Problem 71 - College Physics

Question

You are making pesto for your pasta and have a cylindrical measuring cup \(10.0 \mathrm{~cm}\) high made of ordinary glass \(\left(\beta=2.7 \times 10^{-5}\left(\mathrm{C}^{\circ}\right)^{-1}\right)\) that is filled with olive oil \(\left(\beta=6.8 \times 10^{-4}\left(\mathrm{C}^{\circ}\right)^{-1}\right)\) to a height of \(1.00 \mathrm{~mm}\) below the top of the cup. Initially, the cup and oil are at a kitchen temperature of \(22.0^{\circ} \mathrm{C}\). You get a phone call and forget about the olive oil, which you inadvertently leave on the hot stove. The cup and oil heat up slowly and have a common temperature. At what temperature will the olive oil start to spill out of the cup?

Step-by-Step Solution

Answer

The oil will spill out at 29°C.

Step 1: Understanding the Expansion Concept

Both the glass (measuring cup) and olive oil will expand when they are heated. The volume expansion is directly proportional to the volume of the substance, the temperature change, and the coefficient of volumetric expansion.

Step 2: Apply the Volume Expansion Formula

The volume expansion formula is \( \Delta V = V_0 \beta \Delta T \), where \( \Delta V \) is the change in volume, \( V_0 \) is the original volume, \( \beta \) is the coefficient of volumetric expansion, and \( \Delta T \) is the change in temperature.

Step 3: Calculate Initial Volumes

The height of the measuring cup and oil is given as \(10.0 \mathrm{~cm} \) and \(9.9 \mathrm{~cm} \) (since it's \(1.00 \mathrm{~mm}\) below the top), respectively. Let's assume the cross-sectional area of the cup is \(A\). Therefore, the initial volume of the cup is \(10.0A\) and the initial volume of the oil is \(9.9A\).

Step 4: Determine Conditions for Overflow

Overflow occurs when the volume expansion of the olive oil exceeds the volume expansion capacity provided by the cup. Therefore, solve \((9.9A + 9.9A \cdot 6.8 \times 10^{-4} \cdot \Delta T) = 10.0A + 10.0A \cdot 2.7 \times 10^{-5} \cdot \Delta T\) for \(\Delta T\).

Step 5: Calculate the Temperature Increase

Simplify the equation: \[(9.9 \cdot 6.8 \times 10^{-4} - 10.0 \cdot 2.7 \times 10^{-5}) \Delta T = 0.1 \]and solve for \( \Delta T \).

Step 6: Calculate Final Temperature

Find the final temperature using: \( 22.0^{\circ} \mathrm{C} + \Delta T \). Use the \( \Delta T \) found in the previous step.

Key Concepts

Volume Expansion

When substances, such as glass or olive oil, are heated, they tend to undergo an increase in volume. This phenomenon is known as volume expansion. It occurs because heat provides energy that causes the atoms and molecules within a material to move more vigorously.
This increased movement pushes them slightly further apart, leading to an increase in the overall volume of the material. In the context of the measuring cup and olive oil, both the glass and the oil expand as they are heated.
The challenge, however, is that different materials expand at different rates when subjected to the same temperature increase. This variance is crucial in determining whether the volume of olive oil will eventually exceed the capacity of the glass cup as they are both heated on the stove. Hints of overflow are present when the oil expands more than the cup can accommodate.

Coefficient of Volumetric Expansion

The coefficient of volumetric expansion, denoted as \( \beta \), quantifies how much a material's volume changes with a change in temperature. The value of \( \beta \) depends on the material's properties and is represented in units of inverse degrees Celsius \((^{\circ}C)^{-1}\).
The formula used to calculate volume change, \( \Delta V = V_0 \beta \Delta T \), relies directly on \( \beta \). For example, olive oil has a higher coefficient \((6.8 \times 10^{-4} {^{\circ}C}^{-1})\) compared to ordinary glass \((2.7 \times 10^{-5} {^{\circ}C}^{-1})\).
This indicates that olive oil will expand more in volume compared to the glass container for the same temperature increase. Understanding \( \beta \) is vital to predicting how different materials respond when subjected to heat. It helps calculate when exactly the olive oil begins to spill out, as its rate of expansion is higher than that of the container.

Temperature Increase

Temperature increase plays a significant role in thermal expansion. Every increment in temperature causes materials to expand based on their specific coefficient of volumetric expansion.
In the given scenario, the temperature begins at \( 22.0^{\circ}C \), the room temperature. When the measuring cup and olive oil are placed on the hot stove, they absorb heat and their temperatures rise. The objective is to identify the point where the olive oil, due to its higher rate of expansion, exceeds the capacity of the glass cup and begins to spill over.
Through calculations, \( \Delta T \), the temperature increase needed to cause overflow, is determined using material-specific properties and the initial conditions of the system. This change is crucial to solving the problem of how much hotter the system can get before spillage occurs.

Material Properties

Material properties define how a substance reacts to heat, impacting attributes like expansion. Different materials have unique characteristics—density, specific heat capacity, and coefficients of expansion—which determine how they respond to temperature changes.
In our example, the properties of ordinary glass and olive oil differ significantly. Glass, having a smaller coefficient of volumetric expansion \( (2.7 \times 10^{-5} {^{\circ}C}^{-1}) \), is relatively rigid compared to olive oil, which has a much higher \( \beta \).
This disparity means that, despite both substances being at the same initial and final temperatures, olive oil will expand much more than the glass cup. Understanding these intrinsic properties allows us to use equations for thermal expansion accurately, predicting behavior under changing thermal conditions—ensuring safety and efficiency in practical applications.