A physicist makes a cup of instant coffee and notices that,...

A physicist makes a cup of instant coffee and notices that, as the coffee cools, its level drops $3.00 \mathrm{mm}$ in the glass cup. Show that this decrease cannot be due to thermal contraction by calculating the decrease in level if the $350 \mathrm{cm}^{3}$ of coffee is in a 7.00 -cm-diameter cup and decreases in temperature from $95.0^{\circ} \mathrm{C}$ to $45.0^{\circ} \mathrm{C}$. (Most of the drop in level is actually due to escaping bubbles of air.)

Key Concepts

Thermal Contraction

Thermal contraction refers to the reduction in the volume of a material as it cools. For liquids such as coffee, this contraction is dictated by the material’s volumetric thermal expansion coefficient, which determines the fractional change in volume with temperature change. Although cooling causes some contraction, the magnitude is often too small to account for large observable decreases in height, as in the given scenario.

Volumetric Thermal Expansion Coefficient

The volumetric thermal expansion coefficient, usually denoted as ?, quantifies how a substance's volume changes with temperature. It is defined as the fractional change in volume per degree of temperature change. This concept allows one to compute the expected volume decrease of a liquid when its temperature drops, which can then be translated into an expected change in liquid level within a container.

Cylindrical Geometry and Volume-Height Relationship

In a cylindrical container, the volume of a liquid is the product of the cross-sectional area and the height of the liquid level. Thus, a small change in volume due to thermal contraction can be related to a decrease in the liquid’s height by dividing the change in volume by the area of the cup’s base. This geometric relationship is key in evaluating whether the volume change due to cooling can account for the observed drop in level.

Unit Conversion and Dimensional Analysis

Accurate evaluation of thermal contraction effects requires consistent unit management and dimensional analysis. Converting volume units (such as from cm³ to mm³) and ensuring the geometric dimensions (such as the diameter and height of the cup) are correctly handled in calculations are essential steps. This ensures that the predicted changes in liquid level are physically meaningful and comparable to the observed measurements.

Transcript

Please give Ace some feedback