Problem 17 - University Physics with Modern Physics

Question

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.

Step-by-Step Solution

Answer

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

Step 1: Understand the Provided Information

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

Step 2: 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.

Step 3: 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.

Step 4: 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.

Step 5: 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,\[V_{flask\_expansion} = 9.9 - 8.95 = 0.95 \text{ cm}^3\].

Step 6: 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 \( \beta_{glass} \).

Step 7: Solve the Expression for the Glass Expansion Coefficient

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

Key Concepts

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.