To determine the volume of the air bubble just before it reaches the surface of the lake, we can use the combined gas law, which relates the pressure, volume, and temperature of a gas. The law states that for a given amount of gas, the ratio of the product of pressure and volume to the temperature is constant. This can be expressed as:
P1 × V1 / T1 = P2 × V2 / T2
Where:
- P1 = initial pressure at the bottom of the lake
- V1 = initial volume of the bubble (19.4 cm³)
- T1 = initial temperature (3.80ºC in Kelvin)
- P2 = pressure at the surface of the lake
- V2 = final volume of the bubble (what we want to find)
- T2 = final temperature (22.6ºC in Kelvin)
Step 1: Convert Temperatures to Kelvin
First, we need to convert the temperatures from Celsius to Kelvin. The conversion formula is:
T(K) = T(ºC) + 273.15
For T1:
T1 = 3.80 + 273.15 = 276.95 K
For T2:
T2 = 22.6 + 273.15 = 295.75 K
Step 2: Calculate the Pressures
Next, we need to calculate the pressures at the bottom and the surface of the lake. The pressure at a depth in a fluid can be calculated using the formula:
P = P0 + ρgh
Where:
- P0 = atmospheric pressure (approximately 101.3 kPa)
- ρ = density of water (approximately 1000 kg/m³)
- g = acceleration due to gravity (approximately 9.81 m/s²)
- h = depth of the water (41.5 m)
Calculating the pressure at the bottom (P1):
P1 = 101.3 kPa + (1000 kg/m³ × 9.81 m/s² × 41.5 m)
P1 = 101.3 kPa + 407,415 Pa = 508.715 kPa
At the surface (P2), the pressure is simply the atmospheric pressure:
P2 = 101.3 kPa
Step 3: Apply the Combined Gas Law
Now that we have all the values, we can rearrange the combined gas law to solve for V2:
V2 = V1 × (P1 / P2) × (T2 / T1)
Substituting the known values:
V2 = 19.4 cm³ × (508.715 kPa / 101.3 kPa) × (295.75 K / 276.95 K)
Step 4: Calculate V2
Calculating the ratios:
P1 / P2 = 508.715 / 101.3 ≈ 5.02
T2 / T1 = 295.75 / 276.95 ≈ 1.07
Now, substituting these values back into the equation:
V2 ≈ 19.4 cm³ × 5.02 × 1.07
V2 ≈ 19.4 cm³ × 5.37 ≈ 104.4 cm³
Final Result
Thus, the volume of the air bubble just before it reaches the surface of the lake is approximately 104.4 cm³. This demonstrates how changes in pressure and temperature affect the volume of gases in a fluid environment.
