Problem 30 - Physics

Question

Consult Interactive LearningWare 12.1 at for help in solving this problem. During an all-night cram session, a student heats up a one-half liter \(\left(0.50 \times 10^{-3} \mathrm{~m}^{3}\right)\) glass (Pyrex) beaker of cold coffee. Initially, the temperature is \(18^{\circ} \mathrm{C},\) and the beaker is filled to the brim. A short time later when the student returns, the temperature has risen to \(92^{\circ} \mathrm{C}\). The coefficient of volume expansion of coffee is the same as that of water. How much coffee (in cubic meters) has spilled out of the beaker?

Step-by-Step Solution

Answer

Approximately \( 7.77 \times 10^{-6} \; \text{m}^3 \) of coffee spilled out.

Step 1: Understanding Volume Expansion

When a liquid is heated, its volume changes according to the formula \( \Delta V = \beta V_0 \Delta T \), where \( \Delta V \) is the change in volume, \( \beta \) is the coefficient of volume expansion, \( V_0 \) is the initial volume, and \( \Delta T \) is the change in temperature.

Step 2: Identify Given Values

From the problem, we know:- Initial volume \( V_0 = 0.50 \times 10^{-3} \; \text{m}^3 \).- Initial temperature \( T_i = 18^{\circ} \text{C} \).- Final temperature \( T_f = 92^{\circ} \text{C} \).- \( \beta \approx 2.1 \times 10^{-4} \; \text{C}^{-1} \) for water (and coffee as stated).

Step 3: Calculate Temperature Change

The change in temperature \( \Delta T \) is the difference between the final and initial temperatures: \[ \Delta T = T_f - T_i = 92^{\circ} \text{C} - 18^{\circ} \text{C} = 74^{\circ} \text{C} \]

Step 4: Calculate Volume Change

Substitute the known values into the volume expansion formula:\[ \Delta V = \beta V_0 \Delta T \]\[ \Delta V = (2.1 \times 10^{-4} \; \text{C}^{-1})(0.50 \times 10^{-3} \; \text{m}^3)(74^{\circ} \text{C}) \]Calculate \( \Delta V \):\[ \Delta V = 7.77 \times 10^{-6} \; \text{m}^3 \]

Step 5: Conclusion

Thus, the amount of coffee that has spilled out of the beaker due to heating is approximately \( 7.77 \times 10^{-6} \; \text{m}^3 \).

Key Concepts

Volume Expansion

Volume expansion is an important concept in thermal physics. It describes how the volume of a substance changes when it is subjected to temperature variations. In liquids, this phenomenon is particularly noticeable.
When heated, particles in a liquid gain energy and move apart, causing the liquid to occupy more space. This leads to an increase in volume, known as thermal expansion. The formula for volume expansion is given by:
\[ \Delta V = \beta V_0 \Delta T \]where:

• \( \Delta V \) is the change in volume,
• \( \beta \) is the coefficient of volume expansion,
• \( V_0 \) is the initial volume, and
• \( \Delta T \) is the change in temperature.

This principle explains why the coffee in the beaker spills over when heated.

Coefficient of Volume Expansion

The coefficient of volume expansion (\( \beta \)) is a crucial parameter in understanding how much a substance will expand when its temperature changes. Different materials have different coefficients, which is why they expand at different rates.
For liquids like water and coffee, the coefficient of volume expansion is approximately \( 2.1 \times 10^{-4} \; \text{C}^{-1} \). This value indicates how much the volume of the liquid will increase for each degree Celsius rise in temperature.
Using the coefficient, we can predict the volume increase of the liquid from a temperature change by plugging into:
\[ \Delta V = \beta V_0 \Delta T \]Understanding this coefficient helps in solving practical problems like the spilling coffee exercise, showing how predictive and useful this concept is.

Thermal Physics

Thermal physics explains how temperature and heat energy affect materials. It's the branch of physics that studies the energy transfer between different systems. A fundamental aspect here is understanding how changes in temperature can alter physical properties like volume.
When you heat a material, you essentially supply energy to its particles. They begin to vibrate faster, increasing the space they occupy. This is evident in phenomena like melting, evaporation, and of course, thermal expansion.
In the context of the exercise, thermal physics principles show how heat causes the coffee to expand. Thus, by understanding these concepts, one can predict and explain daily observations like the spilling of liquids upon heating.

Temperature Change

Temperature change is the variation in temperature that an object or substance experiences over time. It's a fundamental factor in various physical changes in materials.
In our coffee example, the temperature rise is crucial for predicting the expansion of the liquid. We calculate this change using the formula:
\[ \Delta T = T_f - T_i \]where:

• \( T_f \) is the final temperature, and
• \( T_i \) is the initial temperature.

Knowing \( \Delta T \), you can use it in the volume expansion formula to find out how much the volume will increase.
Therefore, understanding and calculating temperature change is key to solving problems involving thermal expansion.