The figure shows a glass tube (linear co-efficient of expansion is $\alpha$) completely filled with a liquid of volume expansion co-efficient $\gamma$. On heating length of the liquid column does not change. Choose the correct relation between $\gamma$ and $\alpha$

When length of the liquid column remains constant, then the level of liquid moves down with respect to the container, thus $\mathrm{\gamma }$ must be less than 3$\mathrm{\alpha }$.

Now we can write $\mathrm{V}={\mathrm{V}}_0(1+\mathrm{\gamma \Delta T})$

Since $\mathrm{V}={\mathrm{Al}}_0=\left[{\mathrm{A}}_0(1+2\mathrm{\alpha \Delta T})\right]{\mathrm{l}}_0={\mathrm{V}}_0(1+2\mathrm{\alpha \Delta T})$

Hence ${\mathrm{V}}_0(1+\mathrm{\gamma \Delta T})={\mathrm{V}}_0(1+2\mathrm{\alpha \Delta T})\Rightarrow \mathrm{\gamma }=2\mathrm{\alpha }$