First, let's understand the scenario. The snorkeler has a syringe filled with air, initially at the surface of the water, where the atmospheric pressure is 1 atm and the volume of the air is 20 ml. As the snorkeler dives deeper, the pressure increases and the volume of the air in the syringe decreases to 7.5 ml.
We know that Boyle's law, which states that the product of the pressure and volume for a gas is a constant for a fixed amount of gas at a constant temperature, will apply here. That is, initial pressure times initial volume equals final pressure times final volume.
Mathematically, Boyle's law can be stated as P1 * V1 = P2 * V2, where P1 denotes the initial pressure, V1 - the initial volume, P2 - the final pressure, and V2 - the final volume.
Here, P1 is 1.0 atm, V1 is 20 ml, and V2 is 7.5 ml. We need to find P2.
So, we rearrange Boyle's law and solve for P2:
P2 = P1 * V1 / V2 = 1.0 * 20 / 7.5 ≈ 2.67 atm.
Now, the difference between the initial and final pressure: ΔP = P2 - P1 = 2.67 atm - 1 atm = 1.67 atm.
Given that the pressure increases by 1 atm for every 10 m of depth, we can find the depth change (ΔD) in the problem by using the formula ΔD = ΔP * 10 m/atm.
Therefore, ΔD = 1.67 atm * 10 m/atm ≈ 16.67 m. This represents the depth change from the surface to the location of the snorkeler, implying that the snorkeler has dived approximately 17 meters under the sea level.
Therefore, we can conclude that the snorkeler is at a depth of approximately 17 meters.