Problem 52 - College Physics

Question

A Pyrex beaker that has a capacity of \(1000 \mathrm{~cm}^{3}\) at \(20^{\circ} \mathrm{C}\) contains \(990 \mathrm{~cm}^{3}\) of mercury at that temperature. Is there some temperature at which the mercury will completely fill the beaker? Justify your answer. (Assume that no mass is lost by vaporization and include the expansion of the beaker.)

Step-by-Step Solution

Answer

There is such a temperature; mercury expands enough to fill the beaker as both increase in temperature.

Step 1: Identify Initial Variables

First, note the initial conditions: the beaker has a capacity of 1000 cm³ at 20°C, and it contains 990 cm³ of mercury. We're asked to find the temperature at which the mercury expansion would fill the beaker completely.

Step 2: Thermal Expansion Concept

Understand that both mercury and beaker will expand when heated due to thermal expansion. Each has a different coefficient of volume expansion: mercury ( \( \beta_{\text{Hg}} \)) and Pyrex glass of the beaker ( \( \beta_{\text{beaker}} \)).

Key Concepts

Volume Expansion

Volume expansion occurs when substances increase in volume due to heat. When any object is heated, its particles move faster and spread out, causing it to expand. This change is measured in terms of volume expansion and is crucial in understanding how substances like mercury and containers like beakers behave under changes in temperature.

For example, imagine heating a liquid such as mercury. As it gets warmer, it occupies more space. Similarly, the container, if also heated, will expand but not always at the same rate.

In our exercise, both the mercury and the Pyrex beaker will undergo volume expansion. This means the mercury will increase in volume, as will the beaker itself. The process of volume expansion can significantly impact how systems with confined spaces behave, determining whether a container will hold its contents at higher temperatures.

Coefficient of Expansion

The coefficient of expansion is a numerical value that describes how much a substance expands per degree of temperature increase. Each material has its unique coefficient, which helps predict how much it will expand.

• For our Pyrex glass beaker, this coefficient is notably less than that of mercury. This means that for a similar temperature increase, the beaker expands less compared to mercury.


• The thermal expansion coefficient for mercury (\( \beta_{\text{Hg}} \)) is given as a material constant, and similarly for Pyrex glass (\( \beta_{\text{beaker}} \)).

Understanding these coefficients allows us to calculate the expected expansion for both the mercury and the beaker. With these calculations, it's possible to determine if and when the mercury will completely fill the expanding beaker as the temperature rises.

Temperature Change Effects

The effects of temperature change on objects are significant, especially for liquid-in-solid systems. When the temperature of such a system is increased, both the liquid and the container expand, but not necessarily at the same rate.

Let's apply this to our scenario: As the temperature increases, the mercury's volume increases due to its high coefficient of volume expansion. However, the beaker also expands. The critical factor here is whether the rate of expansion for the mercury exceeds that of the beaker. If the mercury expands faster, there will be a temperature where it completely fills the beaker.

Predicting and calculating these temperature change effects is vital. Using the coefficients for both materials, calculations can determine if the initial space available would be completely occupied at a higher temperature. This demonstration helps highlight the concept that materials respond differently to temperature changes, influencing everyday scenarios like designing thermal containers to accommodate liquids safely.