When a block of iron floats in mercury at 0^oC, a fraction $k_1$, of its volume is submerged, while at the temperature ${60}^0 C$, a fraction k₂, is submerged. If the
coefficient of volume expansion of iron is ${\mathrm{\gamma }}_{\mathrm{Fe}}$ and that of mercury is${\mathrm{\gamma }}_{\mathrm{Fe}}$ , then the ratio ,k₁ / k₂ can be expressed as

 let M be the mass of iron block and V its volume. Then
$$\begin{gathered} \mathrm{Mg}={\mathrm{V\rho }}_{\mathrm{Fe}\left(0\right)}\mathrm{g}={\mathrm{k}}_1{\mathrm{V\rho }}_{\mathrm{Hg}} \\ \mathrm{or} {\mathrm{\rho }}_{\mathrm{Fe}\left(0\right)}={\mathrm{k}}_1{\mathrm{\rho }}_{\mathrm{Hg}\left(0\right)} \\ \mathrm{and} \mathrm{Mg}={\mathrm{V\rho }}_{\mathrm{Fe}\left(60\right)}\mathrm{g}={\mathrm{k}}_2{\mathrm{V\rho }}_{\mathrm{Hg}\left(60\right)}\mathrm{g} \\ \mathrm{or} {\mathrm{\rho }}_{\mathrm{Fe}\left(60\right)}={\mathrm{k}}_2{\mathrm{\rho }}_{\mathrm{Hg}\left(60\right)} \\ \therefore \frac{{\mathrm{\rho }}_{\mathrm{Fe}\left(0\right)}}{{\mathrm{\rho }}_{\mathrm{Fe}\left(60\right)}}=\frac{{\mathrm{k}}_1}{{\mathrm{k}}_2}\times \frac{{\mathrm{\rho }}_{\mathrm{Hg}\left(0\right)}}{{\mathrm{\rho }}_{\mathrm{Hg}\left(60\right)}} \\ \text{ Now }{\mathrm{\rho }}_{\mathrm{Fe}\left(60\right)}=\frac{{\mathrm{\rho }}_{\mathrm{Fe}\left(0\right)}}{\left(1+60{\mathrm{\gamma }}_{\mathrm{Fe}}\right)} \\ \text{ and }{\mathrm{\rho }}_{\mathrm{Hg}\left(60\right)}=\frac{{\mathrm{\rho }}_{Hg\left(0\right)}}{\left(1+60{\mathrm{\gamma }}_{\mathrm{Hg}}\right)} \\ \therefore \frac{{\mathrm{k}}_1}{{\mathrm{k}}_2}=\frac{{\mathrm{\rho }}_{\mathrm{Fe}\left(0\right)}}{{\mathrm{\rho }}_{Fe\left(60\right)}}\times \frac{{\mathrm{\rho }}_{\mathrm{Hg}\left(60\right)}}{{\mathrm{\rho }}_{\mathrm{Hg}\left(0\right)}}=\frac{1+60{\mathrm{\gamma }}_{\mathrm{Fe}}}{1+60{\mathrm{\gamma }}_{\mathrm{Hg}}} \end{gathered}$$