Question

The volume of a glass vessel is $1000cc$ at ${20}^{o}C$. 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 $1.8\times {10}^{-4}$ ${}^o{C}^{-1}$ and $9.0\times {10}^{-6}$ ${}^o{C}^{-1}$ respectively

Answer 1

$V_g=1000cc$, $T_1={20}^{o}C$

$V_{Hg}=?$ ${\gamma }_{Hg}=1.8\times {10}^{-4}{/}^{o}C$

${\gamma }_g=9\times {10}^{-6}{/}^{o}C$

$\Delta T$ remains constant

Volume of remaining space $=V_g^{\prime }-V_{Hg}^{\prime }$

Now

$V_g^{\prime }=V_g(1+{\gamma }_g\Delta T)$ ...(1)

$V_{Hg}^{\prime }=V_{Hg}(1+{\gamma }_{Hg}\Delta T)$ ...(2)

Subtracting (2) from (1)

$V_g^{\prime }-V_{Hg}^{\prime }=V_g-V_{Hg}+V_g{\gamma }_g\Delta T-V_{Hg}{\gamma }_{Hg}\Delta T$

$\Rightarrow \frac{V_g}{V_{Hg}}=\frac{{\gamma }_{Hg}}{{\gamma }_g}\Rightarrow \frac{1000}{V_{Hg}}=\frac{1.8\times {10}^{-4}}{9\times {10}^{-6}}$

$\Rightarrow V_{HG}=\frac{9\times {10}^{-3}}{1.8\times {10}^{-4}}=500cc$.