This week, we are studying the ideal gas law. In this discussion, you will be trying your hand at applying one of the ideal gas laws to a real world situation. Consider a situation that involves an ideal gas law and discuss how you would apply your chosen ideal gas law to the situation. Write your own ideal gas law problem
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Problem:
A scuba diver is exploring a shipwreck at a depth of 30 meters below the surface of the ocean. The diver's air tank has a volume of 10 liters and is initially filled with compressed air at a pressure of 200 bar. The temperature of the air inside the tank is 25°C. Assuming the diver's air tank is an ideal gas, calculate the final pressure inside the tank when fully submerged, considering the increase in pressure due to the water column.
Solution:
To solve this problem, we can apply the combined gas law, which relates the initial and final states of a gas undergoing changes in pressure, volume, and temperature. The combined gas law is given as:
(P1 * V1) / (T1) = (P2 * V2) / (T2)
where P1, V1, and T1 represent the initial pressure, volume, and temperature of the gas, and P2, V2, and T2 represent the final pressure, volume, and temperature of the gas.
Given P1 = 200 bar, V1 = 10 L, and T1 = 25°C, we need to determine P2.
First, let's convert the initial temperature to Kelvin, as it is required in the ideal gas equation:
T1 = 25°C + 273.15 = 298.15 K
At a depth of 30 meters below the ocean's surface, the pressure due to the water column adds an extra pressure equal to the hydrostatic pressure:
P_hydrostatic = ρ * g * h
where ρ is the density of water, g is the acceleration due to gravity, and h is the depth.
Given that the density of water is approximately 1000 kg/m³ and g is approximately 9.8 m/s², we can calculate P_hydrostatic:
P_hydrostatic = 1000 kg/m³ * 9.8 m/s² * 30 m = 294,000 Pa
To convert the hydrostatic pressure from Pascals to bar, we divide by 100,000:
P_hydrostatic = 294,000 Pa / 100,000 = 2.94 bar
The final pressure inside the tank, considering the hydrostatic pressure, can be obtained by substituting P2 = P1 + P_hydrostatic into the combined gas law equation:
(200 bar * 10 L) / (298.15 K) = (P2 * 10 L) / (298.15 K)
Simplifying the equation, we find:
200 bar / 29.815 K = P2
P2 ≈ 6.70 bar
Therefore, the final pressure inside the scuba diver's air tank, when fully submerged at a depth of 30 meters, would be approximately 6.70 bar.
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