Question

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?

Answer 1

Given:
Density of wood at 0 °C, f_w = 880 kgm^​-3
Density of benzene at 0 °C, f_b = 900 kgm^-​3
Coefficient of volume expansion for wood, γ_w = Coefficient of volume expansion for benzene, γ_b = 1.5 × 10^–3 °C^–1
So, initial temperature, T₁ = 0 °C
Let T₂ be the temperature at which the piece of wood will just sink in benzene and $\Delta$T = T₂$-$T₁.
The piece of wood begins to sink when its weight is equal to the weight of the benzene displaced.
Mass = volume $\times$density
$$\begin{gathered} \mathrm{Therefore},Vf{\text{'}}_{w}g=Vf{\text{'}}_{b}g \\ \Rightarrow \frac{f_w}{1+{\gamma }_w\Delta T}=\frac{f_b}{1+{\gamma }_b\Delta T} \\ \Rightarrow \frac{880}{1+1.2\times {10}^{-3}\Delta T}=\frac{900}{1+1.5\times {10}^{-3}\Delta T} \\ \Rightarrow 880+880\times 1.5\times {10}^{-3}\Delta T=900+900\times 1.2\times {10}^{-3}\Delta T \\ \Rightarrow \left(1320-1080\right)\times {10}^{-3}\Delta T=20 \\ \Rightarrow \Delta T=83.3°\mathrm{C} \\ \Rightarrow T_2-T_1\cong {83}^o \\ \Rightarrow T_2-0°\cong {83}^o \\ \Rightarrow T_2\cong 83°\mathrm{C} \end{gathered}$$
Therefore, the piece of wood will just sink in benzene at 83 ^oC.