Question

Helium is mixed with oxygen gas for deep sea divers. Calculate the percent by volume of oxygen gas in the mixture if the diver has to submerge to a depth where the total pressure is \(4.2 \mathrm{~atm} .\) The partial pressure of oxygen is maintained at 0.20 atm at this depth.

Short answer

The volume percentage of oxygen in the mixture is approximately 4.76%.

Step-by-step solution

Understanding the law of partial pressures

According to Dalton's Law of Partial Pressures, the total pressure of a mixture of gases is the sum of the pressures that each gas would exert if it were present alone under the same conditions. Therefore, Total pressure = Partial pressure of Helium gas + Partial pressure of Oxygen gas. In this case, the total pressure is given as 4.2 atm and the partial pressure of Oxygen is 0.20 atm.

Calculating the partial pressure of Helium

The partial pressure of Helium can be calculated by subtracting the partial pressure of Oxygen from the total pressure. So Partial pressure of Helium = Total Pressure - Partial Pressure of Oxygen = \(4.2 \mathrm{~atm} - 0.20 \mathrm{~atm} = 4.00 \mathrm{~atm}\).

Calculating the volume percentage of Oxygen

Percent by volume is usually calculated by the formula Volume percentage = (Volume of component / Total volume) × 100. In gases, partial pressure is directly proportional to volume percentage. So the volume percentage can be calculated as follows: Volume percentage of Oxygen = (Partial pressure of Oxygen / Total Pressure) × 100 = \( (0.20 \mathrm{~atm} / 4.2 \mathrm{~atm}) \times 100 = 4.76 \%\)

Key concepts

Understanding Partial Pressure

The concept of partial pressure is fundamental when studying mixtures of gases. It refers to the pressure that one gas in a mixture would exert if it occupied the entire volume on its own. This is a crucial aspect of Dalton's Law of Partial Pressures, which states that the total pressure in a mixture of gases is the sum of the individual partial pressures.

For instance, consider a container that holds a mixture of nitrogen and oxygen. The pressure exerted by the nitrogen alone, as if there were no other gases present, is the partial pressure of nitrogen. The same applies to oxygen or any other gas in the mixture. Partial pressures are typically measured in atmospheres (atm) or pascals (Pa), and understanding them helps to predict how gases will behave under different conditions.

Chemistry Gas Laws

The behavior of gases and their interactions under varying conditions is described by several key chemistry gas laws. These laws are pivotal in predicting gas behavior and include Boyle's Law (describing the inverse relationship between pressure and volume), Charles's Law (relating volume and temperature), and Avogadro's Law (linking volume and number of moles).

Dalton's Law of Partial Pressures is another important principle, which we encounter in multi-component gas systems. This law is particularly useful in calculating the presence of gases in mixtures, allowing us to determine the contribution of each gas to the total pressure. It’s essential for students to grasp these laws to make accurate predictions and calculations regarding gaseous systems.

Helium-Oxygen Mixture for Divers

When divers submerge into the deep sea, they often rely on a special breathing gas mixture known as heliox, which is a combination of helium and oxygen. This mixture is preferred because helium, being less dense than nitrogen, reduces the risk of nitrogen narcosis, a dangerous condition affecting divers at high pressures. Additionally, helium does not dissolve into the blood as readily as nitrogen, decreasing the chance of decompression sickness ('the bends').

The correct proportion of oxygen in this mixture is also critical. It must be enough to sustain metabolic needs but not too high to avoid oxygen toxicity. For example, maintaining an oxygen partial pressure of 0.20 atm at a depth corresponds to ensuring adequate, but safe, levels of oxygen for the diver.

Determining Percent by Volume

Percent by volume is a common way of expressing the concentration of a component in a mixture of gases. It's calculated by dividing the volume of the component by the total volume of the mixture, then multiplying by 100 to convert to a percentage. In the context of gases, because of Avogadro's Law, we know that equal volumes of gases at the same temperature and pressure contain the same number of molecules. This implies that the percent by volume is proportional to the mole fractions of the gases.

In practice, this means we can use partial pressures to easily calculate percent by volume, as the ratio of a single gas's partial pressure to the total pressure of the mixture reflects this same proportion. Ultimately, understanding percent by volume helps in preparations of gas mixtures for various applications—including medical, industrial, and diving.