An emphysema patient is breathing pure \(\mathrm{O}_{2}\) through a face mask. The cylinder of \(\mathrm{O}_{2}\) contains \(0.60 \mathrm{ft}^{3}\) of \(\mathrm{O}_{2}\) gas at a pressure of \(2200 \mathrm{lb} / \mathrm{in}^{2} .\) (a) What volume would the oxygen occupy at atmospheric pressure (and the same temperature)? (b) If the patient takes in \(8 \mathrm{L} / \mathrm{min}\) of \(\mathrm{O}_{2}\) at atmospheric pressure, how long will the cylinder last?

First, we need to convert the volume of the oxygen cylinder from cubic feet to liters and the pressure from pounds per square inch to Pascals (Pa). 1. Volume conversion (cubic feet to liters): Use the conversion, 1 cubic foot = 28.3168 liters. \(0.60 \, \mathrm{ft}^3 \times \frac{28.3168 \, \mathrm{L}}{1 \, \mathrm{ft}^3} ≈ 16.99 \, \mathrm{L}\) 2. Pressure conversion (pounds per square inch to Pascals): Use the conversion, 1 psi = 6894.76 Pa. \(2200 \, \frac{\mathrm{lb}}{\mathrm{in}^2} \times \frac{6894.76 \, \mathrm{Pa}}{1 \, \frac{\mathrm{lb}}{\mathrm{in}^2}} ≈ 15089529 \, \mathrm{Pa}\) Now we have the volume \(V_1 = 16.99 \, \mathrm{L}\) and pressure \(P_1 = 15089529 \, \mathrm{Pa}\) of the oxygen in the cylinder.

Now, we need to find the volume of the oxygen at atmospheric pressure, given \(P_2 = 101325 \, \mathrm{Pa}\). Using Boyle's Law: \(P_1 V_1 = P_2 V_2\) Rearrange to solve for \(V_2\): \(V_2 = \frac{P_1 V_1}{P_2}\) Plug in the known values: \(V_2 = \frac{15089529 \, \mathrm{Pa} \times 16.99 \, \mathrm{L}}{101325 \, \mathrm{Pa}} ≈ 2537.6 \, \mathrm{L}\) So the volume of the oxygen at atmospheric pressure would be approximately \(2537.6 \, \mathrm{L}\).

The patient's oxygen consumption rate is given: \(8 \, \mathrm{L} / \mathrm{min}\). To find the duration the cylinder will last, we can divide the volume of oxygen at atmospheric pressure by the consumption rate: Duration = \(\frac{\text{Volume of oxygen at atmospheric pressure}}{\text{Oxygen consumption rate}}\) Plug in the known values: Duration = \(\frac{2537.6 \, \mathrm{L}}{8 \, \mathrm{L/min}} ≈ 317.2 \, \mathrm{min}\) The oxygen cylinder will last approximately \(317.2 \, \mathrm{min}\).