Question

An aluminum cup of \(110 \mathrm{~cm}^{3}\) capacity is filled with glycerin at \(22^{\circ} \mathrm{C}\). How much glycerin, if any, will spill out of the cup if the temperature of the cup and glycerin is raised to \(28^{\circ} \mathrm{C}\) ? (The coefficient of volume expansion of glycerin is \(5.1 \times\) \(\left.10^{-4} / \mathrm{C}^{\circ} .\right)\)

Short answer

By the increase in temperature, the volume of glycerin increased by 3.366 cm^3. So, this amount of glycerin will spill out of the cup.

Step-by-step solution

Identify the knowns and unknowns

From the problem, the initial volume of glycerin V0 = 110 cm^3, β = 5.1x10^-4 /°C, initial temperature = 22°, final temperature = 28°C. Our aim is to calculate ∆V.

Calculate the change in temperature

Calculate the change in temperature using the formula ∆T = Final temperature - Initial temperature. Therefore we get ∆T = 28°C - 22°C = 6°C.

Calculate the change in volume

Substitute the values of β, V0 and ∆T in the formula for volume expansion ∆V = βV0∆T. Therefore we get ∆V = 5.1x10^-4 /°C x 110 cm^3 x 6°C = 3.366 cm^3.

Key concepts

Volume Expansion

When objects, especially liquids, change temperature, their volume can change. This behavior is known as "volume expansion." It happens because particles in a material move faster as they get warmer, taking up more space.
Volume expansion is something you notice when heating water, for instance, as it expands slightly.

• For liquids, the volume expansion is a prominent factor when the temperature changes.
• The calculation involves knowing the initial volume and then finding out how much that volume changes when heated.

This concept is crucial in everyday applications, like filling containers with liquids, ensuring there’s enough space for potential expansion. In our exercise, when the temperature of the glycerin rises, its volume increases, potentially causing some to overflow the cup.

Coefficient of Volume Expansion

The coefficient of volume expansion, usually denoted as \( \beta \), is a measure of how much a material's volume changes with temperature. It is specific to each material.

• A larger value of \( \beta \) means the material expands more with each degree of temperature change.
• For glycerin, the coefficient is \( 5.1 \times 10^{-4} / ^{\circ}C \), suggesting it expands significantly as it warms.

You use this coefficient in the formula for volume expansion: \( \Delta V = \beta V_0 \Delta T \). Think of \( \beta \) as a sensitivity factor to temperature changes. This is why in the exercise, we use this coefficient to determine how much the glycerin expanded when the temperature increased.

Temperature Change

Temperature change, symbolized as \( \Delta T \), is simply the difference in initial and final temperatures of a material. This change is a crucial factor when calculating volume expansion.

• It shows how much the environment has changed concerning heat.
• Knowing both the starting and ending temperatures lets you determine \( \Delta T \), which will be used in volume expansion calculations.

In our problem, this change was calculated as \( 6^{\circ}C \), from an initial \( 22^{\circ}C \) to a final \( 28^{\circ}C \). Such calculations enable you to predict or measure how much the glycerin will expand, an important step in managing the liquid in the cup effectively.