A liquid having a coefficient of volume expansion `gamma` is filled in a cylindrical vessel made out of glass having coefficient of lienar expansion `alpha = (gamma)/( 5)` . At room temperature , the level of liquid in the vessel is`l_(0)` and when the temperature is increased by `Delta T`, the level of liquid in the vessel becomes `l~~ l_(0)= ( 1+ n alpha Delta T)`. What is the value of n ? [Given`, alpha Delta T lt lt 1`]

Answer

3

Explanation

Let us assume that the volume of the liquid in the vessel is `V_(0)` and the areaof cross-section of the vessel is `A_(0)`, then
`l_(0)=(V_(0))/(A_(0))`
Similarly,
`l=(V)/(A)=(V_(0)(1+gammaDeltaT))/(A_(0)(1+2alphaDeltaT))`
Using binomial approximation, we get
`l=l_(0)(1+gammaDeltaT)(1-2alphaDeltaT)`
`implies l`
`=l_(0)[1+(gamma-2alpha)DeltaT-2alphagamma(DeltaT)^(2)]`
Neglecting `2alphagamma(DeltaT)^(2)`, since it is insignificant as compared to
`(gamma-2alpha)DeltaT`, we get
`l~~l_(0)[1+(gamma-2alpha)DeltaT]`
`impliesl~~l_(0)[1+(5alpha-2alpha)DeltaT]`
`impliesl~~l_(0)[1+3alphaDeltaT]`
So, n=3