Question

A scuba diver \(60 \mathrm{ft}\) below the ocean surface inhales \(50.0 \mathrm{~mL}\) of compressed air from a scuba tank at a pressure of \(3.00\) atm and a temperature of \(8{ }^{\circ} \mathrm{C}\). What is the pressure of the air, in atm, in the lungs when the gas expands to \(150.0 \mathrm{~mL}\) at a body temperature of \(37{ }^{\circ} \mathrm{C}\), and the amount of gas remains constant?

Short answer

\(P_2 \approx 1.10 \, \mathrm{atm}\)

Step-by-step solution

- Write down the given variables

Identify and list the values provided in the problem:\(V_1 = 50.0 \, \mathrm{mL} \)\(P_1 = 3.00 \, \mathrm{atm} \)\(T_1 = 8^{\circ} \mathrm{C} = 273.15 + 8 = 281.15 \, \mathrm{K} \)\(V_2 = 150.0 \, \mathrm{mL} \)\(T_2 = 37^{\circ} \mathrm{C} = 273.15 + 37 = 310.15 \, \mathrm{K} \)We need to find \(P_2\).

- Use the Combined Gas Law

The Combined Gas Law is given by: \[\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}\]

- Rearrange the equation to solve for \(P_2\)

Rearrange the formula to solve for the unknown pressure \(P_2\): \[P_2 = P_1 \frac{V_1}{V_2} \frac{T_2}{T_1}\]

- Substitute the known values into the equation

Substitute the given values into the rearranged equation: \[P_2 = 3.00 \frac{50.0}{150.0} \frac{310.15}{281.15}\]

- Perform the calculations

Calculate the value step-by-step: \[P_2 = 3.00 \times \frac{50.0}{150.0} \times \frac{310.15}{281.15}\] \[P_2 = 3.00 \times \frac{1}{3} \times 1.1032\] \[P_2 = 3.00 \times 0.3677\] \[P_2 \approx 1.1032 \, \mathrm{atm}\]

Key concepts

gas laws

Gas laws are fundamental principles in chemistry that describe the behavior of gases. They help us understand how gases respond to changes in temperature, pressure, and volume. The major gas laws include Boyle's Law, Charles's Law, and Gay-Lussac's Law. Each of these laws focus on relationships between two variables while keeping the third variable constant.

For example:

• Boyle's Law: This law states that the pressure of a gas is inversely proportional to its volume when the temperature is held constant. The equation is given by: \( P_1V_1 = P_2V_2 \).
• Charles's Law: This law indicates that the volume of a gas is directly proportional to its temperature when the pressure is held constant. The formula is: \( \frac{V_1}{T_1} = \frac{V_2}{T_2} \).
• Gay-Lussac's Law: This law tells us that pressure is directly proportional to temperature when volume is constant: \( \frac{P_1}{T_1} = \frac{P_2}{T_2} \).

However, the Combined Gas Law we used in this problem allows us to consider changes in pressure, volume, and temperature simultaneously. It is a powerful tool when dealing with real-world gas scenarios.

pressure-temperature-volume relationship

Understanding the pressure-temperature-volume relationship is crucial to solving problems involving gases. The Combined Gas Law combines Boyle's, Charles's, and Gay-Lussac's laws. It shows how pressure of a gas is related to its volume and temperature.

In the given exercise, variables undergo changes, such as:

• Initial pressure (P1): The air in the scuba tank at 3.00 atm.
• Final volume (V2): Expanded volume in the lungs at 150.0 mL.
• Initial temperature (T1): Temperature under the ocean at 8°C or 281.15 K.
• Final temperature (T2): Body temperature of 37°C or 310.15 K.

We employ the equation: \[ \frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2} \]. This allows us to connect the initial and final states of the gas. By rearranging and solving this equation, we can determine unknown pressures, volumes, or temperatures when other variables are known.

ideal gas law calculations

The ideal gas law is an extension of the gas laws and introduces the concept of moles and the gas constant. The ideal gas law is represented by the equation \(PV = nRT \), where:

• \text{P}: pressure of the gas (atm)
• \text{V}: volume of the gas (L)
• \text{n}: number of moles of the gas
• \text{R}: gas constant (0.0821 L·atm/(mol·K))
• \text{T}: temperature (K)

The Ideal Gas Law provides a good approximation of the behavior of real gases under many conditions. However, for the scuba diving problem, using the Combined Gas Law is more straightforward because it directly relates pressure, volume, and temperature changes without needing to know the number of moles of gas.

To simplify, always remember these steps:

• Convert all temperatures to Kelvin.
• Ensure all units are consistent (volume in mL or L, pressure in atm).
• Substitute known values into the formula.
• Solve step-by-step to find the unknown variable.

By mastering these fundamental concepts and calculations, you can tackle a wide range of problems involving gases with confidence.