Problem 84 - University Physics with Modern Physics

Question

On a cool \(\left(4.0^{\circ} \mathrm{C}\right)\) Saturday morning, a pilot fills the fuel tanks of her Pitts \(\mathrm{S}-2 \mathrm{C}(\) a two-seat aerobatic airplane) to their full capacity of 106.0 \(\mathrm{L} .\) Before flying on Sunday morning, when the temperature is again \(4.0^{\circ} \mathrm{C},\) she checks the fuel level and finds only 103.4 \(\mathrm{L}\) of gasoline in the 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 \(^{\circ} \mathrm{C} )\) reached by the fuel and the tank on Saturday afternoon? The coefficient of volume expansion of gasoline is \(9.5 \times 10^{-4} \mathrm{K}^{-1}\) , and the tank is made of aluminum. (b) In order 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 approximately \(29.9^{\circ} \text{C}\). The tanks should be filled when it's coolest.

Step 1: Understand the Problem

The fuel level decreased from 106.0 L to 103.4 L due to thermal expansion and subsequent cooling. We need to determine the maximum temperature reached by the fuel when it expanded.

Step 2: Identify Known Values

Initial volume of gasoline, \( V_0 = 106.0 \text{ L} \). Final volume of gasoline, \( V_f = 103.4 \text{ L} \). Coefficient of volume expansion for gasoline, \( \beta = 9.5 \times 10^{-4} \text{ K}^{-1} \). Initial and final temperature are both \(4.0^{\circ} \text{C}\).

Step 3: Calculate Change in Volume

The change in volume, \( \Delta V \), is given by the difference between the initial and final volumes. Thus, \( \Delta V = 106.0 \text{ L} - 103.4 \text{ L} = 2.6 \text{ L} \).

Step 4: Apply the Formula for Volume Expansion

The relation for volume expansion is \( \Delta V = \beta V_0 \Delta T \), where \( \Delta T \) is the change in temperature. Substitute the known values to find \( \Delta T \).

Step 5: Solve for Maximum Temperature Change

Rearrange the formula to solve for \( \Delta T \):\[ \Delta T = \frac{\Delta V}{\beta V_0} = \frac{2.6}{9.5 \times 10^{-4} \times 106.0} \approx 25.9 \text{ K} \].

Step 6: Determine Maximum Temperature

Since the initial temperature is \(4.0^{\circ} \text{C}\), the maximum temperature \( T \) on Saturday was:\[ T = 4.0^{\circ} \text{C} + 25.9 \text{ K} \approx 29.9^{\circ} \text{C} \].

Step 7: Determine Optimal Time to Fill the Tanks

To maximize the amount of fuel available, the pilot should fill the tanks when the temperature is lowest, so it should be filled at the start of the day when it's coolest.

Key Concepts

Coefficient of Volume Expansion

The coefficient of volume expansion is a fundamental concept in the study of thermal expansion. It describes how the volume of a material changes with temperature. For any substance, this coefficient, often denoted as \( \beta \), indicates how much the substance's volume will expand per degree of temperature change.

- **Measurement**: It is measured in units of reciprocal Kelvin \((\text{K}^{-1})\).
- **Applications**: This coefficient is crucial for designing systems that will experience temperature changes, such as fuel tanks.

In our exercise, the coefficient of volume expansion for gasoline was given as \( 9.5 \times 10^{-4} \text{ K}^{-1} \). Understanding this value enabled us to calculate how the gasoline's volume changed with temperature, leading to the observed decrease in fuel due to thermal expansion.

Change in Volume

Volume change due to temperature fluctuation is a direct outcome of thermal expansion. This change, represented as \( \Delta V \), is the magnitude of volume increase or decrease due to a temperature change \( \Delta T \).
To calculate the change in volume, we use the formula:
\[ \Delta V = V_0 \beta \Delta T \]
where \( V_0 \) is the initial volume, \( \beta \) is the coefficient of volume expansion, and \( \Delta T \) is the temperature change.

In the Pitts S-2C airplane scenario, the initial and final volumes of gasoline were 106.0 L and 103.4 L, respectively. Therefore, the change in volume, \( \Delta V \), was calculated as 2.6 L. This calculation helped identify the extent of the gasoline's expansion due to the temperature rise.

Temperature Change Calculation

Calculating temperature change involves understanding how a material's volume changes with thermal conditions. This requires rearranging the formula for volume expansion.
The rearrangement provides the temperature change formula:
\[ \Delta T = \frac{\Delta V}{\beta V_0} \]
This expression allows us to solve for \( \Delta T \) given the change in volume \( \Delta V \), initial volume \( V_0 \), and the coefficient of volume expansion \( \beta \).

In our example, substitution of the given and calculated values (\( \Delta V = 2.6 \text{ L} \), \( \beta = 9.5 \times 10^{-4} \text{ K}^{-1} \), and \( V_0 = 106.0 \text{ L} \)) into the equation yielded \( \Delta T \approx 25.9 \text{ K} \). This temperature change, combined with the initial 4.0°C, indicates that the fuel likely heated to a maximum of around 29.9°C on that hot Saturday afternoon.