Question

Commercially, compressed oxygen is sold in metal cylinders. If a 120 - \(\mathrm{L}\) cylinder is filled with oxygen to a pressure of 132 atm at \(22^{\circ} \mathrm{C},\) what is the mass of \(\mathrm{O}_{2}\) present? How many liters of \(\mathrm{O}_{2}\) gas at \(1.00 \mathrm{~atm}\) and \(22^{\circ} \mathrm{C}\) could the cylinder produce? (Assume ideal behavior.)

Short answer

Mass is approximately 20.88 kg, and it can produce 15840 L of \( \mathrm{O}_2 \) at 1 atm.

Step-by-step solution

Given Information and Ideal Gas Law

We know the cylinder is filled with oxygen at an initial pressure \( P_1 = 132 \) atm, volume \( V_1 = 120 \) L, and temperature \( T_1 = 22^\circ \)C which is equivalent to \( T_1 = 295 \) K (by converting Celsius to Kelvin). The ideal gas law is \( PV = nRT \), where \( n \) is the amount of substance in moles and \( R = 0.0821 \) L atm K⁻¹ mol⁻¹ is the gas constant.

Calculate Moles of \( \mathrm{O}_2 \)

Using the ideal gas law \( PV = nRT \), solve for \( n \):\[ n = \frac{P_1 V_1}{RT_1} = \frac{132 \, \mathrm{atm} \times 120 \, \mathrm{L}}{0.0821 \, \mathrm{L \cdot atm \cdot K^{-1} \cdot mol^{-1}} \times 295 \, \mathrm{K}} \approx 652.58 \, \mathrm{mol} \]

Calculate Mass of \( \mathrm{O}_2 \)

The molar mass of \( \mathrm{O}_2 \) is \( 32.00 \, \mathrm{g/mol} \). Hence, the mass \( m \) can be calculated as:\[ m = n \times \text{molar mass} = 652.58 \, \mathrm{mol} \times 32.00 \, \mathrm{g/mol} = 20882.56 \, \mathrm{g} = 20.88 \, \mathrm{kg} \]

Using Combined Gas Law for Volume at 1 atm

Using \( P_1 V_1/T_1 = P_2 V_2/T_2 \) to find the final volume \( V_2 \) at \( P_2 = 1 \) atm and \( T_2 = 295 \) K:\[ V_2 = \frac{P_1 V_1 T_2}{P_2 T_1} = \frac{132 \, \mathrm{atm} \times 120 \, \mathrm{L}}{1 \, \mathrm{atm}} \approx 15840 \, \mathrm{L} \]

Conclusion

The mass of the oxygen in the cylinder is approximately \( 20.88 \) kg, and it can produce \( 15840 \) liters of \( \mathrm{O}_2 \) gas at \( 1 \) atm and temperature \( 22^\circ \mathrm{C} \).

Key concepts

Molar Mass

In chemistry, molar mass is a fundamental concept that represents the mass of one mole of a substance. It's expressed in grams per mole (g/mol). Understanding molar mass helps in converting the amount of a substance from moles to grams, which is essential in scientific calculations.
For oxygen (\(\mathrm{O}_2\)), the molar mass is \(32.00\, \mathrm{g/mol}\). This is calculated by adding the atomic masses of two oxygen atoms (\(16.00\, \mathrm{g/mol}\) each). Knowing this allows us to determine the mass in a chemical reaction or container when other variables like pressure and temperature are known. In the exercise, calculating the mass of oxygen in the cylinder involves multiplying the number of moles of \(\mathrm{O}_2\) calculated using the ideal gas law by its molar mass.

Pressure

Pressure is the force exerted per unit area by gas particles as they collide with the surfaces of their container. It's a crucial variable in gas behavior and influences how gases interact and react with their surroundings.
Pressure is measured in various units but atmospheres (atm) and pascals (Pa) are common in chemistry.
In our given exercise, the pressure inside the cylinder is \(132\) atm, which signifies a highly compressed gas. This high pressure is necessary for storing large quantities of gas in a relatively small volume. Applying the ideal gas law helps us understand how this pressure, along with volume and temperature, affects the amount of gas (moles) in the cylinder.

Temperature

Temperature reflects the kinetic energy of gas particles and plays an integral role in gas laws. It's essential to measure temperature in Kelvin for gas law computations as Kelvin-scale avoids negative temperatures and provides an absolute scale.
In the problem, the given temperature, initially \(22^{\circ} \mathrm{C}\), converts to \(295 \mathrm{K}\). This conversion is accomplished by adding \(273.15\) to the Celsius temperature.
Temperature directly impacts pressure and volume: as temperature increases, gas particles move faster, increasing pressure if the volume is constant. The exercise uses this fact under the umbrella of the ideal gas law and combined gas law calculations.

Volume

Volume is the space that a gas occupies and is often measured in liters (L) in chemistry. It plays a direct role in influencing gas behavior as described by the gas laws.
The cylinder's volume scenario at \(120\) L describes its capacity under initial conditions of compressed gas at high pressure and moderate temperature.
When dealing with the ideal gas law, volume helps determine the moles of gas present. Later, with the combined gas law, volume shows how gas expands or contracts with changing pressure and temperature. In the exercise, this expanded volume at \(1\) atm and constant temperature helps in expressing the quantity of oxygen in the form that one can use at atmospheric conditions.