Problem 78 - University Physics with Modern Physics

Question

On a cool (4.0\(^\circ\)C) Saturday morning, a pilot fills the fuel tanks of her Pitts S-2C (a two-seat aerobatic airplane) to their full capacity of 106.0 L. Before flying on Sunday morning, when the temperature is again 4.0\(^\circ\)C, she checks the fuel level and finds only 103.4 L of gasoline in the aluminum tanks. She realizes that it was hot on Saturday afternoon and that thermal expansion of the gasoline caused the missing fuel to empty out of the tank's vent. (a) What was the maximum temperature (in \(^\circ\)C) of the fuel and the tank on Saturday afternoon? The coefficient of volume expansion of gasoline is \(9.5 \times 10{^-}{^4} K{^-}{^1}\). (b) To have the maximum amount of fuel available for flight, when should the pilot have filled the fuel tanks?

Step-by-Step Solution

Answer

The maximum temperature was 30.0°C. The pilot should have filled the tanks in the morning when it was cooler.

Step 1: Understanding the Problem

The initial volume of gasoline is 106.0 L at a temperature of 4.0°C, and it reduces to 103.4 L, at the same temperature. The change indicates thermal expansion has taken place, and some gasoline overflowed. The coefficient of volume expansion is provided, which helps in calculating the temperature change that resulted in this overflow.

Step 2: Volume Expansion Formula

The formula for volume expansion is given by \[\Delta V = \beta \cdot V_0 \cdot \Delta T\]where \(\Delta V\) is the change in volume, \(\beta\) is the coefficient of volume expansion, \(V_0\) is the initial volume, and \(\Delta T\) is the change in temperature.

Step 3: Identify the Change in Volume

The change in volume, \(\Delta V\), is calculated as the difference between the initial volume \((V_0 = 106.0\, L)\) and the final volume \((V_f = 103.4\, L)\). Thus, \(\Delta V = V_0 - V_f = 106.0\, L - 103.4\, L = 2.6\, L\).

Step 4: Solve for Temperature Change

Rearrange the volume expansion formula to solve for \(\Delta T\):\[\Delta T = \frac{\Delta V}{\beta \cdot V_0}\]Substitute the known values:\[\Delta T = \frac{2.6\, L}{(9.5 \times 10^{-4}\, K^{-1}) \cdot 106.0\, L}\]Perform the calculation:\[\Delta T \approx 26\, \text{K}\]

Step 5: Calculate Maximum Temperature

To find the maximum temperature \((T_{max})\) on Saturday afternoon, add the temperature change \(\Delta T\) to the initial temperature:\[T_{max} = T_{initial} + \Delta T = 4.0\, ^\circ \text{C} + 26\, K = 30.0\, ^\circ \text{C}\]

Step 6: Determine the Best Time to Fill the Tank

To maximize the amount of fuel available for flight, the pilot should fill the fuel tanks when the temperature is the lowest. By doing this, the thermal expansion effect will be minimized, preventing any overflow when the temperature rises. Therefore, the pilot should have filled the tanks in the morning when it was cooler.

Key Concepts

Volume Expansion Formula

The Volume Expansion Formula is a key concept in understanding how fluids like gasoline respond to temperature changes. When the temperature of a substance increases, its volume generally increases as well. This relationship is captured by the equation:

• \( \Delta V = \beta \cdot V_0 \cdot \Delta T \)

Here, \( \Delta V \) represents the change in volume, \( V_0 \) is the initial volume of the substance, \( \beta \) is the coefficient of volume expansion, and \( \Delta T \) signifies the change in temperature. This formula allows us to calculate how much a substance will expand, based on these parameters.
In the context of the exercise, the initial volume \( V_0 \) is 106.0 L, and due to thermal expansion, the volume decreases to 103.4 L after some gasoline overflowed. By applying the volume expansion formula, we calculate the temperature change that resulted from a hot afternoon, helping us to understand the event's cause.

Coefficient of Volume Expansion

The Coefficient of Volume Expansion, \( \beta \), is a crucial factor that determines how much a given volume of a substance will expand when heated. It is specific to the material in question. For gasoline, the coefficient of volume expansion is provided as \( 9.5 \times 10^{-4} \, K^{-1} \).
This means that for each degree Kelvin (K) temperature increase, the volume of gasoline expands by \( 9.5 \times 10^{-4} \) times its original volume. This coefficient allows us to calculate the temperature change that led to the volume shrinkage observed on Sunday morning. By using the coefficient in the volume expansion formula, students can rearrange the equation to solve for the temperature change \( \Delta T \), illustrating how material and temperature interact.

Aerobatic Airplane Fuel Management

Managing fuel in an aerobatic airplane involves understanding thermal expansion to prevent fuel loss during temperature changes. In aviation, the efficiency and safety of a flight can depend greatly on proper fuel management.
On a hot day, gasoline expands due to temperature increase. If the tanks were completely filled, as in the Pitts S-2C scenario, expanding gasoline can lead to overflow through vents, leading to needless fuel loss.
Pilots can manage this by timing the refueling process. Filling the fuel tanks during cooler parts of the day, like in the morning or late evening, ensures maximum fuel stays in the tanks as temperatures rise. This practice not only optimizes fuel capacity but also maintains the balance and performance of the plane during flight. Hence, understanding and applying thermal expansion principles can critically enhance an aircraft's operational efficiency.