The densities of wood and benzene at 0°C are 880 kg m³ and 900 kg m^–3 respectively. The coefficients of volume expansion are 1.2 × 10^–3 °C^–1 for wood and 1.5 × 10^–3 °C^–1 for benzene. At what temperature will a piece of wood just sink in benzene?

Given:
At T₁ = 0 °C,
Density of Wood : ρ_w = 880 kg m.
Density of Benzene: ρ_b = 900 kg m^–3.
The coefficients of volume expansion of wood:
γ_w =1.2×10^–3°C^–1
The coefficients of volume expansion of benzene:
γ_b=1.5×10^–3 °C^–1.
We need to find the temperature T₂ at which the piece of woo will just sink in benzene.
Here we will be using Archimedes Law:
When the piece of wood just sinks in benzene it will
displace some amount of benzene. In simple words the piece of wood starts to sink when it’s density is equal to or greater than the density of benzene.
Also Density of benzene decreases more rapidly with increase in temperature as compared to that of wood as
γ_g > γ_w.

Formula used:

Numerically:

Where ρ’ and V’ are the changed (Due to volume expansion) Density and Volume respectively.

Formula for Volume expansion of a body is :

Where V’ is the changed volume due to change in temperature ΔT.

For Wood:

For benzene:

Now using Density-Volume Relation:

Changed Density of wood:

Changed Density of benzene:

We Equate the final densities of wood and benzene to obtain the required condition.

At 83.33 ° C the piece of wood will just sink in benzene.