Question

At an underwater depth of \(250 \mathrm{ft}\), the pressure is \(8.38 \mathrm{~atm}\). What should the mole percent of oxygen be in the diving gas for the partial pressure of oxygen in the mixture to be \(0.21 \mathrm{~atm}\), the same as in air at \(1 \mathrm{~atm}\) ?

Short answer

The mole percent of oxygen in the diving gas should be approximately 2.57% for the partial pressure of oxygen in the mixture to be 0.21 atm, the same as in air at 1 atm.

Step-by-step solution

Understand the relationship between total pressure and partial pressure

We can use Dalton's law of partial pressures to understand the relationship between the total pressure of the gas mixture and the partial pressure of individual gases in the mixture. According to Dalton's law: Total pressure = Partial pressure of gas 1 + Partial pressure of gas 2 + ... In this case, we have 2 gases: oxygen and nitrogen (assuming the diving gas is only composed of these 2 gases).

Calculate the partial pressure of nitrogen in the air at 1 atm

In air, the mole fraction of nitrogen is approximately 0.78, and the mole fraction of oxygen is approximately 0.21. Therefore, we can calculate the partial pressure of nitrogen in the air at 1 atm. Partial pressure of nitrogen = mole fraction of nitrogen * total pressure in air Partial pressure of nitrogen = 0.78 * 1 atm = 0.78 atm

Determine the mole fraction of nitrogen in the diving gas

The mole fraction of nitrogen in the diving gas must be the same as in air because the total pressure of the gas mixture is proportional to the depth. Therefore, the mole fraction of nitrogen in the diving gas is also 0.78.

Calculate the partial pressure of nitrogen in the diving gas

Now we can calculate the partial pressure of nitrogen in the diving gas using the mole fraction and total pressure at the depth of 250 ft. Partial pressure of nitrogen in diving gas = mole fraction of nitrogen * total pressure in diving gas = 0.78 * 8.38 atm = 6.5364 atm

Determine the partial pressure of oxygen in the diving gas

According to the problem, the partial pressure of oxygen in the diving gas must be 0.21 atm.

Calculate the total pressure in the diving gas without oxygen

In this step, we will calculate the total pressure in the diving gas without the pressure contribution from oxygen to determine the total pressure due to the other gases. Total pressure without oxygen = Total pressure in diving gas - Partial pressure of oxygen = 8.38 atm - 0.21 atm = 8.17 atm

Determine the mole fraction of oxygen in the diving gas

Now we will calculate the mole fraction of oxygen in the diving gas. Mole fraction of oxygen = Partial pressure of oxygen / Total pressure without oxygen = 0.21 atm / 8.17 atm ≈ 0.0257

Convert the mole fraction to mole percent

Finally, in order to determine the mole percent of oxygen in the diving gas, we can simply multiply the mole fraction by 100: Mole percent of oxygen = 0.0257 * 100 = 2.57 % Thus, the mole percent of oxygen in the diving gas should be approximately 2.57% for the partial pressure of oxygen in the mixture to be 0.21 atm.