Problem 47 - College Physics

Question

One morning, an employee at a rental car company fills a car's steel gas tank to the top and then parks the car a short distance away. (a) That afternoon, when the temperature increases, will any gas overflow? Why? (b) If the temperatures in the morning and afternoon are, respectively, \(10^{\circ} \mathrm{C}\) and \(30^{\circ} \mathrm{C}\) and the gas tank can hold 25 gal in the morning, how much gas will be lost? (Neglect the expansion of the tank.)

Step-by-Step Solution

Answer

Yes, 0.475 gallons will overflow.

Step 1: Understand the Effect of Temperature on Gasoline

When temperature increases, gasoline expands. Since the tank is full in the morning and the temperature rises from 10°C to 30°C, the gasoline will expand and might overflow outside the tank.

Step 2: Volume Expansion Formula

To find out how much gasoline will overflow, use the volume expansion formula. This is given by: \[ V = V_0 (1 + \beta \Delta T) \]where \( V_0 \) is the initial volume, \( \beta \) is the coefficient of volume expansion for gasoline (approximately \( 0.00095 /°C \)), and \( \Delta T \) is the change in temperature.

Step 3: Apply Volume Expansion to Compute Overflow

First, calculate \( \Delta T \):\[ \Delta T = 30°C - 10°C = 20°C \]Substitute \( V_0 = 25 \text{ gallons} \), \( \beta = 0.00095 /°C \), and \( \Delta T = 20°C \) into the formula to find \( V \):\[ V = 25 (1 + 0.00095 \times 20) \]\[ V = 25 (1 + 0.019) = 25 \times 1.019 = 25.475 \text{ gallons} \]

Step 4: Calculate Overflow Volume

Compute the overflow by finding the difference between the expanded volume and the initial volume:\[ \text{Overflow} = V - V_0 \]\[ \text{Overflow} = 25.475 - 25 = 0.475 \text{ gallons} \]

Step 5: Conclusion on Gasoline Overflow

Since the expanded volume is greater than the capacity of the tank (25 gallons), there will be an overflow. The volume of gasoline that overflows is 0.475 gallons.

Key Concepts

Volume Expansion Formula

When substances like gasoline are subjected to heat, they expand. This property is described using the volume expansion formula. The formula helps predict how much expansion occurs when the temperature changes. The volume expansion formula is given by the following equation:\[ V = V_0 (1 + \beta \Delta T) \]
The components of this formula are simple to understand:

• \( V \): Final volume after expansion
• \( V_0 \): Initial volume before any temperature change occurs
• \( \beta \): Coefficient of volume expansion which represents how much a substance expands per degree Celsius temperature change
• \( \Delta T \): The change in temperature, calculated as final temperature minus initial temperature

This formula is a handy tool for calculating how much a liquid will expand when heated, ensuring you can anticipate any overflow, as in the case of a gas tank filled with gasoline.

Temperature Change

Temperature change is a key factor in understanding thermal expansion. It is the difference between the final temperature and the initial temperature. In our scenario with the car gas tank, the temperature change (\( \Delta T \)) is calculated as:

• Afternoon temperature: 30°C
• Morning temperature: 10°C
• Temperature change: \( \Delta T = 30°C - 10°C = 20°C \)

This change plays a crucial role in the volume expansion formula. A higher temperature change means a larger expansion in volume, considering the coefficient of volume expansion remains constant. Thus, knowing the temperature change allows for accurate predictions of how much a substance will expand.

Coefficient of Volume Expansion

The coefficient of volume expansion \( \beta \) is an essential constant in the volume expansion formula. It defines how much a given volume of a substance will expand per degree Celsius of temperature change.
For gasoline, the coefficient of volume expansion is approximately \( 0.00095 /°C \). This value indicates that for every degree Celsius increase in temperature, one cubic unit of gasoline will expand by 0.095% of its original volume.
Understanding \( \beta \) is crucial because it varies for different materials. For example:

• Solids generally have smaller coefficients compared to liquids.
• Gases tend to have higher coefficients of expansion than both solids and liquids given their free particle movement.

In practical scenarios, like filling a steel gas tank, it alerts you to how much a specific liquid will expand if the temperature rises. Thus, it plays a vital role in calculating the amount of overflow or expansion that might occur.