt % ammonia (density of aqueous solution 991.7 kg/m³) and the concentration of ammonia in water at the other end of the film 4.0 mm away is 10 wt % (density 961.7 kg/m³). Water and the organic are insoluble in each other. The diffusion coefficient of NH₃ in water is 1.24 × 10⁻⁹ m²/s.

Estimation of Liquid Diffusivity. It is desired to predict the diffusion coefficient of dilute acetic acid (CH₃COOH) in water at 282.9 K and at 298 K using the Wilke–Chang method. Compare the predicted values with the experimental values in Table 6.3-1.

Estimation of Diffusivity of Methanol in H₂O. The diffusivity of dilute methanol in water has been determined experimentally to be 1.26 × 10⁻⁹ m²/s at 288 K.

Estimation of Diffusivity of Electrolyte NaOH. Dilute NaOH is diffusing in an aqueous solution. Do as follows:

Estimation of Diffusivity of LaCl₃ and Temperature Effect. Estimate the diffusion coefficient of the salt LaCl₃ in dilute aqueous solution at 25°C and at 35°C. Also, calculate the diffusion coefficient for the ions La³⁺ and Cl⁻ at 25°C. What percent increase in rate of diffusion will occur in going from 25 to 35°C if the same dilute concentration difference is present at both temperatures?

Prediction of Diffusivity of Enzyme Urease in Solution. Predict the diffusivity of the enzyme urease in a dilute solution in water at 298 K using the modified Polson equation and compare the result with the experimental value in Table 6.4-1.

Diffusion of Sucrose in Gelatin. A layer of gelatin in water 5 mm thick containing 5.1 wt % gelatin at 293 K separates two solutions of sucrose. The concentration of sucrose in the solution at one surface of the gelatin is constant at 2.0 g sucrose/100 mL solution, and 0.2 g/100 mL at the other surface. Calculate the flux of sucrose in kg sucrose/s·m² through the gel at steady state.

Diffusivity of Oxygen in Protein Solution. Oxygen is diffusing through a solution of bovine serum albumin (BSA) at 298 K. Oxygen has been shown not to bind to BSA. Predict the diffusivity D_AP of oxygen in a protein solution containing 11 g protein/100 mL solution. (Note: See Table 6.3-1 for the diffusivity of O₂ in water.)

Diffusion of Uric Acid in Protein Solution and Binding. Uric acid (A) at 37°C is diffusing in an aqueous solution of proteins (P) containing 8.2 g protein/100 mL solution. Uric acid binds to the proteins, and over the range of concentrations present, 1.0 g mol of acid binds to the proteins for every 3.0 g mol of total acid present in the solution. The diffusivity D_AB of uric acid in water is 1.21 × 10⁻⁵ cm²/s and D_P = 0.091 × 10⁻⁵ cm²/s.

Diffusion of CO₂ Through Rubber. A flat plug 30 mm thick having an area of 4.0 × 10⁻⁴ m² and made of vulcanized rubber is used for closing an opening in a container. The gas CO₂ at 25°C and 2.0 atm pressure is inside the container. Calculate the total leakage or diffusion of CO₂ through the plug to the outside in kg mol CO₂/s at steady state. Assume that the partial pressure of CO₂ outside is zero. From Barrer (B5) the solubility of the CO₂ gas is 0.90 m³ gas (at STP of 0°C and 1 atm) per m³ rubber per atm pressure of CO₂. The diffusivity is 0.11 × 10⁻⁹ m²/s.

Leakage of Hydrogen Through Neoprene Rubber. Pure hydrogen gas at 2.0 atm abs pressure and 27°C is flowing past a vulcanized neoprene rubber slab 5 mm thick. Using the data from Table 6.5-1, calculate the diffusion flux in kg mol/s · m² at steady state. Assume no resistance to diffusion outside the slab and zero partial pressure of H₂ on the outside.

Relation Between Diffusivity and Permeability. The gas hydrogen is diffusing through a sheet of vulcanized rubber 20 mm thick at 25°C. The partial pressure of H₂ is 1.5 atm inside and 0 outside. Using the data from Table 6.5-1, calculate the following:

Loss from a Tube of Neoprene. Hydrogen gas at 2.0 atm and 27°C is flowing in a neoprene tube 3.0 mm inside diameter and 11 mm outside diameter. Calculate the leakage of H₂ through a tube 1.0 m long in kg mol H₂/s at steady state.

Diffusion Through Membranes in Series. Nitrogen gas at 2.0 atm and 30°C is diffusing through a membrane of nylon 1.0 mm thick and polyethylene 8.0 mm thick in series. The partial pressure at the other side of the two films is 0 atm. Assuming no other resistances, calculate the flux N_A at steady state.

Diffusion of CO₂ in a Packed Bed of Sand. It is desired to calculate the rate of diffusion of CO₂ gas in air at steady state through a loosely packed bed of sand at 276 K and a total pressure of 1.013 × 10⁵ Pa. The bed depth is 1.25 m and the void fraction ε is 0.30. The partial pressure of CO₂ is 2.026 × 10³ Pa at the top of the bed and 0 Pa at the bottom. Use a τ of 1.87.

Packaging to Keep Food Moist. Cellophane is being used to keep food moist at 38°C. Calculate the loss of water vapor in g/d at steady state for a wrapping 0.10 mm thick and an area of 0.200 m² when the vapor pressure of water vapor inside is 10 mm Hg and the air outside contains water vapor at a pressure of 5 mm Hg. Use the larger permeability in Table 6.5-1.

Loss of Helium and Permeability. A window of SiO₂ 2.0 mm thick and 1.0 × 10⁻⁴ m² in area is used to view the contents in a metal vessel at 20°C. Helium gas at 202.6 kPa is contained in the vessel. To be conservative use D_AB = 5.5 × 10⁻¹⁴ m²/s from Table 6.5-1.

Numerical Method for Steady-State Diffusion. Using the results from Example 6.6-1, calculate the total diffusion rate in the solid using the bottom nodes and paths of c_2,2 to c_3,2, c_2,3 to c_3,3, and so on. Compare with the other diffusion rates in Example 6.6-1.

Numerical Method for Steady-State Diffusion with Distribution Coefficient. Use the conditions given in Example 6.6-1 except that the distribution coefficient defined by Eq. (6.6-11) between the concentration in the liquid adjacent to the external surface and the concentration in the solid adjacent to the external surface is K = 1.2. Calculate the steady-state concentrations and the diffusion rates.

Spreadsheet Solution for Steady-State Diffusion. Use the conditions given in Example 6.6-1 but instead of using Δx = 0.005 m, use Δx = Δy = 0.001 m. The overall dimensions of the hollow chamber remain as in Example 6.6-1; the only difference is that more nodes will be used. Write the spreadsheet program and solve for the steady-state concentrations using the numerical method. Also, calculate the diffusion rates and compare with Example 6.6-1.

Numerical Method with Fixed Surface Concentrations. Steady-state diffusion is occurring in a two-dimensional solid as shown in Fig. 6.6-4. The grid Δx = Δy = 0.010 m. The diffusivity D_AB = 2.00 × 10⁻⁹ m²/s. At the inside of the chamber the surface concentration is held constant at 2.00 × 10⁻³ kg mol/m³. At the outside surfaces, the concentration is constant at 8.00 × 10⁻³. Calculate the steady-state concentrations and the diffusion rates per m of depth.