On the bottom of a sea (depth:10m) a bubble of a mathane gas if formed (diameter,d bottom=1 cm). What is the diameter of the bubble, when it reaches at the surface? The sea water temperature is 5 C at the bottom and 25 C on the surface.
Try the following:
The pressure on the bubble at a depth of 10 m is Patm + (density of water x gravity x height).
101325 Pa + (1000 kg/m^3 x 9.8 m/s^2 x 10 m) = ?? (The density of water is very near 1000 km/m^3 at 4 degrees C. 5 degrees C won't change it very much.)
The volume of the sphere of methane is 4/3(pi)(r^3).
Now use (P1V1)/T1 = (P2V2)/T2 to convert the volume of methane at the 10 m depth to V2 at the surface of the water where atmospheric pressure is 101325 pa. Then using V2 = 4/3(pi)(r^3) recalculate the radius of the methane bubble at the surface. Check my thinking. Don't forget to use Kelvin for temperature.
To determine the diameter of the bubble when it reaches the surface, we need to apply the ideal gas law and take into account the change in temperature and pressure.
First, let's consider the conditions at the bottom of the sea. The pressure at the bottom can be calculated using the hydrostatic pressure equation:
P1 = P0 + ρgh
Where:
- P1 is the pressure at the bottom
- P0 is the atmospheric pressure at the surface (assume 1 atm)
- ρ is the density of the sea water (assume 1025 kg/m³)
- g is the acceleration due to gravity (assume 9.8 m/s²)
- h is the depth of the sea (10 m)
P1 = 1 atm + (1025 kg/m³ * 9.8 m/s² * 10 m)
P1 ≈ 1 atm + 100500 Pa
Next, let's find the volume of the bubble using the ideal gas law:
V1 = (nRT1) / P1
Where:
- V1 is the volume of the bubble at the bottom
- n is the number of moles of methane gas (assume 1 mole)
- R is the ideal gas constant (8.314 J/(mol·K))
- T1 is the temperature at the bottom of the sea (5 °C + 273.15 K)
V1 = (1 mol * 8.314 J/(mol·K) * (5 °C + 273.15 K)) / (1 atm + 100500 Pa)
V1 ≈ 0.0396 m³
Now, let's find the volume of the bubble at the surface, assuming the volume is constant (since the bubble is not leaking or being compressed):
V1 = V2
Since the bubble rises to the surface, the pressure at the surface is equal to the atmospheric pressure (1 atm). Using the ideal gas law, we can calculate the new temperature at the surface:
T2 = (P2 * V2) / (nR)
Where:
- T2 is the temperature at the surface
- P2 is the pressure at the surface (1 atm)
- V2 is the volume of the bubble at the surface
Rearranging the equation to solve for V2:
V2 = (nRT2) / P2
Substituting the known values:
0.0396 m³ = (1 mol * 8.314 J/(mol·K) * T2) / 1 atm
T2 ≈ 0.0048 K
Next, convert the temperature back to Celsius:
T2 ≈ 0.0048 K - 273.15 K ≈ -273.14 °C
The temperature is extremely low, which means the bubble would have contracted significantly. In reality, at such low temperatures, methane gas may convert to a solid or a liquid. Assuming the bubble remains a gas, we can apply the ideal gas law to find the new diameter:
V2 = (4/3) * π * (d2/2)³
Where:
- V2 is the volume of the bubble at the surface
- d2 is the diameter of the bubble at the surface
Rearranging the equation to solve for d2:
d2 = [(3 * V2) / (4 * π)]^(1/3)
Substituting the known value for V2:
d2 ≈ [(3 * 0.0396 m³) / (4 * π)]^(1/3)
d2 ≈ 0.0904 m
Therefore, when the bubble reaches the surface, its diameter will be approximately 0.0904 meters.