Question

Hyperbaric oxygen therapy is used to treat patients with carbon monoxide poisoning as well as to treat divers with the bends. In hyperbaric oxygen therapy, a patient is placed inside a 7.0-ft cylinder with a 3.0-ft diameter, which is then filled with oxygen gas to a total pressure of 2.50 atm. Assuming the patient takes up 32.0% of the chamber’s volume, what volume of \(\mathrm{O}_{2}(g)\) from a gas cylinder at \(25^{\circ} \mathrm{C}\) and 95 \(\mathrm{atm}\) is required to fill the chamber to a total pressure of 2.50 \(\mathrm{atm}\) at \(25^{\circ} \mathrm{C} ?\) Assume the hyperbaric chamber initially contains air at 1.00 atm before \(\mathrm{O}_{2}(g)\) is added.

Short answer

The volume of oxygen gas required to fill the hyperbaric chamber to a total pressure of 2.50 atm at 25°C and when the patient takes up 32.0% of the chamber's volume can be calculated using the ideal gas law. First, find the volume of the hyperbaric chamber: \(V = \pi (1.5 \,\text{ft})^2 (7.0 \,\text{ft})\). Next, calculate the volume occupied by the patient: \(V_{\text{occupied}} = V * 0.32\). Then, determine the volume available for the gas: \(V_{\text{available}} = V - V_{\text{occupied}}\). Finally, apply the combined gas law to find the volume of \(\mathrm{O}_{2}(g)\) required: \(V_2 = V_1 \frac{n_2 P_1}{n_1 P_2}\).

Step-by-step solution

Determine the chamber's volume

First, we need to calculate the volume of the hyperbaric chamber. It is in the shape of a cylinder with a diameter of 3.0 ft and a height of 7.0 ft. The volume of a cylinder is given by the formula \(V = π r^2 h\) where V is the volume, r is the radius, and h is the height of the cylinder. Given the diameter, we can compute the radius (r) as half of the diameter: \(r = \frac{3.0 \,\text{ft}}{2} = 1.5 \,\text{ft}\) Now we can calculate the volume of the chamber \(V = π (1.5 \,\text{ft})^2 (7.0 \,\text{ft})\)

Calculate the occupied volume by the patient

Now, we determine the volume occupied by the patient. The problem states that the patient takes up 32.0% of the chamber's volume. To calculate this volume, we multiply the total volume of the chamber by 0.32 (32.0%). \(V_{\text{occupied}} = V * 0.32\)

Calculate the volume available for the gas

Next, we subtract the occupied volume from the total chamber volume to find the volume available for the gas. \(V_{\text{available}} = V - V_{\text{occupied}}\)

Apply the ideal gas law

Now, we apply the ideal gas law to determine the amount of \(\mathrm{O}_{2}(g)\) required to fill the chamber to a total pressure of 2.50 atm. Initially, the chamber contains air at 1.00 atm, so we need to add an amount of oxygen gas to reach the desired total pressure of 2.50 atm. Assuming both the initial and final states of the gas are at \(25^{\circ} \text{C}\), the temperature remains constant. Therefore, we can apply the combined gas law as: \(\frac{P_1 V_1}{n_1} = \frac{P_2 V_2}{n_2}\) where \(P_1\) and \(V_1\) are the initial pressure and volume of the chamber, \(P_2\) and \(V_2\) are the final pressure and volume of the chamber, and \(n_1\) and \(n_2\) are the initial and final moles of gas. We want to find the volume of oxygen gas required to fill the chamber to the desired pressure under the given conditions, so we will solve for \(V_2\).

Solve for the volume of oxygen gas required

Rearrange the combined gas law to isolate \(V_2\) and then substitute the known variables. \(V_2 = V_1 \frac{n_2 P_1}{n_1 P_2}\) The desired final pressure, \(P_2 = 2.50 \,\text{atm}\), and the initial pressure, \(P_1 = 1.00 \,\text{atm}\). The initial volume of the chamber, \(V_1\), is \(V_{\text{available}}\). The final volume of the chamber, \(V_2\), is the required volume of oxygen gas. We can find the moles of gas from the ideal gas law, which states: \(PV = nRT\) where \(P\) is pressure, \(V\) is volume, \(n\) is the number of moles of gas, \(R\) is the ideal gas constant, and \(T\) is the temperature in Kelvin. For oxygen gas, we are given the pressure, \(P = 95 \,\text{atm}\), and the temperature, \(T = (25 + 273.15) \,\text{K} = 298.15 \text{K}\). With these values, we can find the moles of \(\mathrm{O}_{2}(g)\) added to the chamber.

Key concepts

Carbon Monoxide Poisoning

Carbon monoxide (CO) poisoning is a serious health risk because CO is a colorless, odorless gas that can be inhaled without notice. It is created by incomplete combustion of carbon-containing fuels, like gasoline, coal, or natural gas. When we breathe in CO, it replaces oxygen in our red blood cells, preventing oxygen from reaching our tissues and organs.
This can lead to symptoms like headaches, dizziness, or even fatalities in severe cases. Hyperbaric oxygen therapy (HBOT) is used to treat CO poisoning by delivering 100% oxygen at high pressure to the patient. This treatment helps to quickly remove carbon monoxide from the bloodstream, restoring normal oxygen levels. The high-pressure oxygen speeds up the displacement of CO from hemoglobin, thus reversing the potential damage caused by CO poisoning.

Ideal Gas Law

The Ideal Gas Law is a fundamental relation used to describe the behavior of gases. It is represented by the equation \(PV = nRT\), where \(P\) is pressure, \(V\) is volume, \(n\) represents the number of moles of gas, \(R\) is the ideal gas constant, and \(T\) is the temperature in Kelvin.
This law assumes that gases behave ideally, which means the interactions between molecules are negligible, and the volume occupied by gas molecules themselves is much smaller compared to the total volume of the container. The Ideal Gas Law allows us to predict how changing one of these variables affects the others, assuming the amount of gas remains constant.

• To apply the Ideal Gas Law effectively, convert all units to match the equation, such as pressure to atm, volume to liters, and temperature to Kelvin.
• Always keep track of units and ensure consistency for accurate calculations.

This concept is crucial for calculating the amount of oxygen needed to fill the hyperbaric chamber under specified conditions.

Gas Cylinder Pressure

Gas cylinder pressure refers to the force exerted by the gas inside a cylinder. This force results from gas molecules colliding with the cylinder walls. In practical scenarios, like filling a hyperbaric chamber, knowing the pressure helps determine how much gas is needed to achieve a certain pressure in the chamber.
For instance, the pressure of oxygen gas in the cylinder is given as 95 atm at the storage conditions. This high pressure indicates that a large quantity of oxygen can be stored in a relatively small volume. When filling a hyperbaric chamber, it is crucial to consider both the initial pressure inside the chamber and the target pressure.

• Managing cylinder pressure is essential for safety to prevent over-pressurization, which could lead to dangerous conditions.
• It's also important to understand how pressure changes will affect the total volume of gas delivered from the cylinder.

Volume Calculations

Calculating volume is essential in determining how much space a gas occupies, especially in processes like filling a hyperbaric chamber. The chamber's volume needs to be known accurately to ensure the correct amount of gaseous oxygen is used.
The volume of a cylinder (such as a hyperbaric chamber) can be determined using the formula \(V = πr^2h\), where \(r\) is the radius and \(h\) is the height.

• Ensure dimensions are in the same units when calculating, such as feet or meters, for consistency.
• Understanding the concept of volume is key to managing space inside the chamber and ensuring it accommodates the necessary quantity of gas after accounting for the patient's presence.

Proper volume calculations help in understanding how additional oxygen gas changes the total volume available for breathing inside the chamber.