Problem 46 - Chemistry: Principles and Reactions

Question

A sample of oxygen gas is collected over water at \(25^{\circ} \mathrm{C}\) (vp \(\mathrm{H}_{2} \mathrm{O}(l)=23.8 \mathrm{~mm} \mathrm{Hg}\) ). The wet gas occupies a volume of \(7.28 \mathrm{~L}\) at a total pressure of \(1.25\) bar. If all the water is removed, what volume will the dry oxygen occupy at a pressure of \(1.07 \mathrm{~atm}\) and a temperature of \(37^{\circ} \mathrm{C}\) ?

Step-by-Step Solution

Answer

Answer: The volume of the dry oxygen gas will be 7.64 liters at a pressure of 1.07 atm and a temperature of 37°C.

Step 1: Convert all units to the proper format

We need to make sure that all pressures, volumes, and temperatures are in the correct units. For this problem, let's use atm for pressure, L for volume, and K for temperature. Convert 1.25 bar to atm: 1 bar = 0.9869 atm, so 1.25 bar = 1.25 * 0.9869 = 1.2336 atm. Next, we'll need to convert the temperature to kelvin: T1 = 25°C + 273.15 = 298.15 K, and T2 = 37°C + 273.15 = 310.15 K.

Step 2: Find the partial pressure of oxygen gas

Since the total pressure of the mixture is given by the sum of the pressures of both oxygen and water vapor, we can find the partial pressure of the oxygen gas as follows: Total pressure = partial pressure of oxygen gas + partial pressure of water vapor 1.2336 atm = partial pressure of oxygen gas + 23.8 mmHg First, we need to convert the water vapor pressure to atm: 23.8 mmHg * (1 atm/760 mmHg) = 0.0313 atm. Now, we can find the partial pressure of oxygen gas: partial pressure of oxygen gas = 1.2336 atm - 0.0313 atm = 1.2023 atm.

Step 3: Use the Ideal Gas Law to find the number of moles of oxygen gas

Now that we have the partial pressure of oxygen gas, we can use the Ideal Gas Law to find the number of moles of oxygen gas present in the wet gas. PV = nRT We know that P = 1.2023 atm, V = 7.28 L, R = 0.0821 (L * atm)/(K * mol), and T1 = 298.15 K. 1.2023 atm * 7.28 L = n * 0.0821 (L * atm)/(K * mol) * 298.15 K Solve for n: n = 0.2997 moles of O2 gas.

Step 4: Use the combined gas law to find the volume of the dry oxygen gas

Now that we know the number of moles of oxygen gas, we can use the combined gas law to determine the volume of the dry oxygen gas at the given temperature and pressure: P1V1/T1 = P2V2/T2 We know that P1 = 1.2023 atm, V1 = 7.28 L, T1 = 298.15 K, P2 = 1.07 atm, and T2 = 310.15 K. (1.2023 atm * 7.28 L) / 298.15 K = (1.07 atm * V2) / 310.15 K Solve for V2: V2 = 7.64 L. So, the volume of the dry oxygen gas will be 7.64 L at a pressure of 1.07 atm and a temperature of 37°C.

Key Concepts

Partial Pressure

When discussing gases in a mixture, it is important to understand the concept of partial pressure. Partial pressure is the pressure that each individual gas in a mixture would exert if it were alone in the container.

For instance, in the scenario where oxygen is collected over water, the total pressure measured is a combination of the oxygen gas and the water vapor. Hence, to find the partial pressure of oxygen specifically, one must account for, or subtract, the pressure exerted by the water vapor. This is essential because gas volumes are often collected over water, and water vapor affects the total pressure.

To find the partial pressure of the oxygen gas in the exercise, you subtract the water vapor pressure (converted into the same unit) from the total pressure. This gives you a clear view of the actual pressure contribution from the oxygen gas itself. Understanding this concept is crucial for solving problems involving gas mixtures, as it simplifies the complexity by focusing on one gas at a time.

Combined Gas Law

The combined gas law integrates Charles's Law, Boyle's Law, and Gay-Lussac's Law into a single expression, which can be extremely useful when dealing with problems that involve changes in conditions like temperature, pressure, and volume simultaneously.

The formula is expressed as: \[ \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \] This equation allows us to solve for one of the variables if the others are known and one state of gas changes into another.

• \(P_1\) and \(P_2\) are the initial and final pressures.
• \(V_1\) and \(V_2\) are the initial and final volumes.
• \(T_1\) and \(T_2\) are the initial and final temperatures, which must be in Kelvin.

In the exercise, this law is used to find the new volume of dry oxygen gas as it changes from one state to another, considering altered conditions of pressure and temperature. By applying the combined gas law, the volume of oxygen under new conditions was calculated.

Oxygen Gas Volume Calculation

To calculate the volume of oxygen gas under new conditions, both pressure and temperature need to be considered. The exercise involves calculating what volume the oxygen will occupy when the water is removed and conditions change.

The first step is to ensure the use of consistent units, typically liters for volume, atm for pressure, and Kelvin for temperature. Once the partial pressure of oxygen is determined, the number of moles can be calculated using the Ideal Gas Law: \[ PV = nRT \] where

• \(P\) is the pressure,
• \(V\) is the volume,
• \(n\) is the number of moles,
• \(R\) is the ideal gas constant,
• \(T\) is the temperature in Kelvin.

By solving for \(n\), the exercise provides the necessary information to explore new conditions using the combined gas law.

The last step involves using the moles calculated and changed conditions to find the new volume of oxygen gas. This gives a complete understanding of how much space the oxygen would occupy at the specified temperature and pressure."}