Step 1: Understanding the Question:
The problem involves a gas bubble undergoing changes in pressure and temperature as it moves vertically through a fluid. We need to find the final volume by relating it to the initial conditions.
Crucially, all gas law calculations must be performed using absolute temperatures (Kelvin).
Step 2: Key Formula or Approach:
We use the Combined Gas Law:
\[ \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \]
First, convert temperatures from Celsius to Kelvin:
\(T_1 = 7 + 273 = 280 K\)
\(T_2 = 27 + 273 = 300 K\)
Step 3: Detailed Explanation:
Step A: List the initial state variables (bottom of the lake):
\(P_1 = 3 \text{ atm}\)
\(V_1 = V_0\)
\(T_1 = 280 \text{ K}\)
Step B: List the final state variables (surface):
\(P_2 = 1 \text{ atm}\)
\(V_2 = V_f\) (unknown)
\(T_2 = 300 \text{ K}\)
Step C: Rearrange the Combined Gas Law to solve for \(V_f\):
\[ V_f = \frac{P_1 \cdot V_0 \cdot T_2}{P_2 \cdot T_1} \]
Step D: Substitute the known values into the equation:
\[ V_f = \frac{3 \cdot V_0 \cdot 300}{1 \cdot 280} \]
Step E: Simplify the numerical fraction:
\[ V_f = V_0 \cdot \frac{900}{280} \]
\[ V_f = V_0 \cdot \frac{90}{28} = V_0 \cdot \frac{45}{14} \]
Step F: Calculate the final decimal value:
\(45 / 14 \approx 3.21428...\)
As the bubble rises, the pressure decreases significantly (from 3 atm to 1 atm), which causes it to expand. Simultaneously, the temperature increases (from 280K to 300K), which also promotes expansion. Both effects work together to increase the volume.
Step 4: Final Answer:
The final volume \(V_f\) of the bubble at the surface is \(3.21 V_0\).