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Question
Answer the following question.
Derive the relation between three coefficients of thermal expansion.
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Solution
- Consider a square plate of side l0 at 0° C and lT at T °C.
∴ lT = l0 (1 + αT)
If area of plate at 0° C is A0, A0 = `l_0^2`
If area of plate at T °C is AT,
`"A"_"T" = l_1^2 = l_0^2 (1 +alpha"T")^2`
or `"A"_"T" = "A"_0(1 + alpha"T")^2` ....(1)
Also,
`"A"_"T" = "A"_0(1 + beta"T")` ....(2) `[because beta = ("A"_"r" - "A"_0)/("A"_0("T" - "T"_0))]` - Using Equations (1) and (2),
`"A"_0(1 + alpha"T")^2 = "A"_0(1 + beta"T")`
∴ `1 + 2alpha"T" + alpha^2"T"^2 = 1 + beta"T"` - Since the values of α are very small, the term α2T2 is very small and may be neglected.
∴ β = 2α - The result is general because any solid can be regarded as a collection of small squares.
- Consider a cube of side l0 at 0 °C and lT at T °C.
∴ lT = l0(1 + αT)
If volume of the cube at 0 °C is V0, V0 = `l_0^3`
If volume of the cube at T °C is
`"V"_"T", "V"_"T" = l_"T"^3 = l_0^3 (1 + alpha"T")^3`
`"V"_"T" = "V"_0(1 + alpha"T")^3` ....(1)
Also,
`"V"_"T" = "V"_0(1 + gamma"T")` ....(2) ...`[therefore gamma = ("V"_"T" - "V"_0)/("V"_0("T" - "T"_0))]` - Using Equations (1) and (2),
`"V"_0 (1 + alpha"T")^3 = "V"_0(1 + gamma"T")`
∴ `1 + 3alpha"T" + 3alpha^2"T"^2 + alpha^3"T"^3 = 1 + gamma"T"` - Since the values of α are very small, the terms with higher powers of α may be neglected.
∴ γ = 3α - The result is general because any solid can be regarded as a collection of small cubes.
- Relation between α, β and γ is given by,
`alpha = beta/2 = gamma/3`
where, α = coefficient of linear expansion,
β = coefficient of superficial expansion,
γ = coefficient of cubical expansion.
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