Question

Consider the following setup, which shows identical containers connected by a tube with a valve that is presently closed. The container on the left has \(1.0 \mathrm{~mol}\) of \(\mathrm{H}_{2}\) gas; the container on the right has \(1.0 \mathrm{~mol}\) of \(\mathrm{O}_{2}\). Which container has the greatest density of gas? Whieh container-has molecules that are moving at a faster average molecular speed? Which container has more molecules? If the valve is opened, will the pressure in each of the containers change? If it does, how will it change (increase, decrease, or no change)? \(2.0 \mathrm{~mol}\) of Ar is added to the system with the valve open. What fraction of the total pressure will be due to the \(\mathrm{H}_{2} ?\)

Short answer

\( \mathrm{O}_2 \) has higher density; \( \mathrm{H}_2 \) has faster molecules; same molecule count; pressure decreases; \( \frac{1}{4} \) of pressure is \( \mathrm{H}_2 \).

Step-by-step solution

Determine the Density of Gases

Density is mass divided by volume. For identical containers, the volume is constant. Density \( \rho \) can be calculated as \( \rho = \frac{m}{V} \). Since each container holds 1 mol of respective gas, the molar mass determines density: \( \mathrm{H}_2 \) (2 g/mol) and \( \mathrm{O}_2 \) (32 g/mol). Thus, \( \mathrm{O}_2 \) has greater density.

Determine Average Molecular Speed

Using the formula for root-mean-square speed: \( \bar{u} = \sqrt{\frac{3RT}{M}} \). Since temperature is constant, \( R \) is the gas constant, and \ T \ is temperature. \( M \) is molar mass: \( \mathrm{H}_2 \) (2 g/mol) and \( \mathrm{O}_2 \) (32 g/mol). Lighter molecules (\( \mathrm{H}_2 \)) move faster.

Count Molecules in Each Container

Since both containers have 1 mol of gas, they contain the same number of molecules, \( 6.022 \times 10^{23} \) molecules (Avogadro's number).

Analyze Pressure Change Upon Opening the Valve

The pressure of an ideal gas is given by \(( P = \frac{nRT}{V} )\). On opening the valve between identical containers, total moles will be distributed between them, leading to equalization. This will result in a decrease in individual pressure of each container as the volume effectively doubles.

Calculate Fraction of Pressure Due to Hydrogen

Total moles in the system becomes: \( 1 \text{ mol } \mathrm{H}_2 + 1 \text{ mol } \mathrm{O}_2 + 2 \text{ mol } \mathrm{Ar} = 4 \). The partial pressure of \( \mathrm{H}_2 \) is due to its mole fraction: \( \frac{1 \text{ mol } \mathrm{H}_2}{4 \text{ mol total}} = \frac{1}{4} \). Thus, \( \mathrm{H}_2 \) makes up \( \frac{1}{4} \) of the total pressure.

Key concepts

Gas Density

Gas density is a crucial concept when analyzing gases. In simple terms, gas density is the mass of the gas divided by its volume. When dealing with gases in identical containers, like in our scenario, the volume remains constant. Therefore, the density primarily depends on the molar mass of the gas.

To determine which container has the greater gas density, we consider the molar masses of the gases. Hydrogen ( H₂) has a molar mass of 2 g/mol, whereas Oxygen (O₂) has a molar mass of 32 g/mol. Given that each container holds 1.0 mol of its respective gas, the density will be higher for the gas with the higher molar mass. So, in this exercise, the O₂ container has greater density due to its higher molar mass. Other observations are:

• Higher molar mass results in higher density for the same molar quantity.
• When comparing gases, always consider both the molar mass and the fixed volume to determine density.

Molecular Speed

Molecular speed gives us insight into how fast molecules move within a gas. The average molecular speed can be calculated using the root-mean-square speed formula: \(\bar{u} = \sqrt{\frac{3RT}{M}}\).This equation shows that molecular speed depends on the temperature (T), the gas constant (R), and the molar mass (M).

Since the temperature and gas constant are constant in our scenario, the molar mass is the main variable affecting speed. Hydrogen molecules are lighter with a molar mass of 2 g/mol compared to Oxygen's 32 g/mol. This difference in mass means that the lighter H₂ molecules will move faster than O₂ molecules. Key points include:

• Lighter molecules travel faster than heavier ones at the same temperature.
• The root-mean-square speed provides a way to calculate average speeds for gas molecules depending on the gas type.

Partial Pressure

Partial pressure helps us understand the individual pressure contributions of different gases within a mixture. According to Dalton's Law of Partial Pressures, the total pressure of a gas mixture is the sum of the partial pressures of each individual gas.

When the valve between the containers is opened, gases mix, and the pressure in each container is influenced by the combined number of moles of different gases. If 2.0 mol of Argon (Ar) is added, the total mol count becomes 4 mol: 1 mol each of H₂ and O₂, and 2 mol of Ar.

The partial pressure from a gas is calculated by its mole fraction in the mixture. Therefore, H₂, with 1 mol out of the total 4 mol, contributes a fraction of \(\frac{1}{4}\) to the total pressure. Consider these points:

• Partial pressure reflects the proportionate pressure each gas exerts within a mixture.
• Adding more gases changes the total pressure and the partial pressure components.

Avogadro's Number

Named after Amedeo Avogadro, Avogadro's number defines the number of constituent particles, usually atoms or molecules, in one mole of a substance. This number is fixed at approximately \(6.022 \times 10^{23}\) particles per mole.

In the context of our exercise, each container holding 1.0 mol of gas means that each has \(6.022 \times 10^{23}\) molecules, despite having different types of gas. This constant allows chemists to count atoms in a macroscopic amount of substances. Important aspects include:

• It serves as a bridge between the macroscopic and molecular scales.
• Avogadro's number is fundamental in calculations involving moles, converting to the number of molecules.