In a U-tube, a liquid is poured to a height h^' in each arm. When left and right arms of the tub

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In a U-tube, a liquid is poured to a height h^' in each arm....

In a U-tube, a liquid is poured to a height $h^{\prime}$ in each arm. When left and right arms of the tube is heated to temperature $T_{1}$ and $T_{2}$ respectively, the height in each arm changes to $h_{1}$ and $h_{2}$ respectively. What is the relation between coefficients of volume expansion of liquid and heights, $h_{1}$ and $h_{2}$ ? (a) $\gamma=\frac{h_{1}-h_{2}}{T_{1} h_{2}-T_{2} h_{1}}$ (b) $\gamma=\frac{h_{1}+h_{2}}{T_{1} h_{2}-T_{2} h_{1}}$ (c) $\gamma=\frac{h_{1}+h_{2}}{T_{1} h_{2}+T_{2} h_{1}}$ (d) $\gamma=\frac{h_{1}-h_{2}}{T_{1} h_{1}-T_{2} h_{2}}$

In a U-tube, a liquid is poured to a height $h^{\prime}$ in each arm. When left and right arms of the tube is heated to temperature $T_{1}$ and $T_{2}$ respectively, the height in each arm changes to $h_{1}$ and $h_{2}$ respectively. What is the relation between coefficients of volume expansion of liquid and heights, $h_{1}$ and $h_{2}$ ?
(a) $\gamma=\frac{h_{1}-h_{2}}{T_{1} h_{2}-T_{2} h_{1}}$
(b) $\gamma=\frac{h_{1}+h_{2}}{T_{1} h_{2}-T_{2} h_{1}}$
(c) $\gamma=\frac{h_{1}+h_{2}}{T_{1} h_{2}+T_{2} h_{1}}$
(d) $\gamma=\frac{h_{1}-h_{2}}{T_{1} h_{1}-T_{2} h_{2}}$

Key Concepts

Hydrostatic Equilibrium

Hydrostatic equilibrium refers to the balance of pressures within a fluid at rest, wherein the weight of the fluid is balanced by the pressure gradient. In systems like U-tubes, differences in the heights of liquid columns result in pressure differences that adjust until equilibrium is achieved. This concept is essential for analyzing how variations in temperature and consequent volume changes result in different liquid heights in the two arms of the tube.

Conservation of Volume in a Closed System

In a system where the container’s geometry is fixed or nearly constant, the total volume of the liquid remains conserved even as its distribution may change due to thermal expansion. This conservation principle is critical in deriving relationships between different parts of the system, such as determining how the liquid redistributes in connected sections when subject to different temperatures.

Coefficient of Volume Expansion

This concept refers to a material property that quantifies the fractional change in volume per unit rise in temperature. It is fundamental in thermodynamics and materials science, as it allows prediction of how much a liquid (or any material) will expand when heated. Understanding this coefficient is crucial when analyzing systems where temperature-induced volume changes affect equilibrium states.

Thermal Expansion

Thermal expansion is the phenomenon where substances change their dimensions, including volume, in response to a change in temperature. In fluids, this volumetric change can lead to observable effects such as differences in height when the liquid is contained in connected vessels. This principle underlies many practical applications and experiments involving temperature variations.

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