Question

A glass vessel measures exactly 10 cm × 10 cm × 10 cm at 0°C. It is filled completely with mercury at this temperature. When the temperature is raised to 10°C, 1.6 cm3 of mercury overflows. Calculate the coefficient of volume expansion of mercury. Coefficient of linear expansion of glass = 6.5 × 10–1 °C–1.

Answer 1

The coefficient of volume expansion of mercury is $\bold{1.8 \times 10^{-4}}$ ° $\bold{C^{-1}}$

Explanation:

At 0°C, glass vessel volume, $Vg = 10 \times 10 \times 10 = 1000 cc$ = mercury volume, $V_{Hg}$

Let the volume of mercury be $V^{\prime}_{Hg}$ at 10 ° C and that of glass $V^{\prime}g$.

Around 10°C, extra mercury volume than glass, due to heating,$V^{\prime} _{Hg} - V^{\prime}_g$ = 1.6 $cm^3$

Thus the temperature changes, ΔT = 10°C

Linear expansion ratio for glass,αg =$6.5 \times 10^{-6}$ °$C^{-1}$  

Hence the volume expansion coefficient for glass, $\gamma g = 3 \times 6.5 \times 10^{-6}$ ° $C^{-1}$

Let the volume expansion coefficient of mercury be γHg.

$\gamma g = 3 \times 6.5 \times 10^{-6}$

$V^{\prime}_{H g}=V_{H g}\left(1+Y_{H g} \Delta T\right)$

$V_{g}^{\prime}=V_{g}\left(1+\mu_{g} \Delta T\right)$

Subtract (2) from (1) that we get,

$\mathrm{V}_{\mathrm{g}}^{\prime}-\mathrm{V}_{\mathrm{Hg}}^{\prime}=\mathrm{V}_{\mathrm{B}}-\mathrm{V}_{\mathrm{Hz}}+\mathrm{V}_{\mathrm{B}} \gamma_{\mathrm{E}} \Delta \mathrm{T}-\mathrm{V}_{\mathrm{Hg}} \gamma_{\mathrm{Hg}} \Delta \mathrm{T}$

$1.6=1000 \times \gamma_{\mathrm{Hg}} \times 10-1000 \times 6.5 \times 3 \times 10-6 \times 10$

$\gamma_{\mathrm{Hg}}=\frac{1.6+19.5 \times 10-2}{10000}$

$\gamma_{\mathrm{Hg}}=\frac{1.6+0.195}{10000}$

$\gamma_{\mathrm{Hg}}=\frac{1.795}{10000}$]

$\gamma_{\mathrm{Hg}}=1.795 \times 10-4$

$\gamma_{\mathrm{Hg}} \cong 1.8 \times 10-4^{\circ} C^{-1}$

Hence, the volume expansion coefficient for mercury is $1.8 \times 10^{-4}$ °$C^{-1}$