Derivation of the relation between coefficient of linear expansion and volume expansion

Introduction:
When an object is heated or cooled, it undergoes dimensional changes. These changes can be quantified by two parameters - the coefficient of linear expansion (α) and the coefficient of volume expansion (β). The coefficient of linear expansion measures the fractional change in length per degree Celsius, while the coefficient of volume expansion measures the fractional change in volume per degree Celsius.

Relation between coefficient of linear expansion and volume expansion:
The relation between the coefficient of linear expansion (α) and the coefficient of volume expansion (β) can be derived by considering a solid object with initial length L and initial volume V. Let's assume that the object is heated by ΔT degrees Celsius.

Derivation:
1. The change in length (ΔL) of the object can be expressed as:
ΔL = αLΔT

2. The final length (L') of the object can be expressed as:
L' = L + ΔL
= L + αLΔT
= L(1 + αΔT)

3. The change in volume (ΔV) of the object can be expressed as:
ΔV = V' - V
= (L' * A) - (L * A)
= A(L' - L)
= A(L(1 + αΔT) - L)
= AαLΔT

(Note: A represents the cross-sectional area of the object)

4. The final volume (V') of the object can be expressed as:
V' = V + ΔV
= V + AαLΔT
= V(1 + AαΔT/V)

5. The coefficient of volume expansion (β) is defined as:
β = (ΔV / VΔT)
= AαLΔT / VΔT
= AαL / V
= α(1/V)

(Note: VΔT represents the initial change in volume)

6. Rearranging the equation, we get:
β = α(1/V)

Conclusion:
The relation between the coefficient of linear expansion (α) and the coefficient of volume expansion (β) is given by β = α(1/V). This relation indicates that the coefficient of volume expansion is directly proportional to the coefficient of linear expansion and inversely proportional to the initial volume of the object.