# Prove that beta equal 2 alpha and prove that gamma equal 3 alpha.

Arun
25757 Points
3 years ago
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) expression

A → ( 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
Sai Soumya
99 Points
3 years ago
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