Question

A glass vessel measures exactly 10 cm $\times$ 10 cm $\times$ 10 cm at $0^{\circ }C$. It is filled completely with mercury at this temperature. When the temperature is raised to ${10}^{\circ }C$, $1.6c{m}^3$ of mercury overflows. Calculate the coefficient of volume expansion of mercury. Coefficient of linear expansion of glass $=6.5\times {10}^{-6}{}^{\circ }{C}^{-1}$.

Answer 1

$V_g=10\times 10\times 10=100cc.$

$\Delta T={10}^{\circ }C$

$V_{Hg}^{\prime }-V_g^{\prime }=1.6c{m}^3$

${\alpha }_g=6.5\times {10}^{-6}{/}^{\circ }C$

${\gamma }_{Hg}=?$

${\gamma }_g=3\times 6.5\times {10}^{-6}{/}^{\circ }C$

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

$V_g^{\prime }=V_g(1+{\gamma }_g\Delta T)$ . . . (2)

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

$1.6=1000\times {\gamma }_{Hg}\times 10-1000\times 6.5\times 3\times {10}^{-6}\times 10$

${\gamma }_{Hg}=\frac{1.6+19.5\times {10}^{-2}}{10000}$

$=\frac{1.6+0.195}{10000}=\frac{1.795}{10000}$

$=1.795\times {10}^{-4}$

$=1.8\times {10}^{-4}/{}^{\circ }C$