The density of liquid gallium at \(30^{\circ} \mathrm{C}\) is \(6.095 \mathrm{~g} / \mathrm{mL}\). Because of its wide liquid range ( 30 to \(2400^{\circ} \mathrm{C}\) ), gallium could be used as a barometer fluid at high temperature. What height (in \(\mathrm{cm}\) ) of gallium will be supported on a day when the mercury barometer reads 740 torr? (The density of mercury is \(13.6 \mathrm{~g} / \mathrm{mL}\).). (a) 322 (b) 285 (c) 165 (d) 210

Fluid density is a fundamental concept in barometric pressure calculations and refers to the mass per unit volume of a fluid. In the context of barometry, it helps determine the weight of the column of fluid that the atmospheric pressure can support. When comparing different fluids, the one with higher density will have a shorter column height for a given pressure. In the original exercise provided, the fluid density is given directly in grams per milliliter (g/mL) for both gallium and mercury. However, to apply the barometric formula properly, we must convert their densities into kilograms per cubic meter (kg/m³), since these are the standard units used in physics for expressing density.

Understanding the pressure-height relationship is pivotal when working with barometers. Barometric pressure can support a column of fluid, and the height of that column is determined by this pressure. Mathematically expressed, the pressure at the bottom of a fluid column is equal to the product of the fluid's height, its density, and the gravitational acceleration (\( P = hgd \)). A crucial aspect of this relationship is its inverse nature; a denser fluid corresponds to a shorter column height for a given pressure. This inverse relationship allows us to compare the heights of different fluids (like gallium and mercury in the exercise) when they are subjected to the same atmospheric pressure.

Unit conversion is an often-overlooked but essential skill, especially in physics problems. For example, to work with the standard SI units in pressure calculations, it's necessary to convert the given pressure from torr to pascals (Pa) and the fluid density from grams per milliliter (g/mL) to kilograms per cubic meter (kg/m³). Also, after calculating the height of the fluid column in meters, we convert to centimeters to match the desired unit for the answer. This step is integral to achieving accuracy in our final result and contributes to a better understanding of the magnitudes involved in the phenomena being studied.

Gravitational acceleration (\( g \), approximately 9.8 m/s² on Earth's surface) is a constant value that enters into the calculation of the pressure exerted by a fluid column. It represents the acceleration due to Earth's gravity and is a crucial factor in the formula relating barometric pressure, fluid density, and column height (\( P = hgd \)). The same value of gravitational acceleration is used regardless of the fluid type, as it's determined by Earth's gravity, not the properties of the fluid.