Question 5.1: In addition to ongoing sources, such as car engines, polluta......
into the air through isolated incidents like the rupture of a gas storage tank. Most gases are stored and transported at high pressures. A common laboratory cylinder of methane, for example, has a volume of 49.0 L and is filled to a pressure of 154 atm. Suppose that all of the CH_4 from this cylinder is released and expands until its pressure falls to 1.00 atm. What volume would the CH_4 occupy?
Strategy The problem involves a change in the conditions, here pressure and volume, of a sample of gas. We will want to use the ideal gas law because it is our model that relates the various properties of a gas to one another. So we begin by assuming that the gas will obey the ideal gas law under both the initial and final conditions. We will further assume that temperature is constant because we are not given any data to suggest otherwise. Because we are working with the same sample of gas throughout, we also know that the number of moles (n) is constant. Collect the constant terms together and work toward a solution.
In this case, n and T are constant, and R is always constant. So,
PV = nRT = constant
Therefore,
P_1V_1 = P_2V_2
Solving this for the final volume (V_2) gives
V_{2}={\frac{P_{1}V_{1}}{P_{2}}}
We know all three terms on the right: P_1 = 154 atm, V_1 = 49.0 L, and P_2 = 1.00 atm. Insert those above and solve:
V_{2}={\frac{(154\mathrm{~atm})(49.0\mathrm{~L})}{(1.00\mathrm{~atm})}}=7550\mathrm{~L}
Analyze Your Answer Our result is in liters, which is an appropriate unit for volume. The actual value is much larger than the initial volume, so we might ask if this seems sensible. Looking at the initial and final conditions, we have lowered the pressure by a large factor, and this should lead to a much larger volume. So our answer seems plausible.
Discussion Because of the high initial pressure, the assumption that the gas behaves ideally may not be very accurate. We’ll examine that assumption later in Example 5.8.
Check Your Understanding A small sample of gas is generated in a laboratory. In a 125-mL vessel, it exerts 115 torr of pressure. Then a valve is opened and the gas expands until its volume is 175 mL. If the temperature is constant throughout this process, what is the new pressure of the gas?