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In an experiment to find the liquid expansion coefficient $(\gamma)$ a column. of experimental liquid at $T\,^oC$ is balanced against another column of experimental liquid at $0\, ^o C$ by taking them in $U-$ tube. The expansion coefficient $(\gamma)$ is ?
$\frac{{{h_T}}}{{\left( {{h_T} - {h_0}} \right)T}}$
$\frac{{{h_0}}}{{\left( {{h_T} - {h_0}} \right)T}}$
$\frac{{\left( {{h_T} - {h_0}} \right)}}{{{h_0}T}}$
$\frac{{{h_T} - {h_0}}}{{{h_T}T}}$
Solution
At bottom
$\rho_{\mathrm{T}} h_{t} g=\rho_{0} h_{0} g \ldots \ldots(1)$
density variation of with temperature.
$\rho_{0}=\frac{M}{V_{0}}$ at $0^{\circ} \mathrm{C}$
$\rho_{T}=\frac{M}{V_{0}(1+\gamma \Delta T)}$ at $\mathrm{T}^{\circ} \mathrm{C}$
$\rho_{T}=\frac{\rho}{1+\gamma \Delta T}$ $…(2)$
Using $(1)$ and $(2)$
$\frac{-\rho_{0}}{1+\gamma \Delta T}-h_{T} g=\rho_{0} h_{0} g$
$\mathrm{h}_{\mathrm{T}}=\mathrm{h}_{0}(1+\gamma \Delta \mathrm{T})$
