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Chapter 12: Problem 37

A spherical brass shell has an interior volume of \(1.60 \times 10^{-3} {m}^{3}\) Within this interior volume is a solid steel ball that has a volume of \(0.70 \times 10^{-3} {m}^{3}\) . The space between the steel ball and the inner surface of the brass shell is filled completely with mercury. A small hole is drilled through the brass, and the temperature of the arrangement is increased by 12 \({C}^{\circ}\). What is the volume of the mercury that spills out of the hole?

Short Answer

Expert verified
The mercury that spills out is approximately \(1.174 \times 10^{-6} \; m^3\).

Step by step solution

01

Understanding Thermal Expansion

We know that when temperature increases, materials expand. For this problem, both the brass shell and steel ball will expand, so the volume they occupy changes with temperature. We need to determine how much the brass shell's interior and the steel ball's volume expand to find how much mercury spills out.
02

Calculate the Volume Expansion of Brass Shell

The volume expansion can be calculated using the formula: \( \Delta V = \beta V_0 \Delta T \) where \( \beta \) is the coefficient of volumetric thermal expansion for brass, \( V_0 \) is the initial volume, and \( \Delta T \) is the temperature change. Assuming \( \beta_{brass} = 57 \times 10^{-6} / ^\circ C \), we find:\[ \Delta V_{brass} = 57 \times 10^{-6} \times 1.60 \times 10^{-3} \times 12 = 1.0944 \times 10^{-6} \; m^3 \]
03

Calculate the Volume Expansion of Steel Ball

Similarly, we use the same formula for the steel ball, but with a different \( \beta \) value. Let's assume \( \beta_{steel} = 36 \times 10^{-6} / ^\circ C \):\[ \Delta V_{steel} = 36 \times 10^{-6} \times 0.70 \times 10^{-3} \times 12 = 0.3024 \times 10^{-6} \; m^3 \]
04

Calculate Initial Mercury Volume

The initial volume of mercury is the difference between the volumes of the brass shell and the steel ball:\[ V_{mercury, initial} = 1.60 \times 10^{-3} - 0.70 \times 10^{-3} = 0.90 \times 10^{-3} \; m^3 \]
05

Calculate Total Volume Change in Cavity

The change in the cavity volume after expansion is the difference between the brass shell expansion and the steel ball expansion:\[ \Delta V_{cavity} = \Delta V_{brass} - \Delta V_{steel} = 1.0944 \times 10^{-6} - 0.3024 \times 10^{-6} = 0.792 \times 10^{-6} \; m^3 \]
06

Calculate Mercury Spill Volume

The mercury expands at a different rate (using \( \beta_{mercury} = 182 \times 10^{-6} / ^\circ C \)):\[ \Delta V_{mercury} = 182 \times 10^{-6} \times 0.90 \times 10^{-3} \times 12 = 1.9656 \times 10^{-6} \; m^3 \]Thus, the mercury that spills out is the difference between the mercury expansion and the cavity expansion:\[ V_{spill} = \Delta V_{mercury} - \Delta V_{cavity} = 1.9656 \times 10^{-6} - 0.792 \times 10^{-6} = 1.1736 \times 10^{-6} \; m^3 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Volumetric Thermal Expansion
When materials experience a temperature change, they typically expand or contract. This process is known as thermal expansion. It occurs because the particles within a material move more energetically when heated, thereby taking up more space.
  • Volumetric thermal expansion specifically refers to the change in volume of a solid, liquid, or gas due to temperature change.
  • The extent of the expansion depends on the material's volumetric thermal expansion coefficient, denoted as \( \beta \).
  • The formula used to calculate this expansion is \( \Delta V = \beta V_0 \Delta T \), where \( \Delta V \) is the change in volume, \( V_0 \) is the initial volume, and \( \Delta T \) is the temperature change.
Understanding this concept is crucial when dealing with enclosures or spaces filled with different materials, such as in this exercise where both brass and steel are considered.
Spherical Geometry
Spherical geometry plays a central role when understanding how spheres behave under temperature changes. In mathematics, a spherical object is defined as having all points on its surface equidistant from its center.
  • In the context of thermal expansion, the spherical geometry determines how the volume of the sphere changes with temperature.
  • For a sphere, if the radius changes with temperature, the new volume can be calculated using the formula for the volume of a sphere: \( V = \frac{4}{3} \pi r^3 \).
  • Through thermal expansion, changes in radial dimensions due to temperature can lead to significant changes in the sphere's volume because of this cubic relationship.
In this scenario, the brass shell forms an outer sphere, and the solid steel ball inside forms an inner sphere, both of which expand volumetrically with increases in temperature.
Temperature Change Effects
Temperature change effects are crucial in evaluating how different materials respond in terms of expansion or contraction.
  • Each material has a unique response to temperature, quantified as the thermal expansion coefficient.
  • This response affects how the internal arrangements or compositions of materials, like the cavity between the brass shell and steel ball, adjust over temperature changes.
  • When temperature rises, the volume of both the brass shell and steel ball increases. However, because different materials expand at different rates, the magnitude in volume change of the enclosing space (cavity) can affect how much mercury might spill out.
These effects explain why some materials, including mercury in this case, may spill when there's more volumetric expansion than the space available inside the cavity.
Brass and Steel Properties
Brass and steel are two commonly used materials with distinct properties, including how they expand with heat.
  • Brass is an alloy primarily made of copper and zinc, known for its noticeable thermal expansion properties, with a coefficient of around \( 57 \times 10^{-6} / ^\circ C \).
  • Steel, primarily an iron-carbon alloy, has a lower expansion coefficient, traditionally close to \( 36 \times 10^{-6} / ^\circ C \), meaning it expands less for the same temperature change compared to brass.
  • The differing thermal expansion rates are significant because they dictate how the respective volumes of the brass shell and steel ball change with temperature, impacting the resultant volume of mercury that spills.
Understanding these material properties is essential in predicting the behavior of assemblies made with brass and steel components when exposed to heat.

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