A bubble of air, 0.010 m^3 in volume, is formed at the bottom of a lake which is 30.0 m deep and where the temperature is 8.0 C. The bubble rises to the surface, where the water temperature is 26 C and where the pressure is atmospheric pressure. What is the volume of the bubble just as it reaches the surface?
To calculate the volume of the bubble just as it reaches the surface, we can use the ideal gas law equation, which states:
PV = nRT
where:
P is the pressure of the gas,
V is the volume of the gas,
n is the number of moles of the gas,
R is the ideal gas constant, and
T is the temperature of the gas.
In this scenario, we are assuming the air inside the bubble is an ideal gas, so we can use the ideal gas law to solve for the volume of the bubble at different depths.
First, we need to find the pressure at the bottom of the lake. The pressure in a fluid increases with depth due to the weight of the water above it. The pressure at a certain depth is given by the equation:
P = P0 + ρgh
where:
P is the pressure at the depth,
P0 is the atmospheric pressure (which we assume is 1 atm),
ρ is the density of the water, and
g is the acceleration due to gravity.
Given that the depth of the lake is 30.0 m, we can use this equation to find the pressure at the bottom of the lake. However, we need to consider the temperature difference between the bottom of the lake and the surface, as well as the density changes with temperature.
Density (ρ) is given by the equation:
ρ = ρ0 (1 + β(T - T0))
where:
ρ is the density at a certain temperature,
ρ0 is the density at the reference temperature (usually 0 degrees Celsius),
β is the volumetric thermal expansion coefficient of water,
T is the temperature, and
T0 is the reference temperature.
In this case, we can use the values for the density of water, the volumetric expansion coefficient, and the temperature difference to find the density at the bottom of the lake.
Once we have the pressure at the bottom of the lake, we can use it as the initial pressure (P) in the ideal gas law equation. We know the volume of the bubble at the bottom, the number of moles (which remains constant), and the temperatures at the bottom and surface.
Therefore, we rearrange the ideal gas law equation to solve for the final volume of the bubble at the surface:
V2 = (P1 * V1 * T2) / (P2 * T1)
where:
V2 is the final volume of the bubble at the surface,
P1 is the pressure at the bottom of the lake,
V1 is the initial volume of the bubble at the bottom,
T2 is the temperature at the surface,
P2 is the atmospheric pressure at the surface, and
T1 is the temperature at the bottom.
By substituting the values into the equation and solving, we can find the volume of the bubble just as it reaches the surface.