# Prove that beta is equal to 2 alpha. And gamma is equal to 3 alpha

Arun
25750 Points
6 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
mary
13 Points
4 years ago
Using solvings show that one beta particle is equal to two alpha particle and one gamma particle is equal to one particle,simultaneously

Rishi Sharma
4 years ago
Dear Student,

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

Thanks and Regards