A glass tube of length `l_(0)` and of uniform cross-section is to be filled with mercury so that the length of the tube unoccoupied by mercury remains same at all temperatures. If `gamma_(g)=3alpha` , calculate length of mercury column.

Explanation

Let `x_(0)` : length of mercury column at `0^(@)C`
`x_(theta)` : length of mercury column at `theta^(@)C`
`A_(0)` : area of cross-section of glass tube at `0^(@)C`
`A_(theta)` : area of cross-section of glass tube at `theta^(@)C`
`A_(theta)=A_(0)(1+beta_(g)theta)=A_(0)(1+2alpha_(g)theta)` (i)
Volume of Hg at `theta^(@)C` , `(V_(m)_(theta)=A_(theta)x_(theta)`
Volume of glass at `theta^(@)C` , `(V_(g))_(theta)=A_(theta)l_(theta)`
Length of glass tube unoccupied by mercury
`Deltal=l_(theta)-x_(theta)=((V_(g))_(theta))/(A_(theta))-((V_(m))theta)/(A_(theta))`
`=((V_(g))_(0)(1+3alpha_(g)theta)-(V_(m))_(0)(1+gamma_(m)theta))/(A_(0)(1+2alpha_(g)theta))`
`=l_(0)(1+3alpha_(g)theta)(1+2alpha_(g)theta)^(-1)-x_(0)(1+gamma_(m)theta)`
`(1+2alpha_(g)theta)^(-1)`
`=l_(0)[1+(3alpha_(g)-2alpha_(g)theta]-x_(0)[(1+(gamma_(m)-2alpha_(g))theta)]`
[neglecting smaller terms]
As `Deltal` is independent of temperature
`(d(Deltal))/(dtheta)=l_(0)(alpha_(g))-xo(gamma_(m)-2alpha_(g))=0`
`x_(0)=(l_(0)alpha_(g))/(gamma_(m)-2alpha_(g))=(l_(0)alpha)/(gamma_(m)-2alpha)`