Thermal Expansion | Physics for JEE Main & Advanced PDF Download

Thermal Expansion

• When a liquid is heated, such as when a thermometer is placed in warm water, it expands. Conversely, it contracts when cooled.
• Similar to liquids, balloons demonstrate thermal expansion. A balloon inflated in a warm environment expands fully, while one in a cold environment shrinks when fully inflated due to air contraction.
• Sealed bottles with tightly screwed metallic lids may require dipping in hot water to loosen the lid, as the expansion of the metallic lid facilitates easier opening.

Most substances expand with heat and contract with cold. This change in dimensions with temperature variation is known as thermal expansion.

Example of Thermal Expansion

Three types of expansion occur in solids:

• Linear expansion: This involves an increase in length when a solid is heated. The fractional change in length (∆l/l) is directly proportional to the change in temperature (∆T).
• Area superficial or superficial expansion
• Volume expansion

Linear expansion is expressed by the formula:

Here, αl represents the coefficient of linear expansion for the specific solid.

The unit of α is per degree Celsius (°C^-1) in the CGS system and per Kelvin (K^-1) in the SI system.

The coefficient of linear expansion for some materials is as follows:

• Superficial or Area expansion involves the increase in the surface area of a substance when heated. A slight change in temperature, ΔT, leads to deformation, where the fractional change in surface area, ΔA/A, is directly proportional to ΔT.
• ΔA/A

  The formula for Area expansion is:

  

  Here, αA represents the coefficient of area expansion of the given solid.

• α_A

  Volume expansion refers to the fractional change in the volume of a substance. A slight variation in temperature, ΔT, causes deformation, where the fractional change in volume, ΔV/V, is directly proportional to ΔT.

• ΔV/V

  Here, α_V is another characteristic of the substance, but it varies with temperature.

The coefficient of volume expansion only becomes constant at high temperatures. For instance, ethyl alcohol has a higher coefficient of volume expansion than mercury and thus expands more for the same temperature rise.

The graph depicts the Coefficient of volume expansion of copper concerning temperature:

It shows how the Coefficient of volume expansion of copper changes with temperature.

Water's Unique Behavior

• Between 0 and 4 degrees Celsius, water behaves unusually when heated.
• As water is cooled, its volume decreases until it reaches around 4°C.
• Below 4°C, water's volume increases, leading to a decrease in density.
• Water has maximum density at 4°C, impacting the freezing of lakes and ponds.
• Energy loss to the atmosphere causes denser water to sink while warmer water rises.
• When cooler water on top cools below 4°C, it becomes less dense and freezes at the surface.

Sample Question

Problem 1: Define the term thermal expansion.

Solution:

Most substances expand with heat and contract when cooled, changing dimensions with temperature shifts.

• Linear expansion
• Area superficial or superficial expansion
• Volume expansion

Problem 2: Calculate the pressure needed to maintain the length of a steel wire when heated by 100°C.

Given:

ΔT = 100°C,

Y = 2 x 10^11 Nm^−2,

α = 1.1 x 10^−5 K^−1.

Thermal strain = 2.2 x 10^8 Pa.

Problem 3: Comparing two wires of the same material and volume but different cross-sectional areas.

If wire 1 lengthens by Δx due to force F, determine the force needed to stretch wire 2 by the same amount.

Young's Modulus and Stress-Strain Relationship

• Young's modulus remains constant for the same material.
• Wire 1 has a cross-sectional area A, while wire 2 has 3A.
• The volumes of wire 1 and wire 2 are equal: V1 = V2.
• Given: A × l1 = 3A × l2, which implies l2 = l1/3.
• Stress-Strain relationship: Y = (F/A)/(Δl/l), where F1 = YA(Δl1/l1) and F2 = Y3A(Δl2/l2).
• Since wire 2 stretches by the same amount, Δl1 = Δl2 = x.
• Substitute to find: F2 = 9F1.

Area Expansion and Coefficients Relationship

• Definition of area expansion: It refers to the increase in the surface area of a substance upon heating, where a slight change in temperature ΔT causes deformation. The fractional change in surface area, ΔA/A, is directly proportional to ΔT.
• Expression for Area expansion: ΔA/A = αA * ΔT.
• Derivation of the relationship between coefficients: αA = 2αl, where αA is the coefficient of area expansion and αl is the coefficient of linear expansion.
• Consider a cube expanding evenly in all directions with temperature rise T. The new area after expansion is related to the linear expansion.
• Equation: ΔA/A ≈ 2(Δl/l), which leads to αA = 2αl, indicating that the coefficient of area expansion is twice the coefficient of linear expansion.