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^–4 °C^–1 and 9.0 × 10^–6 °C^–1 respectively.

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^–4 °C^–1 and 9.0 × 10^–6 °C^–1 respectively.

Philoid · Accepted Answer

Given:
Volume of Glass vessel at T₁ = 20 ° C is : V_g = 1000 cc.

Coefficients of cubical expansion of mercury: γ_Hg =1.8 ×10^–4 °C^–1.

Coefficients of cubical expansion of Glass : γ_g = 9.0 × 10^–6 °C^–1.

Now Let the Volume of mercury at T₁ = 20° C be V_Hg. We need to find V_Hg.

Due to Change in temperature, Volume Expansion takes place.

When mercury is added in the glass, it consumes some amount of volume V_Hg, thus the remaining space is V_g - V_Hg. ----( Initial).

When volume expansion takes place the changed volume of glass and mercury inside it is V’_g and V’_Hg respectively. Thus the remaining space after the volume expansion is V’_g –V’_Hg.--( Final)

The given condition says that the remaining space should not change with temperature which means: Initial = Final

Formula used :
Where V’ is the changed volume at T₂ and V is the initial volume at T₁.

For Glass: V’_g = V_g( 1 + γ_gΔT)

For Mercury : V’_Hg = V_Hg( 1 + γ_HgΔ T)

Substituting we get,

Hence, the volume of mercury poured into the glass it at 20 ° C so that the volume of the remaining space does not change with temperature is 50 cc.