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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.
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Solution
Given: At 0oC, volume of glass vessel, Vg = 10 × 10 × 10 = 1000 cc = volume of mercury, VHg
Let the volume of mercury at 10°C be V'Hg and that of glass be V'g.
At 10oC, 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, αg = 6.5 × 10–6 °C–1
Therefore, the coefficient of volume expansion of glass, γg = 3 × 6.5 × 10–6°C–1
Let the coefficient of volume expansion of mercury be γHg.
We know that
V'Hg = VHg (1 + γHg ΔT) ...(1)
V'g = Vg (1 + γg ΔT) ...(2)
Subtracting (2) from (1) we get,
V'Hg – V'g = VHg – Vg + VHg γHg ΔT – Vg γg ΔT (as VHg = Vg)
\[\Rightarrow 1 . 6 = 1000 \times \gamma_{Hg} \times 10 - 1000 \times 6 . 5 \times 3 \times {10}^{- 6} \times 10\]
\[ \Rightarrow \gamma_{Hg} = \frac{1 . 6 + 19 . 5 \times {10}^{- 2}}{10000}\]
\[ \Rightarrow \gamma_{Hg} = \frac{1 . 6 + 0 . 195}{10000}\]
\[ \Rightarrow \gamma_{Hg} = \frac{1 . 795}{10000}\]
\[ \Rightarrow \gamma_{Hg} = 1 . 795 \times {10}^{- 4} \]
⇒ γHg ≅ 1.8 × 10-4°C-1
Therefore, the coefficient of volume expansion of mercury is 1.8× 10–4 °C–1.
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