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^–6 °C^–1.

Given:
Volume of a glass vessel: V_g = 10cm x 10cm x 10cm = 1000 cm³.
Volume of mercury in the vessel : V_Hg = 1000 cm³.( As it is filled completely in the glass vessel)
Temperature at which glass is filled completely with mercury: T₁ = 0 °C.
Increased temperature : T₂ = 10 ° C.
Change in temperature : Δ T = T₂-T₁ = 10 ° C.
Volume of mercury overflowed:= 1.6 cm³
Coefficient of linear expansion of glass : α_g = 6.5 × 10^–6 °C^–1.
∴ Coefficient of Volume Expansion of Glass: γ_g: 3× α_g = 3× 6.5× 10^-6 °C^–1
We need to find Coefficient of volume expansion of mercury : γ_Hg
Formula used:

Formula for Volume expansion is :

Where V’ is the changed Volume of the body at T₂ and V is the volume of body at T₁.
Now, the Volume of mercury overflown is equal to the difference in changed volumes of mercury and glass respectively.
Numerically:

From the volume expansion formula,
Changed Volume of Mercury: V’_Hg = V_Hg( 1+γ_HgΔ T)
Changed volume of Glass: V’_{g =} V_g(1 +γ_g ΔT)
Substituting the values , we get:

Calculate the coefficient of volume expansion of mercury is :

γ_Hg = 1.795× 10^-4 ° C^-1_.

Note: The Steps marked in Bold are important for the numerical.