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

Given: At 0^o​C, volume of glass vessel, V_g = 10 × 10 × 10 = 1000 cc = volume of mercury, V_Hg
Let the volume of mercury at 10°C be V'_Hg and that of glass be V'_g.
At 10^​oC, the additional volume of mercury than glass, due to heating, V'_Hg – V'_g = 1.6 cm³
So change in temperature, ΔT = 10°C
Coefficient of linear expansion of glass, α_g = 6.5 × 10^–6 °C^–1
Therefore, the coefficient of volume expansion of glass, γ_g = 3 × 6.5 × 10^–6°C^–1​
Let the coefficient of volume expansion of mercury be γ_Hg.
We know that
V'_Hg = V_Hg (1 + γ _Hg ΔT) ...(1)
V'_g = V_g (1 + γ_g ΔT) ...(2)
Subtracting (2) from (1) we get,
V'_Hg – V'_g = V_Hg – V_g + V_Hg γ_Hg ΔT – V_g γ_g ΔT (as V_Hg = V_g)
$$\begin{gathered} \Rightarrow 1.6=1000\times {\gamma }_{\mathrm{Hg}}\times 10-1000\times 6.5\times 3\times {10}^{-6}\times 10 \\ \Rightarrow {\gamma }_{\mathrm{Hg}}=\frac{1.6+19.5\times {10}^{-2}}{10000} \\ \Rightarrow {\gamma }_{\mathrm{Hg}}=\frac{1.6+0.195}{10000} \\ \Rightarrow {\gamma }_{Hg}=\frac{1.795}{10000} \\ \Rightarrow {\gamma }_{Hg}=1.795\times {10}^{-4} \\ \Rightarrow {\gamma }_{\mathrm{Hg}}\cong 1.8\times {10}^{-4}°{\mathrm{C}}^{-1} \end{gathered}$$
Therefore, the coefficient of volume expansion of mercury is 1.8× 10^–4 °C^–1.