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The coefficient of volume expansion is a crucial concept for understanding how substances change in volume with temperature. This coefficient, often denoted as \( \beta \), measures how much a material expands per degree change in temperature. It is expressed in units of \( \mathrm{K}^{-1} \) or \( \mathrm{C}^{-1} \).When a liquid like gasoline heats up, its molecules move more vigorously, leading to an increase in volume. The degree to which the volume changes is proportional to both the temperature change and the coefficient of volume expansion.

• A high \( \beta \) value indicates that the substance will expand significantly even with a small rise in temperature.
• A low \( \beta \) implies that temperature changes have less impact on the material's volume.

For gasoline, \( \beta = 9.5 \times 10^{-4} \mathrm{~K}^{-1} \), meaning it has a relatively high capacity to expand with temperature changes compared to many other liquids.

Calculating the temperature rise that a substance can endure involves applying the formula for volume expansion. When a material like gasoline is contained, any extra expansion due to heat can lead to significant pressure build-up, potentially causing a rupture.To find the allowable increase in temperature \( ( \Delta T ) \), you use:\[\Delta T = \frac{\Delta V}{\beta V_0}\]Where:

• \( \Delta V \) is the change in volume.
• \( \beta \) is the coefficient of volume expansion of the liquid.
• \( V_0 \) is the initial volume of the liquid.

In the case of the gasoline exercise, the maximum tolerable increase in volume (before the container ruptures) is 1% of the container's volume. Thus, by rearranging and simplifying the formula:\[\Delta T = \frac{0.01}{9.5 \times 10^{-4}} \approx 10.53 \mathrm{~K}\]This means the gasoline can safely endure a temperature rise of approximately 10.53°C without risking the container's integrity.

Gasoline expansion is an important consideration in storage and transport. As a fluid, gasoline is subject to changes in volume based on temperature variance, largely due to its relatively high coefficient of volume expansion. In practical scenarios, when gasoline's temperature increases, it expands, potentially leading to overflow or, in enclosed containers, pressure build-up. This is especially important in vehicles and storage units where temperature fluctuations are common.

• In colder climates, as gasoline warms from cold to a more moderate temperature, it may occupy significantly more volume.
• Similarly, in hot environments, the expansion becomes a critical factor in storage design and safety.

Proper design of storage containers must account for this expansion to prevent the risks associated with rupture. In the example, a container can withstand up to a 1% expansion, translating to a temperature rise of about 10.53°C, making understanding these principles essential for safe handling and storage.