at 25.0m below the surface of the sea (density=1.025kg/m^3), where the temperature is 5.00C, a diver exhales an air bubble having a volume of 1.00cm^3. If the surface temperature of the sea is 20.0C, what is the volume of the bubble just before it breaks the surface?
Your sea density is incorrect. It is 1.025*10^3 kg/m^3
First calculate the pressure at 25 m depth, making sure that you add atmospheric pressure Po to (density)*g*(depth).
P(25 m) = 1.01*10^5 + (1.025*10^3)(9.81)*25.0 = 1.01*10^5 + 2.51*10^5 = 3.52*10^5
Prssure at 25 m depth is about 3.5 times larger than at the surface.
The ratio of bubble volume at the surface to the volume at 25 m is
[T(surface)/T(25m)]*[P(25m)/P(surface)]
=(293/278)*(3.5)
Check my numbers.
To calculate the volume of the bubble just before it breaks the surface, we need to consider the change in pressure and the change in temperature of the air inside the bubble.
Here's how you can calculate it step by step:
1. Find the pressure at the depth of 25.0m below the surface of the sea:
- The pressure at any depth in a fluid is given by the formula P = P₀ + ρgh, where P₀ is the initial pressure, ρ is the density of the fluid, g is the acceleration due to gravity, and h is the depth.
- In this case, the initial pressure (P₀) is the atmospheric pressure at the surface, which is typically around 1 atm.
- The density (ρ) of seawater is given as 1.025 kg/m^3.
- The acceleration due to gravity (g) is approximately 9.8 m/s^2.
- The depth (h) is 25.0m.
- Plug these values into the formula to calculate the pressure at the depth of 25.0m below the surface.
2. Find the final pressure at the surface of the sea:
- The pressure at the surface of the sea is also the atmospheric pressure, which is typically around 1 atm.
3. Find the change in pressure:
- Subtract the initial pressure from the final pressure to find the change in pressure.
4. Use the Ideal Gas Law to find the volume change:
- The Ideal Gas Law states that PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature.
- Rearrange the equation to solve for volume (V).
- Since the number of moles (n) and the ideal gas constant (R) are constant, we can write PV = kT, where k is a constant.
- Divide the initial pressure by the initial temperature and multiply by the final temperature to find the new volume.
5. Convert the volume to the appropriate units:
- Convert the volume from cm^3 to m^3 to match the units used for the density.
By following these steps, you'll be able to calculate the volume of the bubble just before it breaks the surface.
To calculate the volume of the bubble just before it breaks the surface, we need to consider the change in pressure and temperature.
Step 1: Convert the given temperatures to Kelvin.
- The temperature at a depth of 25.0m below the surface is 5.00°C = 5.00 + 273.15 = 278.15 K.
- The surface temperature is 20.0°C = 20.0 + 273.15 = 293.15 K.
Step 2: Determine the change in pressure.
The pressure changes with depth and can be calculated using the equation:
ΔP = ρgh
Where:
ΔP is the change in pressure
ρ is the density of the sea
g is the acceleration due to gravity
h is the depth
Given values:
ρ = 1.025 kg/m^3
g = 9.8 m/s^2
h = 25.0 m
ΔP = (1.025 kg/m^3) * (9.8 m/s^2) * (25.0 m)
ΔP = 252.25 Pa
Step 3: Apply Boyle's Law to find the volume at the surface.
Boyle's Law states that the product of the initial pressure and the initial volume is equal to the product of the final pressure and the final volume (assuming constant temperature):
P1 * V1 = P2 * V2
Where:
P1 and V1 are the initial pressure and volume respectively,
P2 and V2 are the final pressure and volume respectively.
The initial pressure can be calculated as:
P1 = atmospheric pressure + ΔP
Given values:
Atmospheric Pressure = 101325 Pa
P1 = 101325 Pa + 252.25 Pa
P1 = 101577.25 Pa
Now we can rearrange Boyle's Law to solve for V2:
V2 = (P1 * V1) / P2
The volume at the depth of 25.0m below the surface is given as 1.00 cm^3, which can be converted to m^3:
V1 = 1.00 cm^3 = 1.00 * 10^-6 m^3
Using the given surface temperature:
P2 = atmospheric pressure = 101325 Pa
V2 = (101577.25 Pa * 1.00 * 10^-6 m^3) / 101325 Pa
V2 = 1.003 cm^3
Therefore, the volume of the bubble just before it breaks the surface is approximately 1.003 cm^3.