Not Boyle's Law

Abstract

To the Editor We are often looking for ways to make the physical behavior of gases more understandable to our trainees, so we were initially pleased to find the 2004 letter by Atlas entitled “A Method to Quickly Estimate Remaining Time of an Oxygen E Cylinder.”1 Although the letter does describe a formula that correctly estimates the desired parameter, the derivation is not valid. Atlas stated that “the pressure–volume relationship for oxygen stored in an E cylinder is characterized by Boyle's Law.” Boyle's Law states that “for a given mass at a constant temperature, the volume of a gas is inversely proportional to the pressure.”2 Boyle's Law does not apply when the mass of a gas is changing and the volume is constant, as is the case when oxygen is being drawn from an E-cylinder. It is easiest to derive the relationship between the pressure and amount of gas in the cylinder using the ideal gas law, PV = nRT, where P is pressure, V is volume, n is the amount of gas (typically in moles), T is temperature, and R is a constant. If we ignore the slight drop in temperature as gas is withdrawn from the cylinder, then we can consider V, R, and T to be constant. We can thus restate the relationship between pressure and amount of gas in an E cylinder as P/n = k. Atlas makes the claim that PiVi = PrVr.. For an E-cylinder, P is initially 1900 psi, but what is n? Although n is typically measured in moles, any measure of amount of gas will suffice. We choose to think of the amount of gas in a cylinder as the volume the gas will occupy when it expands to atmospheric pressure, a convenient unit for clinical purposes. (One mole of an ideal gas occupies 22.4 L at standard temperature and pressure.2) We say that an E-cylinder contains 660 L of oxygen when full. Thus P/n = 1900/660 = 2.88. The constant 2.88 relates any pressure in an E-cylinder, in psi, to the amount of gas expressed, in liters at atmospheric pressure. This can be rearranged algebraically to calculate the amount in liters, n, at any pressure, P, by n = 0.347 × P. For example, if P is 950 psi (half empty), the amount, n, is 0.347 × 950 = 330 L. Because we are trying to determine how long the amount will last at a given flow rate, we can convert the amount in the cylinder, n, to time, t, by noting that the amount consumed is time, t, times oxygen flow rate, Q. Substituting this yields the relationship Q × t = 0.347 × P, which can be rearranged to solve for t: t = 0.347 × P/Q. This is the same formula that Atlas published as equation 4, although Atlas rounded the numbers in a coarser manner, and expressed t in hours. Atlas started his derivation with an incorrect representation of Boyle's Law: pressure (initial) × volume (initial) = pressure (residual) × volume (residual). The volume of the cylinder does not change, the amount of gas changes, so the statement is incorrect. Atlas stated that the flow, Q, times remaining time, t, equals the initial amount (e.g., 600 L—note different rounding). This is also incorrect. It is the residual amount that is equal to Q × t. Atlas correctly used 2000 psi as the initial pressure (again, note slightly different rounding), but then inexplicably uses 600 L as the residual amount (it is the initial amount). This produces the following equation: 2000 × Q × t = 600 × P. Solving for t generates t = 0.3 × P/Q, which is correct. Thus, Atlas derived the correct equation, excluding rounding differences, by virtue of 3 offsetting errors.