Question

A sample of oxygen gas is collected by displacement of water at ${25}^{\circ }\mathrm{C}$ and 1.02 atm total pressure. If the vapor pressure of water is $23.756\mathrm{mm}\mathrm{Hg}$ at ${25}^{\circ }\mathrm{C}$ what is the partial pressure of the oxygen gas in the sample?

Short answer

The partial pressure of oxygen gas in the sample is calculated using Dalton's Law of Partial Pressures. First, convert the vapor pressure of water to atm: $23.756\frac{mmHg}{1}$ × $\frac{1atm}{760mmHg}$ = 0.03126 atm. Then, subtract the vapor pressure of water from the total pressure: 1.02 atm - 0.03126 atm = 0.98874 atm. Therefore, the partial pressure of the oxygen gas in the sample is 0.98874 atm.

Step-by-step solution

Convert given data to the same units

In this problem, the given pressure data is in different units, so first, we need to convert them to the same unit, let's use atm. We know that 1 atm = 760 mmHg. So, we have: Vapor pressure of water = $23.756\frac{mmHg}{1}$ × $\frac{1atm}{760mmHg}$ = 0.03126 atm Total pressure = 1.02 atm

Apply Dalton's Law of Partial Pressures

According to Dalton's Law of Partial Pressures, the total pressure is the sum of the partial pressures of each gas (oxygen and water vapor) in the sample: Total pressure = Partial pressure of oxygen gas + Partial pressure of water vapor Rearrange the equation to solve for the partial pressure of oxygen gas: Partial pressure of oxygen gas = Total pressure - Partial pressure of water vapor

Calculate the partial pressure of oxygen gas

Now, we can plug in the given values and solve for the partial pressure of oxygen gas: Partial pressure of oxygen gas = 1.02 atm - 0.03126 atm = 0.98874 atm The partial pressure of the oxygen gas in the sample is 0.98874 atm.

Key concepts

Dalton's Law of Partial Pressures

When dealing with mixtures of gases, we often need to understand each gas's contribution to the overall pressure. This is where Dalton's Law of Partial Pressures becomes essential. It states that within a mixture of non-reacting gases, the total pressure exerted is equal to the sum of the partial pressures of individual gases. A gas's partial pressure is the pressure that gas would exert if it were alone in the container.

The law is mathematically expressed as: $P_{total}=P_1+P_2+P_3+...$, where $P_{total}$ is the total pressure of the gas mixture and $P_1,P_2,P_3,...$ are the partial pressures of the gases. In other words, if you know the total pressure and the partial pressures of all but one component, you can easily find the unknown partial pressure by rearranging the equation accordingly.

Conversion of Pressure Units

Understanding how to convert between different pressure units is crucial when solving problems involving gases, as measurements can be given in any number of units. The most common pressure units in chemistry are atmospheres (atm), millimeters of mercury (mmHg), and Pascals (Pa). Conversion factors between these units are essential and must be memorized or readily accessible.

To convert from mmHg to atm, a widely used conversion rate is $1atm=760mmHg$. Using this conversion factor, you would divide the pressure in mmHg by 760 to find the pressure in atm. Conversely, to convert from atm to mmHg, you would multiply the pressure in atm by 760. It's all about understanding which factor to use based on the desired end unit.

Vapor Pressure of Water

Vapor pressure is a measure of a liquid's tendency to evaporate. It is the pressure exerted by the vapor of the liquid when the liquid and vapor are in dynamic equilibrium. This indicates that the rate of evaporation of the liquid equals the rate of condensation from the vapor phase. For water, the vapor pressure increases with temperature, meaning that water will evaporate more quickly at higher temperatures.

At any given temperature, the vapor pressure of water is a fixed quantity. Knowing this value can be particularly useful when calculating the partial pressure of a gas collected over water, as seen in the exercise. The vapor pressure of water at ${25}^{\circ }C$ subtracted from the total pressure gives the partial pressure of the gas of interest. Weather prediction, cooking, and understanding humidity levels are just a few applications where knowledge of water's vapor pressure is beneficial.