First slide

Thermal expansion

Question

The coefficient of linear expansion of crystal in one direction is $α_{1}$ and that in every direction perpendicular to it is $α_{2}$ . The coefficient of cubical expansion is

Moderate

A

$α_{1} + α_{2}$
B

$2 α_{1} + α_{2}$
C

$α_{1} + 2 α_{2}$
D

None of these

Solution

$V = V_{0} (1 + γ ∆ θ)$
$L^{3} = L_{0} (1 + α_{1} ∆ θ) {L^{2}}_{0} {(1 + α_{2} ∆ θ)}^{2}$
= ${L^{3}}_{0} (1 + α_{1} ∆ θ) {(1 + α_{2} ∆ θ)}^{2}$
Since ${L^{3}}_{0} = V_{0} and L^{3} = V$
Hence 1+ $γ ∆ θ = (1 + α_{1} ∆ θ) {(1 + α_{2} ∆ θ)}^{2}$
= $(1 + α_{1} ∆ θ) (1 + 2 α_{2} ∆ θ)$
= (1+ $α_{1} ∆ θ + 2 α_{2} ∆ θ$ )
$\Rightarrow γ = α_{1} + 2 α_{2}$

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