Question

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 × 10–6 °C–1 and 9.0 × 10–6 °C–1 , respectively.

Answer 1

The volume of mercury should be poured into it at this temperature so that the volume of the remaining space does not change with temperature is 50 cc.

Explanation:

At T = 20°C, Glass vessel capacity,  $V_g$ = 1000 cc.

Let mercury be in volume$V_{Hg}$.

Cubic expansion coefficient for mercury, $\gamma H_g = 1.8 × 10–$4 /°C

Cubic Glass Expansion Coefficient, $\gamma g = 9 \times 10{-6}$  /°C

Temperature change is the same for glass and mercury.

Let the amount of glass and mercury be after heat rise V’g and V’Hg respectively.

Amount of space left after temperature change ,$(V{\prime}_g - V{\prime}_{Hg})$ = Volume of space left over (initial)$(V_g - V_{Hg})$

We know:  

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

$\mathrm{V}_{\mathrm{Hg}}^{\prime}=\mathrm{V}_{\mathrm{Hg}}\left(1+\gamma_{\mathrm{Hg}} \Delta \mathrm{T}\right)$------------------(2)

Subtracting (2) from (1) , we obtain:

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

$\frac{V_{g}}{V_{H g}}=\frac{\gamma_{H g}}{\gamma_{g}}$

$\frac{1000}{V_{H g}}=\frac{1.8 \times 10^{-4}}{9 \times 10^{-6}}$

$V_{H B}=\frac{1000 \times 9 \times 10^{-6}}{1.8 \times 10^{-4}}$

$&V_{H g}=\frac{9000 \times 10^{-6}}{1.8 \times 10^{-4}}$

$&V_{H g}=\frac{9 \times 10^{-3}}{1.8 \times 10^{-4}}$

$&V_{H B}=\frac{9 \times 10^{1}}{1.8}$

$&V_{H g}=\frac{900}{18}$

$&\mathrm{V}_{\mathrm{Hg}}=50$

The volume of mercury to be poured into the glass vessel therefore is 50 cc.