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Chapter 17: Problem 25

A glass flask whose volume is 1000.00 \(\mathrm{cm}^{3}\) at \(0.0^{\circ} \mathrm{C}\) is completely filled with mercury at this temperature. When flask and mercury are warmed to \(55.0^{\circ} \mathrm{C}, 8.95 \mathrm{cm}^{3}\) of mercury overflow. If the coefficient of volume expansion of mercury is \(18.0 \times 10^{-5} \mathrm{K}^{-1}\) . compute the coefficient of volume expansion of the glass.

Short Answer

Expert verified
The coefficient of volume expansion of the glass is approximately \(1.727 \times 10^{-5} \, \text{K}^{-1}\).

Step by step solution

01

Understand the Problem

We have a glass flask and mercury that both expand when heated. The problem is to find the coefficient of volume expansion of the glass, which will account for the volume change of mercury that did not overflow.
02

Calculate the Change in Mercury Volume Due to Heating

The formula for the change in volume due to thermal expansion for mercury is given by \[\Delta V = \beta \cdot V_0 \cdot \Delta T\]where \(\beta\) is the expansion coefficient of mercury, \(V_0\) initial volume, and \(\Delta T\) temperature change. So, \[\Delta V_\text{mercury} = 18.0 \times 10^{-5} \cdot 1000.00 \, \text{cm}^3 \cdot (55.0 - 0.0) \, ^{\circ}\text{C}\] Calculate this value.
03

Result of Mercury Expansion Calculation

The change in volume of the mercury, \(\Delta V_\text{mercury}\), is given by \[\Delta V_\text{mercury} = 18.0 \times 10^{-5} \cdot 1000 \cdot 55 = 9.9 \, \text{cm}^3\]
04

Determine Actual Volume Change of the Flask

Since 8.95 cm^3 of mercury overflowed, the actual change in volume that the flask can accommodate is smaller than the volume change of mercury which is:\[\Delta V_\text{flask} = \Delta V_\text{mercury} - \text{overflow} = 9.9 \, \text{cm}^3 - 8.95 \, \text{cm}^3 = 0.95 \, \text{cm}^3\]
05

Calculate the Coefficient of Volume Expansion for the Glass

Use the formula for volume expansion of the flask:\[\Delta V_\text{flask} = \beta_\text{glass} \cdot V_{0,\text{glass}} \cdot \Delta T\]Solving for \(\beta_\text{glass}\):\[0.95 = \beta_\text{glass} \cdot 1000.00 \cdot 55\]Rearrange to find:\[\beta_\text{glass} = \frac{0.95}{1000 \cdot 55}\] Calculate this value.
06

Result of Glass Coefficient Calculation

The coefficient of volume expansion for the glass, \(\beta_\text{glass}\), is \[\beta_\text{glass} = \frac{0.95}{55000} \approx 1.727 \times 10^{-5} \, \text{K}^{-1}\]

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Key Concepts

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

Coefficient of Volume Expansion
The coefficient of volume expansion is a measure that describes how the volume of a material changes as temperature changes. It is denoted by \( \beta \) and is calculated using:\[\Delta V = \beta \cdot V_0 \cdot \Delta T\]where:
  • \( \Delta V \) is the change in volume,
  • \( V_0 \) is the initial volume,
  • \( \Delta T \) is the temperature change, and
  • \( \beta \) is the coefficient of volume expansion.
This coefficient helps predict and calculate changes in a material's volume as it undergoes temperature changes. In practical applications like the glass flask and mercury problem, it shows why certain materials expand more than others upon heating.
Physics Problem Solving
Solving physics problems involves understanding the given conditions and using appropriate formulas to find unknown values. In thermal expansion problems, typically as shown in the example of the glass flask:
  • You begin by identifying what you know (e.g., initial volume, temperature change, known coefficient of expansion).
  • Then, calculate step-by-step, keeping track of units and conversions.
  • Follow the structure of the provided equations, and always verify your results by thinking if they make physical sense.
Clear understanding and organization are crucial. Practice will also help improve your efficiency and accuracy when tackling similar physics problems.
Material Properties
Material properties, such as the coefficient of volume expansion, define how substances react to environmental changes like temperature. Different materials behave differently:
  • Metals typically expand more than glass as they have higher coefficients of expansion.
  • Mercury, a liquid, has a distinct expansion coefficient which makes it suitable for thermometers.
  • Properties like strength, elasticity, and thermal expansion are essential when selecting materials for applications.
Understanding material properties helps in applications like preventing overflow in containers by calculating expansion under heat and selecting proper materials for structures or components.
Temperature Effects
Temperature changes can significantly affect the volume of materials. As temperature increases:
  • Molecules move faster and occupy more space, causing expansion.
  • In the context of a tightly filled container, this expansion can cause overflow, as observed with the glass flask and mercury.
  • Temperature effects are crucial in natural phenomena and engineering applications.
  • Engineers must consider these effects when designing systems to ensure safety and functionality in temperature variations.
Recognizing how temperature affects different materials allows for informed decisions in both everyday contexts and scientific solutions.

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Most popular questions from this chapter

(a) A wire that is 1.50 \(\mathrm{m}\) long at \(20.0^{\circ} \mathrm{C}\) is found to increase in length by 1.90 \(\mathrm{cm}\) when warmed to \(420.0^{\circ} \mathrm{C}\) . Compute its average coefficient of linear expansion for this temperature range. (b) The wire is stretched just (zero tension) at \(420.0^{\circ} \mathrm{C}\) . Find the stress in the wire if it is cooled to \(20.0^{\circ} \mathrm{C}\) without being allowed to contract. Young's modulus for the wire is \(20 \times 10^{11} \mathrm{Pa}\) .

One end of an insulated metal rod is maintained at \(100.0^{\circ} \mathrm{C}\) and the other end is maintained at \(0.00^{\circ} \mathrm{C}\) by an ice-water mixture. The rod is 60.0 \(\mathrm{cm}\) long and has a cross-sectional area of 1.25 \(\mathrm{cm}^{2}\) . The heat conducted by the rod melts 8.50 \(\mathrm{g}\) of ic. 0 min. Find the thermal conductivity \(k\) of the metal.

On a cool \(\left(4,0^{\circ} \mathrm{C}\right)\) Saturfay moming, a pilot fills the fuel tanks of her Pitts \(S-2 C\) (a two-seat aerobatic airplane) to their full capacity of 106.0 L. Before flying on Sunday morning, when the temperature is again \(4.0^{\circ} \mathrm{C}\) , she checks the fuel level and finds only 103.4 \(\mathrm{L}\) of gasoline in the tanks. She realizes that it was hot on Saturday afternoon, and that thermal expansion of the gasoline caused the missing fuel to empty out of the tank's vent. (a) What was the maximum temperature (in "C) reached by the fuel and the tank on Saturday aftemoon? The coefficient of volume expansion of gasoline is \(9.5 \times 10^{-4} \mathrm{K}^{-1}\) , and the tank is made of aluminum. (b) In order to have the maximum amount of fuel available for flight, when should the pilot have filled the fuel tanks?

A wood ceiling with thermal resistance \(R_{1}\) is covered with a layer of insulation with thermal resistance \(R_{2} .\) Prove that the effective thermal resistance of the combination is \(R=R_{1}+R_{2}\) .

Like the Kelvin scale, the Rankine scale is an absolute temperature scale: Absolute zero is zero degrees Rankine \(\left(0^{\circ} \mathrm{R}\right)\) . However, the units of this scale are the same size as those of the Fahrenheit scale rather than the Celsius scale. What is the numerical value of the triple-point temperature of water on the Rankine scale?
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