Prove that, if we neglect extremely small quantities, the change in volume of a solid upon expansion through a temperature rise a temperature rise ∆T is given by ∆V= 3αV∆T, where α is the coefficient of linear expansion.

To understand how the change in volume of a solid upon expansion due to a temperature rise is expressed as ∆V = 3αV∆T, we need to delve into the concepts of thermal expansion and how they relate to volume changes in solids. Let's break this down step by step.

Understanding Thermal Expansion

When a solid is heated, its particles gain energy and move more vigorously. This increased movement causes the solid to expand. The extent of this expansion can be quantified using the coefficient of linear expansion, denoted as α. This coefficient indicates how much a unit length of a material expands per degree of temperature increase.

Linear Expansion and Its Relation to Volume

For a solid object, the linear expansion can be expressed mathematically as:

• ΔL = αL₀ΔT

Here, ΔL is the change in length, L₀ is the original length, and ΔT is the change in temperature. This equation tells us how much a single dimension of the solid will change with temperature.

From Linear to Volumetric Expansion

Now, to find the change in volume, we need to consider that a solid has three dimensions: length, width, and height. If we assume that the solid expands uniformly in all three dimensions, we can apply the linear expansion formula to each dimension:

• ΔL₁ = αL₀ΔT (for length)
• ΔL₂ = αW₀ΔT (for width)
• ΔL₃ = αH₀ΔT (for height)

Assuming the original dimensions are equal (for simplicity, let’s say L₀ = W₀ = H₀ = L), the change in volume (ΔV) can be calculated as follows:

Calculating the Change in Volume

The original volume (V₀) of the solid can be expressed as:

• V₀ = L₀ × W₀ × H₀ = L³

After the temperature increase, the new dimensions will be:

• L' = L₀ + ΔL = L + αLΔT
• W' = W₀ + ΔW = L + αLΔT
• H' = H₀ + ΔH = L + αLΔT

Thus, the new volume (V') becomes:

• V' = L' × W' × H' = (L + αLΔT) × (L + αLΔT) × (L + αLΔT)

Expanding this expression using the binomial theorem and neglecting higher-order terms (since αΔT is small), we find:

• V' ≈ L³ + 3αL²ΔT

Now, substituting V₀ back into the equation gives us:

• V' ≈ V₀ + 3αV₀ΔT

Therefore, the change in volume (∆V) can be expressed as:

• ∆V = V' - V₀ = 3αV₀ΔT

Final Expression

In conclusion, we arrive at the formula:

• ∆V = 3αVΔT

This equation illustrates that the change in volume of a solid due to thermal expansion is directly proportional to the original volume, the coefficient of linear expansion, and the change in temperature. This relationship is crucial in various applications, from engineering to everyday objects, where temperature variations can significantly affect material properties.