An air bubble has a volume of 1.51 cm3 when it is released by a submarine 102 m below the surface of a lake. What is the volume of the bubble when it reaches the surface? Assume that the temperature of the air in the bubble remains constant during ascent.
P*V = constant as the bubble rises
Calculate the initial-to-final pressure ratio.
That equals the final/initial volume ratio.
To calculate the volume of the bubble when it reaches the surface, we need to use Boyle's Law, which states that the pressure and volume of a gas are inversely proportional, assuming constant temperature.
Boyle's Law equation can be stated as:
P1 * V1 = P2 * V2
where P1 and V1 are the initial pressure and volume of the gas, and P2 and V2 are the final pressure and volume.
In this case, the initial pressure (P1) is the pressure at a depth of 102 m below the surface. The pressure at this depth can be calculated using the hydrostatic pressure formula:
P1 = P0 + (ρ * g * h)
Where P0 is the atmospheric pressure, ρ is the density of the liquid (which is water in this case), g is the acceleration due to gravity, and h is the depth.
The standard atmospheric pressure at sea level is approximately 101325 Pa, and the density of water is approximately 1000 kg/m³. The acceleration due to gravity, g, is 9.8 m/s².
Using these values, we can calculate P1 as:
P1 = 101325 Pa + (1000 kg/m³ * 9.8 m/s² * 102 m)
Now that we have P1, we can rearrange the Boyle's Law equation to solve for V2:
V2 = (P1 * V1) / P2
Since the temperature of the air in the bubble remains constant, we can assume that the final pressure (P2) is equal to the atmospheric pressure at the surface, which is 101325 Pa.
Now we can substitute the known values into the equation to find V2:
V2 = (P1 * V1) / P2
Finally, we can plug in the values 1.51 cm³ for V1 and the calculated value for P1, and solve for V2 to find the volume of the bubble when it reaches the surface.