Obtain relation between coefficient of volume expansion `(alpha_(V))` and coefficient of linear expansion `(alpha_(l))`.

Explanation

Suppose there is a cube of side of length .l.. When its temperature is increased by `DeltaT`, it expands equally in all dimensions.
Hence from `V=l^(3)`
`DeltaV=(l+Deltal)^(3)-l^(3)`
`=l^(3)+3l^(2)Deltal+3l(Deltal)^(2)+(Deltal)^(3)-l^(3)`
But `(Deltal)^(2)` and `(Deltal)^(3)` are much more less than l, hence by neglecting them
`DeltaV=3l^(2)Deltal` . . .(1)
But, from linear expansion.
`Deltal=alpha_(l)lDeltaT` . . .(2)
`:.` By using value of equation (1) in equation (2),
`DeltaV=3l^(2)(alpha_(l)lDeltaT)`
`=3l^(3)alpha_(l)DeltaT`
`DeltaV=3Valpha_(l)DeltaT" "` . . .(3)
`" "(becausel^(3)=" volume of cube")`
By comparing equation (3) with general equation of volume expansion `DeltaV=alpha_(V)VDeltaT`.
`alpha_(V)=3alpha_(l)` which is relation between coefficient of volume and linear expansion.