Problem 79 - University Physics with Modern 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 \([\beta=6.8 \times\) \(10^{-4}\left(\mathrm{C}^{\circ}\right)^{-1} ]\) to a height of 2.00 \(\mathrm{mm}\) below the top of the cup. Initially, the cup and oil are at room temperature \(\left(22.0^{\circ} \mathrm{C}\right) .\) 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 start to spill when the temperature is significantly increased from 22°C, exact calculation requires known radius 'r' due to generalized setup.

Step 1: Understand the Expansion Concepts

When heated, both the glass cup and the olive oil expand. However, they expand at different rates due to the different coefficients of volume expansion: \(\beta_{glass} = 2.7 \times 10^{-5} /^{\circ}C\) for glass and \(\beta_{oil} = 6.8 \times 10^{-4} /^{\circ}C\) for oil. The oil will start to spill out when its expansion exceeds the expanded capacity of the cup.

Step 2: Calculate Initial Conditions

Initial volume of the glass cup \(V_{cup} = \pi r^2 h\) is needed, but since the radius \(r\) isn't specified, the specific numerical volume calculation isn't necessary. The key parameter is the 2.00 mm height (\(h = 10.0\; \mathrm{cm} - 0.2\; \mathrm{cm}\)) filled with oil.

Step 3: Determine Volume Change Requirement

The oil spills out when the expansion of its volume surpasses the expansion of the available space in the glass cup. Given the height to overflow is 2.00 mm, convert this to cm for consistency (0.2 cm).

Step 4: Set Up Expansion Equation

When both the oil and the cup reach temperature \(T\), the change in oil volume \(\Delta V_{oil} = V_{0,oil} \beta_{oil} \Delta T\) and the change in glass volume \(\Delta V_{glass} = V_{0,glass} \beta_{glass} \Delta T\) should equate to the initial difference in volume due to the 2.00 mm (0.2 cm) gap.

Step 5: Solve Expansion Equation for Temperature

Substitute \(V_{0,oil} = V_{0,glass}\) at \(22^{\circ}C\) since initially, the oil and the cup are almost full at the same volume.Solve for \(\Delta T\): \(V_{0} \beta_{oil} \Delta T - V_{0} \beta_{glass} \Delta T = 0.2 \quad cm\).Simplify and solve: \((6.8 \times 10^{-4} - 2.7 \times 10^{-5})V_{0} \Delta T = 0.2\) and solve for \(\Delta T\).

Step 6: Calculate Final Answer

Given oil's initial height as \(9.8\,\mathrm{cm}\):\[\Delta T = \frac{0.2}{(6.8 \times 10^{-4} - 2.7 \times 10^{-5}) \cdot \pi \cdot r^2 \cdot 9.8}=\text{Requires necessary cup radius "r" for final computation. Compute based on these assumptions.} \]Finally, find \(T=22^{\circ}C + \Delta T\) to determine when the oil begins to spill.

Key Concepts

Coefficient of Volume Expansion

The Coefficient of Volume Expansion is a crucial concept in understanding how materials respond to temperature changes. It is denoted by the symbol \( \beta \) and expresses how the volume of a substance increases per degree of temperature rise.

In the given exercise, two different coefficients are involved:- The coefficient for the glass cup \( (\beta_{glass}) \) is \( 2.7 \times 10^{-5} \, \text{C}^{\circ}\)\(^{-1}\).- The coefficient for the olive oil \( (\beta_{oil}) \) is \( 6.8 \times 10^{-4} \, \text{C}^{\circ}\)\(^{-1}\).

These values indicate that olive oil expands much more than glass per degree Celsius increase. The coefficient is essentially a measure of how sensitive a material is to temperature changes in terms of volume expansion. This factor is pivotal in predicting when materials like the olive oil will overflow when confined, as oil expands more quickly compared to the glass cup that holds it.

Temperature Change

Temperature change \((\Delta T)\) plays a critical role in calculating how much a material will expand under heat. This change is crucial in identifying at what point the olive oil will spill from the glass cup.
In the exercise, you start with an initial temperature of \(22.0^{\circ} \text{C}\). As the temperature increases, both the glass and oil expand, but at different rates due to their distinct coefficients of expansion.

To find \(\Delta T\), where the oil starts spilling, you employ the expansion equation for both the oil and the glass. By equating the increase in the oil’s volume to the expansion capacity of the glass, considering the initial 2.00 mm gap, you solve for \(\Delta T\). This process involves setting up the equation:

• \(V_{0,oil} \beta_{oil} \Delta T - V_{0,glass} \beta_{glass} \Delta T = 0.2 \text{ cm}\), considering the volume difference due to the gap.

Here, \(\Delta T\) is solved based on these conditions to find out how high the temperature gets before the oil overflows.

Glass and Oil Interaction

The interaction between glass and oil in this scenario is driven by their respective expansion properties. You are dealing with a scenario where both materials are subject to the same temperature increase but respond differently.

The glass cup and the oil initially have the same volume capacity but distinct coefficients mean: - Glass and oil expand, but oil's volume increases more rapidly. - The cup, made of ordinary glass, with a lower expansion coefficient, allows it to expand less.

In practical terms, this difference means the oil, which is just below the top of the glass initially, expands faster and may exceed the containment of the cup if heated sufficiently.

This interaction emphasizes the importance of considering material properties in containers where precise volume and temperature control is essential. It is vital in both everyday scenarios, like cooking, and more technical applications, such as in designing laboratory equipment.