A thin long glass tube of uniform cross-section contains 1 m long mercury thread at `0^(@)C.` At `100^(@)C`, the mercury thread increases by 16.5 mm. If the coefficient of real expansion of mercury is `0.000182^(@)C^(-1)`, find the coefficient of linear expansion of glass.

Explanation

Let the areas of cross-section of the glass tube be `A_(0)cm^(2) " and " A_(100)cm^(2) " at " 0^(@)C " and " 100^(@)C` respectively.
Volume of mercury at `0^(@)C, V_(0)=100 times A_(0) cm^(3) " …(1)"`
and volume of mercury at `100^(@)C`, `" " V_(100)=101.65 times A_(100)cm^(3) " ...(2)"`
Also from surface expansion of glass we get,
`" " A_(100)=A_(0)(1+100beta)`
`" "=A_(0)(1+100 times 2alpha_(g)) " ...(3)"`
`" "`[where `alpha_(g)=` coefficient of linear expansion of glass]
From definition, `gamma` for mercury `=(V_(100)-V_(0))/(V_(0) times 100) " ...(4)"`
Substituting the values of `gamma, V_(0), V_(100)` and using (3), we have,
`0.000182=(101.65 times A_(100)-100 times A_(0))/(100 times A_(0) times 100)`
`" "=(101.65A_(0)(1+200alpha_(g))-100A_(0))/(100 times A_(0) times 100)`
`" "=(101.65(1+200alpha_(g))-100)/(10^(4))`
or, `" "1.82=101.65(1+200alpha)-100`
`therefore " " 1+200alpha=101.82/101.65 " or, " alpha=8.3 times 10^(-6@)C^(-1).`