The methane from the landfills, Problem 10.70 , is saturated with water. What volume of "wet" methane would you have to collect at \(20.0^{\circ} \mathrm{C}\) and 742 torr to be sure that the sample contains \(244 \mathrm{~mL}\) of dry methane (also at 742 torr \() ?\)

When it comes to gases, one rule that stands as a cornerstone is Dalton's Law of Partial Pressures. In a mixture of non-reacting gases, such as methane and water vapor in our exercise, the total pressure exerted is equal to the sum of the partial pressures of each individual gas.

Dalton's Law can be represented by the formula:
\[ P_{total} = P_{1} + P_{2} + ... + P_{n} \]
where \( P_{total} \) is the total pressure of the gas mixture, and \( P_{1}, P_{2}, ..., P_{n} \) represent the partial pressures of each gas component.

Understanding this law is crucial for solving problems involving 'wet' gases. In the given problem, we deduce the specific volume of 'wet' methane needed to ensure we have the desired volume of dry methane, considering that the presence of water vapor affects the total pressure of the gas sample collected from the landfill.

The Ideal Gas Law is a pivotal equation in chemistry and physics that describes the behavior of ideal gases. It is represented by the equation:
\[ PV = nRT \]
Here, \( P \) stands for pressure, \( V \) for volume, \( n \) for number of moles, \( R \) for the ideal gas constant, and \( T \) for temperature in Kelvin.

In our exercise, this law helps to find the volume of 'wet' methane at a certain temperature and pressure. The law assumes that the gas behaves ideally, which means the gas particles do not interact with each other and occupy no volume. However, it's important to note that in real-world applications, such as landfill methane collection, gases are not perfectly ideal. Despite this, the Ideal Gas Law provides a good approximation which allows us to perform necessary calculations for practical situations.

Vapor pressure is a term frequently encountered in the study of gases and liquids. It's the pressure exerted by a vapor in thermodynamic equilibrium with its condensed phases at a given temperature.

In the context of the exercise, water's vapor pressure at \(20.0^{\text{o}} C\) is a crucial value needed to calculate the volume of 'wet' methane. Vapor pressure can typically be found in reference tables and indicates the tendency of a substance to evaporate. At equilibrium, vapor pressure remains constant, and it is unique to each substance at a particular temperature.

Knowing the vapor pressure of water allows us to subtract it from the total pressure, isolating the partial pressure of methane in the 'wet' gas sample. This step is essential for accurately applying Dalton's Law and ultimately using the Ideal Gas Law to solve for the desired volume of methane.