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

At T = 20°C, the volume of the glass vessel, V_g = 1000 cc.
Let the volume of mercury be V_Hg .
Coefficient of cubical expansion of mercury, γ_Hg = 1.8 × 10^–4 /°C
Coefficient of cubical expansion of glass, γ_g = 9 × 10^–6 /°C
​Change in temperature, ΔT, is same for glass and mercury.
Let the volume of glass and mercury after rise in temperature be V'_g and V'_Hg respectively.
Volume of remaining space after change in temperature,(V'_g – V'_Hg) = Volume of the remaining space (initial),(V_g​​ – V_Hg)
We know: V'_g = V_g (1 + γ_g ΔT) ...(1)
V'_Hg = V_Hg (1 + γ _Hg ΔT) ...(2)

Subtracting (2) from (1), we get:
$$\begin{gathered} V{\text{'}}_{\mathrm{g}}-V{\text{'}}_{\mathrm{Hg}}=V_{\mathrm{g}}-V_{\mathrm{Hg}}+V_{\mathrm{g}}{\gamma }_{\mathrm{g}}∆T-V_{\mathrm{Hg}}{\gamma }_{\mathrm{Hg}}∆T \\ \Rightarrow V_{\mathrm{g}}{\gamma }_{\mathrm{g}}∆T-V_{\mathrm{Hg}}{\gamma }_{\mathrm{Hg}}∆T=0 \\ \Rightarrow \frac{V_{\mathrm{g}}}{V_{\mathrm{Hg}}}=\frac{{\gamma }_{\mathrm{Hg}}}{{\gamma }_{\mathrm{g}}} \\ \Rightarrow \frac{1000}{V_{\mathrm{Hg}}}=\frac{1.8\times {10}^{-4}}{9\times {10}^{-6}} \\ \Rightarrow V_{\mathrm{Hg}}=\frac{9\times {10}^{-3}}{1.8\times {10}^{-4}} \\ \Rightarrow V_{\mathrm{Hg}}=50\mathrm{cc} \end{gathered}$$
Therefore, the volume of mercury that should be poured into the glass vessel is 50 cc.