Problem 97 - Physics

Question

Concept Questions An aluminum can is filled to the brim with a liquid. The can and the liquid are heated so their temperatures change by the same amount. (a) In general, what factors determine how the volume of an object changes when it is heated? (b) Aluminum has a smaller coefficient of volume expansion than the liquid does. Which, if either, will expand more, the can or the liquid? Provide a reason for your answer. (c) How is the volume of liquid that spills over related to the changes in the volume of the can and the liquid?

Step-by-Step Solution

Answer

(a) Initial volume, temperature change, and material's volumetric expansion coefficient. (b) The liquid, due to its larger expansion coefficient. (c) Spilled volume equals liquid's expansion minus can's expansion.

Step 1: Understanding Factors in Volume Change

When an object is heated, its volume changes due to thermal expansion. The key factors affecting this change are the initial volume of the object, the temperature change, and the material's coefficient of volumetric expansion \( \beta \). The volume change \( \Delta V \) can be calculated using the formula \( \Delta V = V_0 \beta \Delta T \), where \( V_0 \) is the initial volume and \( \Delta T \) is the temperature change.

Step 2: Comparing Expansion Coefficients

The coefficient of volumetric expansion \( \beta \) determines how much an object's volume changes with temperature. In this scenario, the liquid in the can has a larger \( \beta \) than aluminum, meaning the liquid will expand more than the can for the same temperature change. Therefore, even if both are exposed to the same temperature change, the difference in \( \beta \) values indicates that the liquid will expand more than the aluminum can.

Step 3: Consequence of Differential Expansion

As the liquid expands more than the can, the excess volume of the liquid will spill over. The volume of liquid that spills over is equal to the difference between the volume expansion of the liquid and the volume expansion of the can. Mathematically, this is expressed as \( \Delta V_{liquid} - \Delta V_{can} \), assuming the can was initially filled to the brim.

Key Concepts

Coefficient of Volumetric Expansion

In the fascinating world of thermal expansion, the coefficient of volumetric expansion, denoted as \( \beta \), plays a pivotal role. It tells us how much an object's volume changes with a change in temperature. This coefficient is distinct for different materials, which is why substances react differently when exposed to identical temperature changes.

• The coefficient of volumetric expansion does not have a constant value across all materials. This means, for instance, metals generally expand differently compared to liquids when heated.


• For practical scenarios, the larger the value of \( \beta \), the more a material will expand for a given temperature increase.


• Understanding \( \beta \) helps in predicting and managing the physical behavior of materials under temperature variations, which is especially crucial in industrial applications.

By knowing the \( \beta \) of the liquid and the can, we can precisely determine which expands more when both are heated, as illustrated by the aluminum can and liquid scenario.

Volume Change Formula

The volume change formula is a simple yet powerful tool in analyzing thermal expansion. It's used to compute the change in volume \( \Delta V \) of a material when it undergoes a temperature change \( \Delta T \). The formula is given by:\[ \Delta V = V_0 \beta \Delta T \]
In this formula:

• \( V_0 \) represents the initial volume of the substance before heating.


• \( \beta \) is the coefficient of volumetric expansion, highlighting the material's responsiveness to temperature changes.


• \( \Delta T \) refers to the change in temperature.

Using this formula, you can predict how much a material's volume will increase or decrease when its temperature changes. For example, if you know the initial volume, coefficient of expansion, and temperature change, you can find the expanded volume of the liquid and the aluminum can separately. This understanding allows us to foresee the behavior of materials under thermal stress.

Differential Expansion

Differential expansion occurs when different materials exposed to the same temperature change expand by different amounts due to their distinct coefficients of volumetric expansion. This concept is crucial, especially in systems involving more than one material.

• In our scenario, the aluminum can and the liquid have different expansion coefficients. The liquid's higher \( \beta \) means it expands more than the can as both are heated.


• The difference in expansion leads to an overflow of liquid, as it expands beyond the confines of the can.


• This process is calculated by finding the difference between the liquid's expansion and the can's expansion: \( \Delta V_{liquid} - \Delta V_{can} \).

Understanding differential expansion is critical in designing containers and structures where mixed materials are involved, such as preventing leakage in thermal systems or ensuring structural integrity in construction.