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The coefficient of volume expansion (\beta) is a measure of how the volume of a substance changes as the temperature changes. It shows the sensitivity of the liquid's volume to temperature variations. In mathematical terms, if a liquid has an initial volume V, and the temperature changes by \Delta T, the corresponding change in volume (\Delta V) can be expressed as \( \Delta V = V \beta \Delta T \). This formula tells us that for every degree change in temperature, the volume changes by a fraction of its original volume, which is determined by \beta. For instance, if \beta = 0.001 per degree Celsius, and the temperature increases by 10 degrees, then the volume of the liquid will increase by 1% of its original volume.

To grasp the volume-height relationship in a barometer, think of the liquid column as having a constant cross-sectional area A. In this scenario, the volume of the liquid V is given by the product of its height h and the cross-sectional area A, i.e., V = Ah. When the temperature changes, both volume and height change, but the cross-sectional area remains constant. Given that the volume changes with temperature as \( \Delta V = V \beta \Delta T \), we can express this volume change in terms of the corresponding height change \Delta h. Since \Delta V = A \Delta h, we can substitute the relationships into one another to find \Delta h. This results in the equation \( \ \Delta h = h\beta \Delta T \), illustrating how the height of the liquid column in a barometer relates directly to changes in temperature and the coefficient of volume expansion.

In a barometer, the height of the liquid column is sensitive to temperature changes, reflecting the physical principles of thermal expansion. When the temperature of the liquid increases, the liquid expands. The expansion can be quantified using the coefficient of volume expansion (\beta), governing how the liquid's height changes. For a constant pressure scenario assumed in the barometer, as the temperature goes up or down by \Delta T, the height change \Delta h of the liquid column is given by \( \Delta h = h\beta \Delta T \). This relationship underscores a direct link between the height variation of the liquid column and temperature shifts. In essence, a higher temperature results in a longer column, while a drop in temperature shortens it, assuming the pressure is constant. This principle helps in understanding and predicting how barometers, and similar liquid systems, behave under varying thermal conditions.