Question

Hyperbolic Oxygen Therapy Hyperbolic oxygen chambers are used to treat divers suffering from decompression sickness (the "bends") with pure oxygen at greater than atmospheric pressure. Other clinical uses include treatment of patients with thermal burns and CO poisoning. What is the pressure in a chamber with a volume of \(4.85 \times 10^{3} \mathrm{L}\) that contains \(15.00 \mathrm{kg}\) of \(\mathrm{O}_{2}(g)\) at per ef \(298 \mathrm{K} ?\)

Short answer

Answer: The pressure inside the hyperbaric oxygen chamber is approximately 2.34 atm.

Step-by-step solution

Convert mass of oxygen to moles

To convert the mass of oxygen to moles, we'll use the formula n = m / M, where n is the amount of the gas in moles, m is the mass of the gas, and M is the molar mass of the gas. The molar mass of oxygen is 32 g/mol, and we are given 15 kg of oxygen, which is equal to 15,000 g. So, the number of moles of oxygen in the chamber is: n = \(\frac{15,000 \; \mathrm{g}}{32 \; \mathrm{g/mol}} = 468.75 \; \mathrm{moles}\)

Use the Ideal Gas Law to find the pressure

Now we have all the necessary values to solve for the pressure in the chamber using the ideal gas law: \(PV = nRT\) Here, P is the pressure we want to find, V is the volume (\(4.85 \times 10^3 \; \mathrm{L}\)), n is the amount of the gas in moles (468.75 moles), R is the ideal gas constant with a value of 0.0821 \(\frac{\mathrm{L}\cdot\mathrm{atm}}{\mathrm{mol}\cdot\mathrm{K}}\), and T is the temperature (298 K). We will now rearrange the equation to solve for P: \(P = \frac{nRT}{V}\) Substitute the given values into the equation: \(P = \frac{(468.75 \; \mathrm{moles}) (0.0821 \; \frac{\mathrm{L}\cdot\mathrm{atm}}{\mathrm{mol}\cdot\mathrm{K}}) (298 \; \mathrm{K})}{4.85 \times 10^{3} \; \mathrm{L}}\) Calculate the pressure: \(P \approx 2.34 \; \mathrm{atm}\) So, the pressure inside the hyperbaric oxygen chamber is approximately 2.34 atm.

Key concepts

Hyperbaric Oxygen Therapy

Hyperbaric Oxygen Therapy (HBOT) involves breathing pure oxygen in a pressurized chamber. This therapy is primarily utilized for treating conditions like decompression sickness
and carbon monoxide poisoning. The high pressure helps dissolve more oxygen in the blood, which can accelerate the healing process and improve blood circulation.
\(
\)- **Uses of HBOT**: - **Decompression Sickness**: Common in divers, it occurs when nitrogen bubbles form in the bloodstream after a rapid ascent. - **Thermal Burns**: Oxygen can aid in tissue healing and antibacterial action. - **Carbon Monoxide Poisoning**: High oxygen levels help detach CO from hemoglobin.
\(
\)HBOT settings vary depending on the condition being treated, but usually involve pressures greater than the normal atmospheric level. Understanding the principles of gas behavior is important in configuring HBOT conditions effectively. This makes gas laws, such as the Ideal Gas Law, vital in the accurate calculation of pressure and dosage within the chamber.

Decompression Sickness

Decompression sickness (DCS), also known as "the bends," is a potential risk for divers. It arises when dissolved gases, primarily nitrogen, form bubbles in the body's tissues during rapid ascent.
These bubbles can cause joint pain, dizziness, and in severe cases, paralysis or death.
\(
\)- **Causes of DCS**: - During a dive, increased pressure causes more nitrogen to dissolve into the blood. - A rapid decrease in pressure, such as surfacing too quickly, leads to the formation of gas bubbles. - These bubbles can obstruct blood flow and stretch tissues.
\(
\)**Preventing DCS**:- Divers should ascend slowly and perform safety stops, giving the body time to expel nitrogen naturally.- Training and using dive tables or computers can help plan dives safely.
\(
\)**Treatment**:- Immediate recompression in a hyperbaric chamber, like the exercise outlined above, is the most effective treatment.
- Oxygen-rich environments enhance nitrogen elimination from the body.

Molar Mass

Molar mass is a key concept in chemistry, representing the mass of a given substance (molecule, atom, or ion) per mole of that substance, expressed in g/mol. It allows for the conversion of mass
to moles using the formula: \[m = n \times M\]where \(m\) is the mass, \(n\) the number of moles, and \(M\) the molar mass. In the context of the original problem, we calculated the number of moles of oxygen using its molar mass
(32 g/mol) and the given mass of 15,000 g.
\(
\)- **Why is Molar Mass Important?**: - It provides a bridge between the macroscopic and atomic levels of chemistry, allowing chemists to count particles by weighing them. - Essential for stoichiometric calculations in a wide range of chemical reactions.
\(
\)Knowing the molar mass of substances enables accurate measurements in laboratory settings and predictive modeling in engineering applications that involve gases like oxygen.

Gas Laws in Chemistry

Gas laws are fundamental principles that describe the behavior of gases. The Ideal Gas Law combines several simpler gas laws to relate pressure, volume, temperature, and amount of a gas:\[PV = nRT\]where \(P\) is pressure, \(V\) is volume, \(n\) is the number of moles, \(R\) is the gas constant, and \(T\) is temperature.
\(
\)- **Core Gas Laws**: - **Boyle’s Law**: Pressure and volume are inversely proportional at constant temperature. - **Charles’s Law**: Volume and temperature are directly proportional at constant pressure. - **Avogadro’s Law**: Volume and amount (in moles) are directly proportional at constant temperature and pressure.
\(
\)**Applications**:- Predicting how gases will react under different conditions.- Calculating the pressure exerted by gases, as done in the original exercise using the Ideal Gas Law.
\(
\)Understanding these laws is crucial for scientific research and practical applications, like designing hyperbaric chambers or predicting the behavior of gases in chemical reactions.