Coefficient of linear expansion along one edge of a cube-shaped crystal is `alpha_(1)`, but that along a perpendicular direction is `alpha_(2)`. Find the coefficient of volume expansion of the crystal.

Explanation

Let `l_(0)=` length of a side of the crystal at `0^(@)C`, and t = rise in temperature
`therefore` Final volume of the crystal
`" "=l_(0)(1+alpha_(1)t) times l_(0)^(2)(1+alpha_(2)t)^(2)`
`" "=l_(0)^(3)(1+alpha_(1)t)(1+alpha_(2)t)^(2)`
`" "=V_(0)(1+alpha_(1)t)(1+alpha_(2)t)^(2) " "[because l_(0)^(3)=V_(0)]`
Now, `V=V_(0)(1+gammat), " where " gamma` is the coefficient of volume expansion of the crystal.
`therefore " " 1+gammat=(1+alpha_(1)t))(1+alpha_(2)t)^(2)=(1+alpha_(1)t)(1+2alpha_(2)t)`
`" "=1+t(alpha_(1)+2alpha_(2))` [neglecting higher powers]
or, `" "gamma=alpha_(1)+2alpha_(2).`