Question

When a 50.0 -m-long metal pipe is heated from \(10.0^{\circ} \mathrm{C}\) to \(40.0^{\circ} \mathrm{C}\), it lengthens by \(2.85 \mathrm{~cm}\). a) Determine the linear expansion coefficient. b) What type of metal is the pipe made of?

Short answer

Answer: The linear expansion coefficient of the metal pipe is \(1.9\times 10^{-5}\,\text{C}^{-1}\), and it is most likely made of aluminum.

Step-by-step solution

Identify the given values

We are given the following values: Initial length of the pipe, \(L_{1} = 50.0\,\text{m}\) Final length of the pipe, \(L_{2}=50.0\,\text{m} + 2.85\,\text{cm}\) Initial temperature, \(T_{1} = 10.0^{\circ}\mathrm{C}\) Final temperature, \(T_{2} = 40.0^{\circ}\mathrm{C}\) Convert the final length of the pipe to meters for consistency. \(L_{2}=50.0\,\text{m} + 0.0285\,\text{m}=50.0285\,\text{m}\)

Calculate the change in length and temperature

We need to calculate the change in length and temperature. Use the formula: \(\Delta L = L_{2} - L_{1}\) \(\Delta T = T_{2} - T_{1}\) \(\Delta L = 50.0285\,\text{m} - 50.0\,\text{m} = 0.0285\,\text{m}\) \(\Delta T = 40.0^{\circ}\mathrm{C} - 10.0^{\circ}\mathrm{C} = 30.0^{\circ}\mathrm{C}\)

Determine the linear expansion coefficient

The formula for linear expansion is given by: \(\alpha = \frac{\Delta L}{L_{1}\Delta T}\) Plug in the values to calculate the linear expansion coefficient: \(\alpha = \frac{0.0285\,\text{m}}{(50.0\,\text{m})(30.0^{\circ}\mathrm{C})}\) \(\alpha = 1.9\times 10^{-5}\,\text{C}^{-1}\) a) The linear expansion coefficient is \(1.9\times 10^{-5}\,\text{C}^{-1}\).

Identify the type of metal

By comparing the linear expansion coefficient found above with known expansion coefficients of common metals, we can find that the value is close to the expansion coefficient of aluminum, which has a linear expansion coefficient of approximately \(2.3\times 10^{-5}\,\text{C}^{-1}\). b) The type of metal is most likely aluminum.

Key concepts

Linear Expansion Coefficient

The linear expansion coefficient \(\alpha\) is a crucial concept in thermal expansion. It describes how much a material's length changes with temperature. When a material is heated, its particles move more vigorously, causing the material to expand. This expansion can be quantified using the linear expansion coefficient, which is unique to each material. To find the linear expansion coefficient, you can use the formula:

• \(\alpha = \frac{\Delta L}{L_{1}\Delta T}\)

Here, \(\Delta L\) is the change in length, \(L_{1}\) is the initial length, and \(\Delta T\) is the temperature change.
The calculated linear expansion coefficient helps predict how a material will behave under thermal stress. This is vital in applications like construction or manufacturing, where precise measurements are crucial.
A small linear expansion coefficient indicates that a material does not expand significantly with temperature changes, whereas a larger coefficient suggests greater sensitivity to thermal variations.

Metal Properties

Metals possess specific properties that influence their behavior under temperature changes. These include thermal conductivity, specific heat, and importantly, the linear expansion coefficient. The exercise shows that the metal pipe in question likely consists of aluminum due to its measured expansion.

• Thermal Conductivity: This is the ability of a metal to conduct heat.
• Specific Heat: This refers to the amount of heat necessary to change a substance's temperature.
• Linear Expansion Coefficient: Determines the extent to which a metal expands upon heating.

Different metals have varying linear expansion coefficients. For example, aluminum has a coefficient of approximately \(2.3\times 10^{-5}\,\text{C}^{-1}\), making it quite sensitive to temperature changes. This property makes it a popular choice for applications needing quick thermal response.
When selecting a metal for a specific application, consider these properties to ensure that the metal will perform well under the expected conditions.

Temperature Change

Understanding temperature changes is essential when working with materials that expand or contract. Temperature change is the difference between the initial and final temperatures of an object, and it plays a key role in determining the extent of thermal expansion.

• Initial Temperature (\(T_{1}\)): The starting temperature of an object before any changes.
• Final Temperature (\(T_{2}\)): The temperature of the object after heating or cooling has occurred.
• Temperature Change (\(\Delta T\)): Calculated using \(T_{2} - T_{1}\).

In the exercise, the temperature change is \(30.0^{\circ}\mathrm{C}\) when the metal pipe is heated from \(10.0^{\circ}\mathrm{C}\) to \(40.0^{\circ}\mathrm{C}\).
Accurate temperature measurements ensure precise calculations in thermal expansion exercises. These calculations are crucial in applications ranging from industrial design to household installations, where thermal effects might influence material properties and performance.