Question

A 2.00 -L sample of ${\mathrm{O}}_2(g)$ was collected over water at a total pressure of 785 torr and ${25}^{\circ }\mathrm{C}$ . When the ${\mathrm{O}}_2(g)$ was dried (water vapor removed), the gas had a volume of 1.94 $\mathrm{L}$ at ${25}^{\circ }\mathrm{C}$ and 785 torr. Calculate the vapor pressure of water at ${25}^{\circ }\mathrm{C}.$

Short answer

The vapor pressure of water at ${25}^{\circ }\mathrm{C}$ can be determined by calculating the partial pressures of the oxygen gas and water vapor and applying Dalton's Law. First, calculate the number of moles of oxygen gas using the total pressure, initial gas mixture volume, temperature, and ideal gas constant. Then, find the partial pressure of the oxygen gas, and finally, calculate the partial pressure of the water vapor. The vapor pressure of water at ${25}^{\circ }\mathrm{C}$ is equal to the partial pressure of the water vapor: $P_{H_{2}O}=P_{total}-P_{O_2}$.

Step-by-step solution

Calculate the partial pressure of the oxygen gas

To calculate the partial pressure of the oxygen gas, we will use Dalton's Law, which states that the total pressure of a gas mixture is the sum of the partial pressures of its components: $$P_{total}=P_{O_2}+P_{H_{2}O}$$, where $P_{total}$ is the total pressure, $P_{O_2}$ is the partial pressure of the oxygen gas, and $P_{H_{2}O}$ is the partial pressure of water vapor. We will first determine the partial pressure of the oxygen gas, which can be calculated using the following formula: $$P_{O_2}{V}_{O_2}=n_{O_2}RT$$, where $P_{O_2}$ is the partial pressure of the oxygen gas, $V_{O_2}$ is the volume of the dried oxygen gas, $n_{O_2}$ is the number of moles of oxygen gas, $R$ is the ideal gas constant, and $T$ is the temperature in Kelvin. We are given: - The volume of the dried oxygen gas, $V_{O_2}$ = 1.94 L - The temperature, $T$ = 25°C = 298.15 K - The total pressure, $P_{total}$ = 785 torr - The ideal gas constant, $R$ = 62.36 L torr / K mol We can now solve for $P_{O_2}$: $$P_{O_2}=\frac{n_{O_2}RT}{V_{O_2}}$$

Calculate the number of moles of oxygen gas

To calculate the number of moles of oxygen gas, we will use the initial volume of the gas mixture (2.00 L) with the given temperature and total pressure. We will use the following formula: $$n_{O_2}=\frac{P_{total}{V}_{O_{2_{initial}}}}{RT}$$, where $V_{O_{2_{initial}}}$ is the initial volume of the gas mixture. Now we plug in the values: $$n_{O_2}=\frac{785\text{torr}\times 2.00\text{L}}{62.36\frac{\text{L torr}}{\text{K mol}}\times 298.15\text{K}}$$

Calculate the partial pressure of oxygen gas

Now we will compute the partial pressure of the oxygen gas using the number of moles of oxygen gas obtained from step 2: $$P_{O_2}=\frac{n_{O_2}RT}{V_{O_2}}$$

Calculate the partial pressure of water vapor

Once we have the partial pressure of the oxygen gas, we can use Dalton's Law to find the partial pressure of the water vapor: $$P_{H_{2}O}=P_{total}-P_{O_2}$$

Obtain the vapor pressure of water at 25°C

The partial pressure of the water vapor calculated in step 4 is equal to the vapor pressure of water at 25°C. Therefore, the vapor pressure of water at 25°C is: $$P_{H_{2}O}=P_{total}-P_{O_2}$$

Key concepts

Dalton's Law of Partial Pressure

Understanding the behavior of gas mixtures is made simpler with Dalton's Law of Partial Pressure. This law tells us that the total pressure exerted by a mixture of non-reactive gases is equal to the sum of the partial pressures of the individual gases. In simple terms, each gas in a mixture acts independently and contributes to the total pressure based on its portion.

• Formula: $P_{\text{total}}=P_1+P_2+\dots +P_n$
• Application: It's particularly useful when dealing with gases collected over water, where the gas is mixed with water vapor, leading to its total pressure being a combination of the gas and the water vapor pressures.

In the context of our exercise, Dalton's Law helps us separate the total pressure of the collected gas into the pressure of the oxygen and the pressure of the water vapor.

Ideal Gas Law

The Ideal Gas Law is a critical equation in understanding the macroscopic properties of gases. It connects the four essential gas variables: pressure ($P$), volume ($V$), temperature ($T$), and the number of moles ($n$). The law is typically represented as $PV=nRT$, where $R$ is the ideal gas constant.

• Formula: $PV=nRT$
• Assumptions: The law assumes that gases behave ideally, meaning there are no interactions between the gas molecules, and the volume occupied by the gas molecules themselves is negligible.

This law helps us determine unknown variables when others are known, making it invaluable for solving problems involving gases collected under changing conditions.

Oxygen Gas

Oxygen gas, symbolized as $O_2$, is crucial in biological and chemical processes. In our exercise, it represents the main constituent of the collected gas sample.

• Properties: As a diatomic molecule, oxygen is colorless, tasteless, and essential for respiration and combustion processes.
• Behavior: When dealing with gas collections, understanding the behavior of $O_2$ under varying conditions like pressure and temperature is key for precise calculations.

In the exercise, isolated $O_2$ helps us determine pressures separately from the water vapor when the gas is dried.

Water Vapor

Water vapor is the gaseous form of water and is always present to some extent when gases are collected over water. Its pressure, known as vapor pressure, varies with temperature.

• Vapor Pressure: This is the pressure exerted by the water vapor in a mixture and fluctuates with temperature. At ${25}^{\circ }C$, the vapor pressure can be calculated using Dalton's Law.
• Relevance: This component must be carefully considered, as it affects the total pressure in systems where gas is collected over water.

In the exercise, knowing the vapor pressure allows us to adjust the total pressure by accounting for the water vapor, isolating the pressure of $O_2$.

Gas Collection over Water

When gases are collected over water, they are inevitably mixed with water vapor, adding complexity to pressure calculations.

• Method: This process involves capturing gas in an inverted container filled with water, where the collected gas displaces the water.
• Considerations: Because water vapor comes into play, the pressure of the collected gas must be adjusted for the water vapor's partial pressure.

In our exercise, this method requires us to consider both the gas and water vapor pressures to accurately ascertain the actual pressure of oxygen and water vapor contributions.