Question

A sample of oxygen gas is saturated with water vapor at \(27^{\circ} \mathrm{C} .\) The total pressure of the mixture is 772 torr, and the vapor pressure of water is 26.7 torr at \(27^{\circ} \mathrm{C}\). What is the partial pressure of the oxygen gas?

Short answer

The partial pressure of the oxygen gas in the mixture is \(745.3 \,\text{torr}\).

Step-by-step solution

Identifying the given values

Let's first list down the given values: - Total pressure of the mixture (P_total): 772 torr - Vapor pressure of water (P_water): 26.7 torr Now we will use Dalton's Law of Partial Pressures to find the partial pressure of the oxygen gas (P_O2) in the mixture.

Applying Dalton's Law of Partial Pressures

Dalton's Law of Partial Pressures states that the total pressure of a mixture of gases is equal to the sum of the partial pressures of the individual gases. In our case, the total pressure is the sum of the partial pressures of the oxygen gas and the water vapor. Mathematically, this can be written as: \[P_{total} = P_{O_2} + P_{water}\] Now, we can solve for the partial pressure of oxygen (P_O2):

Calculating the partial pressure of oxygen gas

Rearrange the equation to solve for P_O2 and then plug in the given values: \[P_{O_2} = P_{total} - P_{water}\] \[P_{O_2} = 772 \,\text{torr} - 26.7 \,\text{torr}\] Finally, do the subtraction: \[P_{O_2} = 745.3 \,\text{torr}\]

Final Answer

The partial pressure of the oxygen gas in the mixture is 745.3 torr.

Key concepts

Dalton's Law of Partial Pressures

Understanding Dalton's Law of Partial Pressures is essential for solving problems related to the partial pressure of gases in a mixture. This law is a cornerstone of gas laws in chemistry, indicating that in a mixture of non-reacting gases, the total pressure exerted is equal to the sum of the partial pressures of individual gases. Simplified, it’s the idea that each gas in a mixture behaves independently and contributes to the total pressure as if it were alone in the container.

Let's take the given exercise problem as an example. We have a mixture of oxygen gas and water vapor, and we know the total pressure and the vapor pressure of water. To find the partial pressure of oxygen, according to Dalton's Law, we subtract the known vapor pressure of water from the total pressure of the mixture. It’s a straightforward application of the law that allows us to see that each gas's partial pressure is unaffected by the presence of other gases.

Gas Laws in Chemistry

Gas laws in chemistry refer to the various laws that describe the behavior of gases, and they hold the key to predict and explain how gases will react when subjected to changes in volume, temperature, and pressure. These laws are vital to understanding the physical properties of gases and their interactions.

In our example of the oxygen and water vapor mixture, we employ one of these laws—Dalton's Law—to determine the partial pressure. Besides Dalton's Law, there are other important laws such as Boyle's Law, which relates pressure and volume, Charles's Law, linking volume and temperature, and Avogadro's Law, which connects volume and the amount of gas. These laws are interrelated and can be combined into the General Gas Law, which is instrumental in many chemical calculations and real-life applications.

Vapor Pressure

Vapor pressure is a significant concept when discussing the partial pressure of gases, as it pertains to the pressure exerted by the vapor of a liquid when it is in equilibrium with its liquid or solid form. It indicates how volatile a liquid is and varies with temperature—the higher the temperature, the more molecules have enough energy to escape into the vapor phase, increasing the vapor pressure.

In the context of our problem, we are given the vapor pressure of water at a specific temperature. This value is crucial as it is used in the Dalton's Law of Partial Pressures equation to find the partial pressure of oxygen. Understanding vapor pressure helps in explaining phenomena such as boiling point and evaporation. Knowing that it can change with temperature explains why adjusting the temperature during an experiment is an approach to controlling the behavior of gases and vapors.