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Prove that beta equal 2 alpha and prove that gamma equal 3 alpha.
Denote by α : the coefficient of linear expansion β : the coefficient of surface expansion γ : the coefficient of volumetric expansion Then a length increases as L → L ( 1 + α ΔT) But this means that for isotropic (same in every direction) expansion a surface (length x length) increases as A → A ( 1 + α ΔT)( 1 + α ΔT) ≈ A (1 +2 α ΔT) where we have neglected the (usually very small) square term (α ΔT)² . Comparing with the (definition of β) expressionA → ( 1 + βΔT) , we see the relation β = 2α . Likewise V → V ( 1 + γ ΔT) from the definition of volumetric expansion coefficient. But also we can approximate (volume = length x length x length) V → V ( 1 + α ΔT)³ ≈ V ( 1 + 3 α ΔT) , neglecting higher powers of α ΔT. Hence γ = 3 α Hope it helps
You can find the answer in class 11 NCERT book in the chapter Kinetic Theory of Gases.It has been briefly explained Otherwise you can refer to any reference book
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