Problem 88 - University Physics with Modern Physics

Question

On a cool \(\left(4,0^{\circ} \mathrm{C}\right)\) Saturfay moming, a pilot fills the fuel tanks of her Pitts \(S-2 C\) (a two-seat aerobatic airplane) to their full capacity of 106.0 L. Before flying on Sunday morning, when the temperature is again \(4.0^{\circ} \mathrm{C}\) , 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 "C) reached by the fuel and the tank on Saturday aftemoon? 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

(a) The maximum temperature was 29.9°C. (b) She should have filled the tanks on Saturday afternoon.

Step 1: Determine the Change in Volume

The change in volume of the gasoline can be calculated as the difference between the initial volume filled and the volume remaining after the thermal expansion. \[\Delta V = 106.0 \, \text{L} - 103.4 \, \text{L} = 2.6 \, \text{L}.\]

Step 2: Use the Formula for Volume Expansion

The formula for volume expansion due to temperature change is given by:\[\Delta V = \beta V_0 \Delta T,\]where \(\beta\) is the coefficient of volume expansion and \(\Delta T\) is the change in temperature. Rearrange to solve for \(\Delta T\):\[\Delta T = \frac{\Delta V}{\beta V_0}.\]

Step 3: Plug in Known Values to Find Temperature Change

Plug in the known values: \(\Delta V = 2.6 \, \text{L}\), \(\beta = 9.5 \times 10^{-4} \, \text{K}^{-1}\), and \(V_0 = 106.0 \, \text{L}\). Calculate \(\Delta T\):\[\Delta T = \frac{2.6 \, \text{L}}{9.5 \times 10^{-4} \, \text{K}^{-1} \times 106.0 \, \text{L}} \approx 25.9 \, \text{K}.\]

Step 4: Calculate Maximum Temperature

The maximum temperature reached can be calculated by adding the temperature change to the initial temperature:\[T_{\text{max}} = 4.0\,^{\circ}\mathrm{C} + 25.9 \, \text{K} = 29.9\,^{\circ}\mathrm{C}.\]

Step 5: Determine Optimal Filling Time

To have the maximum amount of fuel available for flight, the pilot should have filled the tanks when the temperature was at its maximum, on Saturday afternoon.

Key Concepts

Coefficient of Volume Expansion

When substances are heated, they often expand in volume. The "Coefficient of Volume Expansion" is a measure that describes how much a specific volume of material will increase in volume for each degree of temperature increase. This coefficient is symbolized by \( \beta \) and is expressed in units of \( \text{K}^{-1} \).

In the case of liquids, the coefficient of volume expansion is usually larger compared to solids. This is because the particles in liquids are generally more free to move and spread out when heated. For example, gasoline has a coefficient of volume expansion of \( 9.5 \times 10^{-4} \ \text{K}^{-1} \), indicating how it can noticeably expand with even slight temperature changes.

To calculate how much a liquid like gasoline expands, the volume change formula \( \Delta V = \beta V_0 \Delta T \) is used, where:

• \( \Delta V \) is the change in volume
• \( \beta \) is the coefficient of volume expansion
• \( V_0 \) is the original volume
• \( \Delta T \) is the change in temperature

Keep in mind: A higher \( \beta \) means the substance is more sensitive to temperature changes.

Temperature Change

Temperature change is central to understanding thermal expansion. In physics problems involving thermal expansion, calculating how the temperature affects the volume requires precise calculations based on known properties. For aluminum tanks and gasoline, this means considering both the tank’s and fuel’s responses to heat.

Here's how temperature change plays a role:

• The change in temperature (\( \Delta T \)) can be calculated by rearranging the volume expansion formula to \( \Delta T = \frac{\Delta V}{\beta V_0} \)
• Knowing how much the volume changed, along with the initial volume and the coefficient of volume expansion, you can determine the temperature change.

For instance, in our problem, when the volume of gasoline changes by 2.6 L upon warming, given a coefficient of \( 9.5 \times 10^{-4} \ \text{K}^{-1} \), we find \( \Delta T \approx 25.9 \ \text{K} \). This tells us how much warmer the environment was that caused the fuel to expand."
Understanding these details about temperature change helps in predicting how materials behave in different thermal conditions.

Aluminum Expansion

Aluminum is widely used in tank construction, including those for airplanes, due to its lightweight and durable nature. However, like all materials, aluminum expands when heated, although not as significantly as fluids like gasoline.

Here's what you should know about aluminum expansion:

• Aluminum's coefficient of linear expansion is typically about \( 23 \times 10^{-6} \ \text{K}^{-1} \), which is much lower than gasoline's volume expansion coefficient.
• Aluminum's response to temperature is more about slight expansions that can affect joint fittings and seals.
• In practical terms, the aluminum tank itself might expand slightly, contributing to more space for an expanding fluid inside, but any outlet or vent could loosen, leading to fuel loss.

Despite its lower expansion rate, the aluminum tank’s slight expansion still plays a role in fuel management strategies, especially in high-temperature conditions.