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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 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, 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 + γ Hg 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)
1.6=1000×γHg×10-1000 × 6.5×3×10-6×10γHg=1.6+19.5×10-210000γHg=1.6+0.19510000γHg=1.79510000γHg=1.795×10-4γHg1.8×10-4°C-1
Therefore, the coefficient of volume expansion of mercury is 1.8× 10–4 °C–1.

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