The volume of a glass vessel is 1000 cc at `20^@C`. What volume of mercury should be poured into it at this temperature so that the volume of the remaining space does not change with temperature? Coefficients of cubical expansion of mercury and glass are `1.8 xx 10^(-4)C^(-1)` and `9.0xx10^(-6)C^(-1).` respectively.

Explanation

`V_g=1000cc` V_Hg=?`
`T_1 =20^0C`
`gamma_Hg= 1.8 xx 10^-4/^0C`
`gamma_g= 9 xx 10^-6/^0C`
Here `Delta T` remains constant
Volume of remaining space `= V'_g-V'_Hg`
`Now V'_g =V'_g(1+gamma_g Delta T)` ........(1)
`V'_Hg = V_Hg (1+ gamma_Hg Delta T)` ........(2)
`Subtracting (2) from (1)`
`(V'_g-V'_Hg) =(V_g- V_Hg + ((V_g) (gamma_g) Delta T))`
`-((V_Hg) (gamma_Hg) Delta T)`
`rArr V_g / V_Hg = gamma_Hg / gamma_g`
`rArr 1000 / V_Hg =(1.8 xx 10^-4)/ (9 xx 10^-6)`
`rArr V_Hg = (9 xx 10^-3)/( 1.8 xx 10^-4)= 50 cc`