Thermal Expansion Formula, Definition, Types, Solved Examples

Thermal Expansion Formula: Thermal expansion is a fundamental concept in the world of materials science and engineering. It describes the tendency of materials to change in size, volume, or shape in response to changes in temperature.

Definition of Thermal Expansion

Thermal Expansion Formula: Thermal expansion occurs when a material's temperature increases, causing its particles to move more vigorously. As they gain energy, these particles push against one another and spread out. This increased kinetic energy translates into an expansion in the material's dimensions. In essence, thermal expansion is a direct result of the basic principles of thermodynamics.

Thermal Expansion Formula

1. Linear Expansion Formula:

Linear expansion occurs when a material expands in only one dimension (typically length). The formula for calculating the change in length (∆L) is as follows:

∆L = α * L * ∆T

Where:

- ∆L is the change in length.

- α is the coefficient of linear expansion for the material.

- L is the original length.

- ∆T is the change in temperature.

2. Volume Expansion Formula:

Volume expansion occurs when a material expands in all three dimensions, leading to an increase in volume. The formula for calculating the change in volume (∆V) is:

∆V = β * V * ∆T

Where:

- ∆V is the change in volume.

- β is the coefficient of volume expansion for the material.

- V is the initial volume.

- ∆T is the change in temperature.

3. Coefficient of Linear Expansion (α):

The coefficient of linear expansion, denoted by α, is a material-specific constant that represents how much a material expands per degree of temperature change. It's measured in inverse temperature units, typically 1/°C. α is a property intrinsic to the material and can be found in reference materials.

4. Coefficient of Volume Expansion (β):

The coefficient of volume expansion, denoted by β, is another material-specific constant, representing the change in volume per degree of temperature change. Like α, it's measured in inverse temperature units and is a material property.

Types of Thermal Expansion

There are three main types of thermal expansion:

1. Linear Expansion : This occurs when a material expands in only one dimension, typically length. The change in length (∆L) is directly proportional to the original length (L) and the change in temperature (∆T) according to the equation ∆L = α * L * ∆T, where α is the coefficient of linear expansion.
2. Volume Expansion: Some materials expand in all three dimensions, causing an increase in volume. The equation for volume expansion is ∆V = β * V * ∆T, where ∆V is the change in volume, V is the initial volume, and β is the coefficient of volume expansion.
3. Anisotropic Expansion: In some materials, the coefficients of expansion can vary significantly in different directions. This is known as anisotropic expansion.

Applications of Thermal Expansion Formula

Thermal expansion is practically significant in various industries and everyday life, including:

• Construction: Architects and engineers account for thermal expansion when designing structures.
• Transportation: Railway tracks and bridges have expansion joints to accommodate changes in length due to temperature fluctuations.
• Thermometers: Many thermometers use the expansion of a liquid column to measure temperature.

Solved Examples of Thermal Expansion Formula

Let's explore some practical examples to illustrate the concept of thermal expansion:

Example 1: Linear Expansion of a Steel Rod

A steel rod with a length of 2 meters is subjected to a temperature increase of 50°C. Calculate the change in length.

Solution:

Given: L = 2 meters, ∆T = 50°C, α (for steel) = 12 × 10^(-6) /°C

Using the linear expansion formula: ∆L = α * L * ∆T

∆L = (12 × 10^(-6) /°C) * (2 meters) * (50°C) = 0.0012 meters or 1.2 millimeters

Example 2: Volume Expansion of Water

Suppose a container holds 10 liters of water, and the temperature increases by 20°C. Find the change in volume.

Solution:

Given: V = 10 liters, ∆T = 20°C, β (for water) = 207 × 10^(-6) /°C

Using the volume expansion formula: ∆V = β * V * ∆T

∆V = (207 × 10^(-6) /°C) * (10 liters) * (20°C) = 0.0414 liters or 41.4 milliliters

Example 3: Expansion of a Copper Wire

A copper wire with an initial length of 5 meters undergoes a temperature increase of 60°C. Calculate the change in length.

Solution:

Given: L = 5 meters, ∆T = 60°C, α (for copper) = 16.6 × 10^(-6) /°C

Using the linear expansion formula: ∆L = α * L * ∆T

∆L = (16.6 × 10^(-6) /°C) * (5 meters) * (60°C) = 0.05 meters or 50 millimeters

Example 4: Volume Expansion of Air in a Balloon

A balloon has an initial volume of 2 liters at room temperature (25°C). If it's heated to 80°C, find the change in volume.

Solution:

Given: V = 2 liters, ∆T = 55°C (80°C - 25°C), β (for air) = 3.67 × 10^(-3) /°C

Using the volume expansion formula: ∆V = β * V * ∆T

∆V = (3.67 × 10^(-3) /°C) * (2 liters) * (55°C) = 0.4046 liters or 404.6 milliliters

Example 5: Thermal Expansion of a Steel Beam

A steel beam used in construction has an original length of 10 meters. If the temperature drops by 30°C, calculate the change in length.

Solution:

Given: L = 10 meters, ∆T = -30°C (temperature drop), α (for steel) = 12 × 10^(-6) /°C

Using the linear expansion formula: ∆L = α * L * ∆T

∆L = (12 × 10^(-6) /°C) * (10 meters) * (-30°C) = -0.0036 meters or -3.6 millimeters

In this case, the negative sign indicates a contraction due to the decrease in temperature.

Example 6: Expansion of a Glass Bottle

A glass bottle has an initial volume of 500 milliliters and is placed in a freezer at -20°C. If it's taken out and allowed to warm to room temperature at 25°C, find the change in volume. use Thermal Expansion Formula

Solution:

Given: V = 500 milliliters, ∆T = 45°C (25°C - (-20°C)), β (for glass) = 5.0 × 10^(-6) /°C

Using the volume expansion formula: ∆V = β * V * ∆T

∆V = (5.0 × 10^(-6) /°C) * (500 milliliters) * (45°C) = 0.1125 milliliters

In this example, the glass bottle experiences a minor expansion when it warms up.

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