A thread of liquid is in a uniform capillary tube of length L, as measured by a ruler. The temperature of the tube and thread of liquid is raised by $\mathrm{\Delta T}$. If $\mathrm{\gamma }$ be the coefficient of volume expansion of the liquid and $\mathrm{\alpha }$ be the coefficient of linear expansion of the material of the tube, then the increase $\mathrm{\Delta L}$, in the length of the thread, again measured by the ruler will be

Since a ruler is used, the scale used does not expand with the tube. If the radius of the capillary be r, the increase due to thermal expansion is given by $\mathrm{dr}=\mathrm{r\alpha dT}$ for a temperature rise of dT. Since area of cross section is $\mathrm{A}={\mathrm{\pi r}}^2$, we see that$\mathrm{dA}/\mathrm{A}=2\mathrm{dr}/\mathrm{r}\text{ or }\mathrm{dA}=\mathrm{A}\left(2\mathrm{\alpha }\right)\mathrm{dT}.$ Thus if the temperature is increased from T to T + dT, the cross-sectional area changes from A to A$(1+2\mathrm{\alpha dT})$. The volume expansion of the liquid gives ${\mathrm{V}}^{\mathrm{\prime }}=\mathrm{V}+\mathrm{dV}=\mathrm{V}(1+\mathrm{\gamma dT})$ , where $\mathrm{\gamma }$ is the coefficient of the volume expansion of the liquid. This causes change in length of thread and final length becomes L' = L + dL. The mass of liquid is constant; hence, ${\mathrm{L}}^{\mathrm{\prime }}{\mathrm{A}}^{\mathrm{\prime }}={\mathrm{V}}^{\mathrm{\prime }}=\mathrm{V}(1+\mathrm{\gamma dT})=\mathrm{LA}(1+\mathrm{\gamma dT})$.

But ${\mathrm{A}}^{\mathrm{\prime }}=\mathrm{A}(1+2\mathrm{\alpha dT})$

Hence, ${\mathrm{L}}^{\mathrm{\prime }}=\mathrm{L}\left(\frac{1+\mathrm{\gamma dT}}{1+2\mathrm{\alpha dT}}\right)$

         $=\mathrm{L}\left[1+(\mathrm{\gamma }-2\mathrm{\alpha })\mathrm{dT}-2\mathrm{\alpha \gamma }(\mathrm{dT}{)}^2\right]$

The last term is negligible.

$\text{ Hence, }{\mathrm{L}}^{\mathrm{\prime }}=\mathrm{L}[1+(\mathrm{\gamma }-2\mathrm{\alpha }\left)\mathrm{dT}\right]$

                 $\mathrm{\Delta L}=\mathrm{L}(\mathrm{\gamma }-2\mathrm{\alpha })\mathrm{\Delta T}$