Question

Normal atmospheric pressure will push a column of mercury up an evacuated glass tube (a barometer) to a height of \(76 \mathrm{~cm}(760 \mathrm{~mm} \mathrm{Hg})\), which we call one atmosphere. Suppose water were used as the liquid in a barometer instead of mercury. How high up would the atmosphere push a column of water in centimeters and in feet? Also, why would mercury give a more accurate indication of the atmospheric pressure then water? Some data you may need: Density of mercury, \(13.6 \mathrm{~g} / \mathrm{mL} ;\) Density of water, \(1.00 \mathrm{~g} / \mathrm{mL}\); Boiling point of mercury, \(357^{\circ} \mathrm{C}\); Boiling point of water, \(100^{\circ} \mathrm{C}\).

Short answer

The height of the water column would be approximately 1034 cm (33.9 ft). Mercury is a better choice for barometers because it has a high density, which results in a shorter column height for easier measurement; it does not stick to the glass surface, providing accurate and reliable measurements; and it has a higher boiling point, allowing it to remain a liquid at a wide range of temperatures and making it suitable for various environments.

Step-by-step solution

Determine the pressure exerted by mercury in a barometer

Normal atmospheric pressure is defined as one atmosphere, which corresponds to a 76 cm (760 mm) mercury column. We can use this data to find the pressure exerted by mercury Using the formula P = ρgh: P = (13.6 g/mL × 1000 kg/m³) × (9.81 m/s²) × (0.76 m) P = 101325 Pa (approximately) We can use this pressure to find the height of the water column.

Determine the height of the water column

Now, we will use the pressure formula to find the height of the water column: P = ρgh h = P / (ρg) h = 101325 Pa / ((1 g/mL × 1000 kg/m³) × (9.81 m/s²)) h ≈ 10.34 m Now convert the height to centimeters: h = 10.34 m × (100 cm/m) h ≈ 1034 cm

Convert the height to feet

Next, convert the height from centimeters to feet: h = 1034 cm × (1 ft / 30.48 cm) h ≈ 33.9 ft So, the height of the water column is approximately 1034 cm (33.9 ft).

Explain why mercury gives a more accurate indication of pressure than water

Mercury is a better choice for barometers for several reasons: 1. Mercury has a high density, resulting in a relatively shorter column height than water, which makes it easier to measure and read accurately. 2. Mercury does not stick to the glass surface (unlike water), so the measurements are more accurate and reliable. 3. The boiling point of mercury is much higher than that of water, which means it remains a liquid at a wide range of temperatures, making it suitable for various environments.

Key concepts

Barometer

A barometer is an instrument used to measure atmospheric pressure. It works on the principle that atmospheric pressure can support a column of fluid, and the height of that fluid column reflects the pressure of the atmosphere. As air molecules collide with the surface of the liquid, they exert a force on it, allowing it to rise up a tube until the internal fluid pressure matches the external air pressure.

A classic example is the mercury barometer where a glass tube is filled with mercury and inverted into a mercury reservoir. The space above the mercury in the tube is nearly a vacuum, and the external atmospheric pressure holds the mercury up in the tube. Barometers can also be designed with other fluids like water, oil, or alcohol. When selecting a fluid, it's essential to consider characteristics like the fluid's density and how it interacts with materials like glass, as these impact the accuracy and practicality of the device.

Improvements in digital technology have led to modern electronic barometers that use sensors to detect pressure changes, making them more convenient for various applications. However, traditional mercury barometers are still used for their precision and historical significance.

Density

Density is a fundamental concept in understanding barometric measurements. It refers to the mass per unit volume of a substance and is expressed in units such as grams per milliliter (g/mL) or kilograms per cubic meter (kg/m³). The density of a fluid used in a barometer, like mercury or water, directly influences the height of the fluid column the atmospheric pressure can support.

Mercury, with a density of 13.6 g/mL, is significantly denser than water, which has a density of 1.00 g/mL. This higher density allows for a shorter column height to balance the atmospheric pressure. A mercury column of 76 cm can achieve equilibrium with atmospheric pressure, while a much taller column of water would be required due to its lower density.

Density also affects the sensitivity of pressure measurements. Fluids with higher densities change their height less markedly with small pressure differences, allowing for more precise measurement as the scale can be more finely divided. These characteristics make the understanding of density crucial in pressure calculations and the design of barometers.

Pressure Calculations

Pressure calculations play a pivotal role when working with barometers and understanding atmospheric conditions. The fundamental equation to remember is the hydrostatic equation for pressure, given as:
\[ P = \rho g h \]
where \( P \) represents the pressure exerted by the fluid column, \( \rho \) is the density of the fluid, \( g \) is the acceleration due to gravity (approximately 9.81 m/s²), and \( h \) is the height of the column.

For a given atmospheric pressure, the density of the fluid determines the height of the fluid column. In other words, the denser the fluid, the shorter the column needed to exert the same pressure as atmospheric pressure. On the other hand, fluids with lower density such as water require a much taller column to achieve the same pressure. This is why water in a barometer requires a height of around 10.34 meters to indicate standard atmospheric pressure, while mercury only requires a height of 76 centimeters.

Understanding these calculations helps to interpret the measurements taken by a barometer, as well as aids in understanding the science behind how atmospheric pressure affects the world around us, such as weather patterns, wind, and even the breathing mechanism in living organisms.