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When comparing gases at Standard Temperature and Pressure (STP), the number of molecules in a container depends on the conditions defined by the Ideal Gas Law. The formula \( PV = nRT \) helps us determine the moles of gas present. Using this equation, we can find that at STP, 1 mole of any gas occupies 22.4 liters.Therefore, each 1-liter flask of \(N_2\) and \(CH_4\) contains the same number of moles. Since both gases are in equal volume and under identical conditions, they have an equal number of molecules. This happens because of Avogadro's principle, which states that equal volumes of gases at the same temperature and pressure contain an equal number of molecules.Hence, despite \(N_2\) and \(CH_4\) having different molecular structures and molar masses, their amount in moles and, therefore, the number of molecules are equivalent in their respective flasks.

Density is a measure of how much mass is contained in a given volume. The formula for the density of a gas is generally given by the equation:\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]To determine the density of \(N_2\) and \(CH_4\) under the same conditions, we need to consider their molar masses. - For \(N_2\), the molar mass is 28 g/mol.- For \(CH_4\), the molar mass is 16 g/mol.Because both gases occupy a 1-liter volume at STP, we can use these molar masses to calculate their densities. Multiply the number of moles of each gas (at STP, 1 mole in 22.4 liters) by the respective molar mass, and then divide by the flask volume.Consequently, the density of \(CH_4\) is less than that of \(N_2\), due to its smaller molar mass. This explains why lighter gases have lower densities when compared to heavier gases under the same conditions.

The average kinetic energy of gas molecules is directly related to temperature, irrespective of the type of gas or its molar mass. It is given by the expression:\[ KE_{\text{avg}} = \frac{3}{2} k_B T \]where \(k_B\) is Boltzmann's constant and \(T\) is the temperature. Because both \(N_2\) and \(CH_4\) are at STP, they are at the same temperature, leading to identical average kinetic energies of their molecules.This concept emphasizes that temperature, not the type or mass of gas molecules, dictates the kinetic energy. Therefore, any gas at a given temperature (in an ideal setup) will have molecules with the same average kinetic energy. Thus, understand that while the masses of \(N_2\) and \(CH_4\) molecules differ, their movements at the same temperature are just as energetically active.

Graham's Law of Effusion describes how different gases diffuse or effuse through small openings, such as a pinhole. It is mathematically stated as:\[ \frac{\text{Rate of effusion of gas 1}}{\text{Rate of effusion of gas 2}} = \sqrt{\frac{Molar mass_{2}}{Molar mass_{1}}} \]Given the problem setup, for \(N_2\) and \(CH_4\), we will compare their rates of effusion. Since \(N_2\) has a greater molar mass (28 g/mol) compared to \(CH_4\) (16 g/mol), \(CH_4\) will effuse faster.Graham's Law highlights that lighter gases effuse more quickly than heavier ones due to their lower molar masses. This property is why gases like helium and hydrogen, being very light, diffuse through materials at a faster rate than heavier gases.