One of the oldest membrane materials used for dialysis is porous cellophane, a thin transparent sheet made of regenerated cellulose. Typical values of parameters for commercial cellophane membranes are as follows: thickness \(=80 \mathrm{~m}\), porosity \(=0.45\), tortuosity \(=5.0\), and pore diameter =40 Å (Seader and Henley, 2006).

(a) Estimate the effective diffusivity of urea through this membrane at \(298 \mathrm{~K}\). The diffusivity of urea in dilute aqueous solution at this temperature is \(1.38 \times 10^{-5} \mathrm{~cm}^{2} / \mathrm{s}\). The molecular diameter of urea is 5.28 Å.

(b) The artificial kidney of Example 1.3 uses a cellophane membrane. If the urea concentration difference across the membrane is \(25 \mathrm{mg} / \mathrm{dL}\), estimate the membrane area required.

Data From Example 1.3:-

The primary functions of the kidneys are to remove waste products (such as urea,uric acid, and creatinine), and to maintain the fluid and salt balance in the blood.Blood consists of two parts: blood cells, mostly red (45% by volume), and plasma(55% by volume). Urea, uric acid, creatinine, and water are all found in the plasma.If the kidneys fail, wastes start to accumulate, and the body becomes overloaded with fluid. Fortunately, patients with renal failure can use an external dialysis machine, also known as an artificial kidney, to clean the blood. The cleaning of the blood in the artificial kidney is due to the difference in toxin concentrations between the blood and the dialysis fluid. Semipermeable membranes in the machine selectively allow toxins to pass from the blood to the dialysis fluid. During a dialysis procedure, a patient was connected to the machine for 4 h. The blood was pumped through the artificial kidney at the rate of 1200 mL/min.
The partially cleansed blood was returned to the patient’s body, and the wastes removed were collected in the used dialysis fluid. During the procedure, the patient’s kidneys were completely inactive. A total of 1540 g of urine was collected with an urea concentration of 1.3% by weight. A sample of the blood plasma was analyzed before the dialysis and found to contain 155.3 mg/dL of urea. The specific gravity of the plasma was measured at 1.0245. Calculate:

(a) The urea removal efficiency by the artificial kidney.

(b) The urea concentration in the plasma of the cleansed blood, in mg/dL.