Question

Consider two thermometers $T_1$ and $T_2$ of equal length which can be used to measure temperature over the range ${\theta }_1$ and ${\theta }_2$. $T_1$ contains mercury as thermometric liquid while $T_2$ contains bromine. The volumes of the two liquids are the same at the temperature ${\theta }_1$. The volumetric coefficients of expansion of mercury and bromine are $18\times {10}^{-5}{K}^{-1}$ and $108\times {10}^{-5}{K}^{-1}$, respectively. The increase in length of each liquid is the same for the same increase in temperature. If the diameters of the capillary tubes if the two thermometers are $d_1$ and $d_2$ respectively, then the ratio $d_1:d_2$ would be closest to.

• A. $6.0$
• B. $2.5$
• C. $0.5$
• D. $0.4$

Answer 1

The correct option is D $0.4$

Increase in length of each liquid is same

$\frac{\Delta V_{hg}}{\pi d_1^2}=\frac{\Delta V_{br}}{\pi d_2^2}$

$\frac{\Delta V_{hg}\Delta \theta }{\pi d_1^2}=\frac{\Delta V_{br}\Delta \theta }{\pi d_2^2}$

$\frac{d_1^2}{d_2^2}=\frac{{\gamma }_{hg}}{{\gamma }_{br}}=\frac{1}{6}$

$\frac{d_1}{d_2}=0.4$