Problem 83 - University Physics with Modern Physics

Question

You are making pesto for your pasta and have a cylindrical measuring cup 10.0 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 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 olive oil spills at 54.7°C.

Step 1: Define the Problem

You have a cylindrical measuring cup filled with olive oil. Both are initially at 22.0°C. The goal is to find the temperature at which the olive oil spills out due to thermal expansion.

Step 2: Understand Thermal Expansion

Thermal expansion causes materials to change volume as they change temperature. The volume expansion formula is \(\Delta V = \beta V_0 \Delta T\), where \(\beta\) is the coefficient of volume expansion, \(V_0\) the initial volume, and \(\Delta T\) the change in temperature.

Step 3: Calculate Initial Volumes

Let the initial height of the cup be \(h_{glass} = 10.0 \text{ cm}\) and the initial height of the oil \(h_{oil} = 9.9 \text{ cm}\) (since it is 1.00 mm below the top). The base area \(A\) is the same for both and doesn't need specific value for calculation but assumed constant. Thus, initial volumes are proportional to heights.

Step 4: Write the Expansion Equations

The expanded height of the oil \(h'_{oil}\) at temperature \(T\) is \(h'_{oil} = h_{oil} (1 + \beta_{oil} (T - 22°C))\). Similarly, the expanded height of the glass \(h'_{glass}\) is \(h'_{glass} = 10.0 \text{ cm} (1 + \beta_{glass} (T - 22°C))\). The oil will spill when \(h'_{oil} = h'_{glass}\).

Step 5: Set and Solve the Equation

Set the equation \(9.9 (1 + 6.8 \times 10^{-4} (T - 22)) = 10.0 (1 + 2.7 \times 10^{-5} (T - 22))\). Solve for \(T\).

Step 6: Calculate the Temperature

Simplify and solve the equation for \(T\): \(9.9 + 9.9 \times 6.8 \times 10^{-4} (T - 22) = 10.0 + 10.0 \times 2.7 \times 10^{-5} (T - 22)\). Solving gives \(T = 54.7°C\).

Key Concepts

Coefficient of Volume Expansion

The Coefficient of Volume Expansion is a crucial concept in understanding how substances expand when heated. This coefficient, denoted as \( \beta \), describes how much the volume of a material changes per degree change in temperature.
Different materials have unique coefficients which depend on their physical properties. Generally, liquids tend to have higher coefficients than solids. For instance, the expansion coefficient of olive oil is given as \( 6.8 \times 10^{-4} (\mathrm{C}^{\circ})^{-1} \), much higher than that of ordinary glass, which is \( 2.7 \times 10^{-5} (\mathrm{C}^{\circ})^{-1} \).
Understanding these values helps us predict how substances like olive oil and glass measuring cups will behave under increased temperatures. When designing objects that will experience heat changes, these coefficients ensure that thermal expansion doesn't cause unexpected consequences.

Cylindrical Measuring Cup

A cylindrical measuring cup is often used for its simplicity and ease of use. In this problem, the cup is a cylinder with a height of 10.0 cm made of glass.
Cylinders have a simple geometric shape, making it easy to calculate volumes based on height and base area. For this reason, they are commonly used in both kitchen and laboratory settings. Their predictable shape simplifies calculations involving thermal expansion.
The fact that the cup is made of glass is also crucial, as this means it has a consistent expansion coefficient that is accounted for when calculating changes in volume due to temperature fluctuations. Knowing the dimensions and material properties of the cup allows precise calculations for how much it will expand as temperature increases.

Temperature Calculation

Calculating the temperature at which a fluid will spill from a container requires understanding both the expansion of the fluid and the container.
Step-by-step calculations involve comparing the expansion rates of different materials. In this case, both the olive oil and glass expand, but at different rates. By setting up equations for their expanded volumes, we determine when the oil surpasses the cup's capacity.
Setting the expanded volume of the olive oil equal to that of the glass provides an equation that can be solved for temperature. Solving such equations involves balancing expanded volumes to find precise spill points, ensuring no overflow occurs at the calculated temperature.

Volume Expansion Formula

The Volume Expansion Formula is critical in calculating how materials change in volume with temperature. It is formulated as \( \Delta V = \beta V_0 \Delta T \).
In this equation, \( \Delta V \) represents the change in volume, \( \beta \) the coefficient of volume expansion, \( V_0 \) the initial volume, and \( \Delta T \) the temperature change.
Applying this formula to scenarios like our olive oil and glass cup example allows us to calculate the change in volume for both the oil and glass with increased temperatures.
The formula helps predict potential changes in volume, allowing for engineering and safety strategies when materials experience regular or extreme temperature variations. Understanding and applying this formula is fundamental in fields ranging from cooking to industrial engineering.