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

Short Answer

Expert verified
The coefficient of volume expansion of the glass is approximately \(1.73 \times 10^{-5} K^{-1}\).

Step by step solution

01

Understand the Provided Information

We have a glass flask with a volume of 1000.00 cm to evaluate the expansion of mercury first.
02

Calculate the Expansion of Mercury

The change in volume due to expansion is given by \( \Delta V = V_0 \beta_{mercury} \Delta T \). Here, \( V_0 = 1000.00 \text{ cm}^3 \) is the initial volume, \( \beta_{mercury} = 18.0 \times 10^{-5} K^{-1} \) is the coefficient of volume expansion of mercury, and \( \Delta T = 55.0 - 0.0 = 55.0 \text{ K} \) is the change in temperature. We need to compute the expanded volume of mercury.
03

Substitute Values to Find Change in Mercury Volume

Substitute the known values into the formula: \[\Delta V = 1000.00 \times 18.0 \times 10^{-5} \times 55.0\] Calculate \( \Delta V \) to find how much the volume increases.
04

Calculate the Numerical Result for Mercury Expansion

Perform the calculations: \[\Delta V = 1000.00 \times 18.0 \times 10^{-5} \times 55.0 = 9.9 \text{ cm}^3\] Thus, mercury's volume would increase by 9.9 cm³ due to heating.
05

Determine Flask's Expansion

Since 8.95 cm³ of mercury overflowed, the volume expansion of the flask itself must account for the difference between the mercury expansion and overflow. Therefore,\[V_{flask\_expansion} = 9.9 - 8.95 = 0.95 \text{ cm}^3\].
06

Calculate the Coefficient of Volume Expansion of the Glass

We use the formula \( \Delta V_{flask} = V_0 \beta_{glass} \Delta T \) where \( \Delta V_{flask} = 0.95 \text{ cm}^3 \), \( V_0 = 1000.00 \text{ cm}^3 \), and \( \Delta T = 55.0 \text{ K} \). Solving for \( \beta_{glass} \):\[\beta_{glass} = \frac{0.95}{1000.00 \times 55.0}\]. Substitute the known values to find \( \beta_{glass} \).
07

Solve the Expression for the Glass Expansion Coefficient

Perform the calculation for \( \beta_{glass} \): \[\beta_{glass} = \frac{0.95}{1000.00 \times 55.0} = \frac{0.95}{55000} \approx 1.73 \times 10^{-5} K^{-1}\].

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Key Concepts

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

Thermal Expansion
Thermal Expansion refers to the tendency of a material to change its volume in response to a change in temperature. When materials are heated, their particles begin to move more vigorously. This increased motion causes the material to expand, which is particularly noticeable in solids, liquids, and gases.
In the case of this exercise, the glass flask and the mercury inside it both experience thermal expansion. Since the coefficient of volume expansion differs between different materials, not all components expand at the same rate when subjected to the same temperature change. This differing expansion is the reason we observe overflow in the mercury from the flask.
  • The coefficient of volume expansion (\(\beta)\) is a property that quantifies how much a material's volume changes per degree change in temperature.
  • In the formula \(\Delta V = V_0 \beta \Delta T\), \\(\Delta V\) is the change in volume, \\(V_0\) is the initial volume, \\(\beta\) is the coefficient of volume expansion, and\(\Delta T\) is the temperature change.
In practical terms, engineers and designers must consider thermal expansion in applications such as building bridges, where temperatures can vary dramatically between seasons.
Physics Problem Solving
In tackling physics problems, particularly those involving thermal expansion like the one in this exercise, the approach involves understanding given information and applying relevant formulas to find unknowns.
The step-by-step process follows a structured approach:
  • Understand the problem: Identify what is known and what needs to be found. Here, we know the initial volume of the flask and mercury, the overflow amount, and the coefficient of mercury's expansion.
  • Calculate known variables: Calculate the change in volume (\(\Delta V)\) for mercury using the formula \\(\Delta V = V_0 \beta \Delta T\). This helps determine how much the volume of mercury increases when heated.
  • Analyze results: By calculating the difference in volume expansions between the mercury and the flask, you find how much the mercury overflows due to expansion.
  • Solve for unknowns: Determine the coefficient of volume expansion of the glass by rearranging formulas accordingly and substituting known quantities.
This logical way of breaking down the problem helps to simplify complex physics concepts, ensuring that all parts are understood and solved correctly.
Temperature Change
Temperature Change is a critical element affecting thermal expansion. In this exercise, it's calculated as the difference between the final and initial temperatures of the system. This change directly influences how much the volume of the materials expand.
Understanding \(\Delta T\) (the change in temperature) is key in applying the thermal expansion formulas as it helps quantify how significant the expansion will be.
  • \(\Delta T\) for our exercise is 55.0 K, calculated simply as \\(55.0 - 0.0\).
  • Temperature changes affect different materials in distinct ways depending on their expansion coefficients.
  • Accurate measurement of temperature change ensures precise assessments of how materials like glass and mercury will react when heated.
In practical scenarios, knowing how materials expand with temperature change is crucial for designing systems that accommodate this expansion, ensuring that they maintain structural integrity and functionality across various temperatures.

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Most popular questions from this chapter

BIO Temperatures in Biomedicine. (a) Normal body temperature. The average normal body temperature measured in the mouth is 310 K. What would Celsius and Fahrenheit thermometers read for this temperature? (b) Elevated body temperature. During very vigorous exercise, the body’s temperature can go as high as 40\(^\circ\)C. What would Kelvin and Fahrenheit thermometers read for this temperature? (c) Temperature difference in the body. The surface temperature of the body is normally about 7 C\(^\circ\) lower than the internal temperature. Express this temperature difference in kelvins and in Fahrenheit degrees. (d) Blood storage. Blood stored at 4.0\(^\circ\)C lasts safely for about 3 weeks, whereas blood stored at -160\(^\circ\)C lasts for 5 years. Express both temperatures on the Fahrenheit and Kelvin scales. (e) Heat stroke. If the body’s temperature is above 105\(^\circ\)F for a prolonged period, heat stroke can result. Express this temperature on the Celsius and Kelvin scales.

A constant-volume gas thermometer registers an absolute pressure corresponding to 325 mm of mercury when in contact with water at the triple point. What pressure does it read when in contact with water at the normal boiling point?

A carpenter builds an exterior house wall with a layer of wood 3.0 cm thick on the outside and a layer of Styrofoam insulation 2.2 cm thick on the inside wall surface. The wood has \(k\) = 0.080 W/m \(\cdot\) K, and the Styrofoam has \(k\) = 0.027 W /m \(\cdot\) K. The interior surface temperature is 19.0\(^\circ\)C, and the exterior surface temperature is -10.0\(^\circ\)C. (a) What is the temperature at the plane where the wood meets the Styrofoam? (b) What is the rate of heat flow per square meter through this wall?

A 4.00-kg silver ingot is taken from a furnace, where its temperature is 750.0\(^\circ\)C, and placed on a large block of ice at 0.0\(^\circ\)C. Assuming that all the heat given up by the silver is used to melt the ice, how much ice is melted?

A U.S. penny has a diameter of 1.9000 cm at 20.0\(^\circ\)C. The coin is made of a metal alloy (mostly zinc) for which the coefficient of linear expansion is \(2.6 \times 10{^-}{^5} K{^-}{^1}\). What would its diameter be on a hot day in Death Valley (48.0\(^\circ\)C)? On a cold night in the mountains of Greenland (-53\(^\circ\)C)?

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