Problem 26 - General Chemistry: Principles and Modern Applications

Question

A 34.0 L cylinder contains \(305 \mathrm{g} \mathrm{O}_{2}(\mathrm{g})\) at \(22^{\circ} \mathrm{C} .\) How many grams of \(\mathrm{O}_{2}(\mathrm{g})\) must be released to reduce the pressure in the cylinder to 1.15 atm if the temperature remains constant?

Step-by-Step Solution

Answer

The cylinder needs to release 253.44 g of O2 gas to reduce the pressure to 1.15 atm.

Step 1: Calculate the Initial Number of Moles

First, calculate the initial number of moles of O2 gas using the Ideal Gas Law. The initial pressure is 1 atm by virtue of it being a standard container, and the volume is 34.0 L. The temperature is 22 degrees Celsius, which converted to Kelvin is 295.15 K, and the gas constant R is 0.0821 L∙atm/K∙mol. Thus, \(n_1 = PV / RT = (1 atm × 34.0 L) / (0.0821 L∙atm/K∙mol × 295.15 K) = 1.402 mol\) . We also know that the O2 cylinder initially has 305 g of O2, which corresponds to \(305 g / 32.00 g/mol = 9.53 mol\).

Step 2: On Determine the Final Number of Moles

Next the number of moles of O2 gas that would create a pressure of 1.15 atm is calculated. With the volume (34.0 L), temperature (295.15 K), and gas constant (0.0821 L∙atm/K∙mol) unchanged, the new pressure is 1.15 atm. Hence, \(n_2 = PV / RT = (1.15 atm × 34.0 L) / (0.0821 L∙atm/K∙mol × 295.15 K) = 1.611 mol\).

Step 3: Calculate the Released O2

The initial number of moles minus the final number of moles gives the number of moles of O2 gas that must be released to reduce the pressure to 1.15 atm. Therefore, \(n = n_1 - n_2 = 9.53 mol - 1.61 mol = 7.92 mol\).

Step 4: Convert Moles into Grams

The last step is to convert the number of moles (7.92 mol) into grams using the molar mass of O2 (32.00 g/mol). Therefore, the mass of O2 that needs to be released is \(7.92 mol × 32.00 g/mol = 253.44 g\).

Key Concepts

Molar Calculations

When dealing with gases, molar calculations play a pivotal role in understanding how much of a substance is present. Moles are the measure of quantity used in chemistry, linking mass to a specific number of particles. For oxygen, which has a molar mass of 32.00 g/mol, we can convert a given mass into moles using the formula:

• Number of moles = given mass (g) / molar mass (g/mol).

For instance, if you have 305 grams of oxygen, by dividing this mass by 32.00 g/mol, you find that there are approximately 9.53 moles of oxygen. This conversion helps bridge the gap between the macroscopic measurements we can see and touch, and the microscopic particles and their behaviors under different conditions.
By calculating moles, it becomes easier to plug values into the Ideal Gas Law, correlating pressures, volumes, and temperatures to predict and analyze gas behaviors.

Pressure Change

Understanding how pressure changes in a gas cylinder is crucial, especially when conditions such as temperature and volume remain constant. The Ideal Gas Law, \[ PV = nRT \], allows us to see that if the temperature and volume are unchanged, the pressure directly changes with the moles of gas present.
This means more moles equal higher pressure and vice versa.

• Initial pressure is determined by the initial number of moles.
• To adjust the pressure, moles of gas need to be added or removed.

By calculating the difference in moles required to achieve a target pressure, we can then relate these moles back to mass to understand how much substance must be added or released.

Gas Cylinders

Gas cylinders serve as critical storage containers for gases such as oxygen, which are stored under high pressure. Key to working with gas cylinders is understanding their volume capacity and how temperature and pressure interplay within them.
Using the Ideal Gas Law, one can predict the behavior of gases in the cylinder, including how many grams of gas should be released to decrease pressure when the gas expands to fill a room or is consumed.
Given the cylinder's volume remains constant, any change in the amount of gas (measured in moles) is directly reflected in pressure changes. Understanding this helps ensure safe and efficient use of gas cylinders, preventing overpressure situations and ensuring optimal release of the gas for usage needs.