91Ó°ÊÓ

The coefficient of volume expansion, denoted as \( \beta \), is a constant that describes how much a substance's volume changes with temperature. It is an intrinsic property of materials and is usually expressed in \( \text{°C}^{-1} \) or \( \text{K}^{-1} \). This coefficient tells us how much one unit of volume will expand per degree of temperature increase.

In our exercise, we used the coefficient of volume expansion for water, which is \( 207 \times 10^{-6} \text{°C}^{-1} \). This means that for every °C the temperature increases, the water's volume increases by \( 207 \times 10^{-6} \) times its original volume.

Understanding this concept is essential because it allows us to predict how substances will behave when heated or cooled. For instance, materials with a high coefficient expand more significantly than those with a low coefficient, which can be crucial when designing containers or structures.

When we talk about temperature change, we refer to the difference between the initial and final temperatures of an object or substance. It is represented by \( \Delta T \) in the volume expansion formula.

In our example, the coffee's temperature changed from \( 18^{\circ} \text{C} \) to \( 92^{\circ} \text{C} \), giving a \( \Delta T \) of \( 92^{\circ} \text{C} - 18^{\circ} \text{C} = 74^{\circ} \text{C} \). This calculation is straightforward, but it's a critical step because the temperature change determines how much the coffee will expand.

Temperature changes can have various effects, from causing liquids to overflow to affecting the strength and integrity of materials. Therefore, understanding and calculating \( \Delta T \) is a fundamental skill in physics and engineering.

Volume calculation regarding thermal expansion is about determining how much the volume of a substance changes when its temperature changes. This is where the formula \( \Delta V = \beta V_0 \Delta T \) becomes vital.

In the exercise, we had:

• \( \beta = 207 \times 10^{-6} \text{°C}^{-1} \)
• Initial volume \( V_0 = 0.50 \times 10^{-3} \text{ m}^3 \)
• Temperature change \( \Delta T = 74^{\circ} \text{C} \)

Substituting these values into the formula, we calculated the increase in volume as \( \Delta V = 7.659 \times 10^{-6} \text{ m}^3 \).

This resultant \( \Delta V \) tells us that approximately \( 7.659 \times 10^{-6} \text{ m}^3 \) of coffee overflowed due to the temperature increase. These calculations help when precision is essential, such as in a lab setting or industrial process, ensuring that components will handle temperature changes without failure.