Question

If a barometer were built using water \(\left(d=1.0 \mathrm{g} / \mathrm{cm}^{3}\right)\) instead of mercury \(\left(d=13.6 \mathrm{g} / \mathrm{cm}^{3}\right),\) would the column of water be higher than, lower than, or the same as the column of mercury at 1.00 atm? If the level is different, by what factor? Explain.

Short answer

At 1.00 atm, the column of water in a water-based barometer would be higher than the column of mercury in a mercury-based barometer. The difference between their column heights is due to the different densities of water and mercury. The height of the water column is approximately 10,336 cm, while the height of the mercury column is approximately 76 cm. Therefore, the water column is higher by a factor of approximately 136.

Step-by-step solution

Understand the concept of hydrostatic pressure

In a fluid at rest, the pressure at any given depth depends on fluid density and the height of the column above that point. The hydrostatic pressure is given by the formula: \[P = hρg\], where \(P\) is the hydrostatic pressure, \(h\) is the height of the fluid column, \(ρ\) is the fluid density, and \(g\) is the acceleration due to gravity.

Calculate the height for a mercury barometer at 1.00 atm

Using the hydrostatic pressure formula for mercury, we can find the height of the column by replacing pressure with the atmospheric pressure (1.00 atm), and using mercury's density and gravity's acceleration: \[ h_{Hg} = \frac{P_{atm}}{ρ_{Hg} \times g} \] Given that the density of mercury is 13.6 g/cm³, and 1 atm = 101,325 Pa (Pascals) and g = 9.81 m/s², we can now calculate the height: \[ h_{Hg} = \frac{101,325}{13.6 \times 10^{-3} \times 9.81} \] \[ h_{Hg} \approx 76.0 \mathrm{cm} \]

Calculate the height for a water barometer at 1.00 atm

Similarly, we will calculate the height of the water column using the same formula: \[ h_{H_2O} = \frac{P_{atm}}{ρ_{H_2O} \times g} \] Given that the density of water is 1.0 g/cm³, we can now calculate the height in a similar fashion as the mercury column: \[ h_{H_2O} = \frac{101,325}{1.0 \times 10^{-3} \times 9.81} \] \[ h_{H_2O} \approx 10,336.0 \mathrm{cm} \]

Compare the heights and find the factor by which they differ

Now that we have the heights of both mercury and water columns, we can compare them: \[ h_{H_2O} \approx 10,336.0 \mathrm{cm} \] (Water column) \[ h_{Hg} \approx 76.0 \mathrm{cm} \] (Mercury column) The water column is higher by a considerable amount. To find the factor by which they differ, divide the height of the water column by the height of the mercury column: \[ factor = \frac{h_{H_2O}}{h_{Hg}} \] \[ factor \approx \frac{10,336.0}{76.0} \] \[ factor \approx 136 \] So, at 1.00 atm, the column of water is higher than the mercury column by a factor of approximately 136. This difference is due to the significantly lower density of water compared to mercury, resulting in a much taller column height needed to balance the atmospheric pressure.

Key concepts

Fluid Density

Fluid density is an essential concept when understanding hydrostatic pressure and how different liquids behave under similar conditions. Density (\(\rho\)) is defined as the mass of a fluid per unit volume. It plays a crucial role in determining how tall a fluid column needs to be to exert a certain amount of pressure.

Water, with a density of 1.0 g/cm³, is much less dense than mercury, which has a density of 13.6 g/cm³. This means that, for water to exert the same pressure as mercury, it must have a much taller column. This is because the same amount of pressure requires more volume when the fluid is less dense.

In practical applications, this principle explains why fluids of lower density, like water, require larger columns in measuring atmospheric pressure, compared to dense fluids like mercury. Understanding density helps in the design of devices that measure pressure, such as barometers.

Barometer

A barometer is an instrument used to measure atmospheric pressure. Typically, a barometer consists of a column of fluid whose height changes in response to the pressure of the atmosphere. The most common type of barometer uses mercury as the fluid, because of its high density.

Mercury barometers are compact because mercury's high density means that the column doesn't need to be very tall to measure normal atmospheric pressure. In contrast, when using a fluid like water, the column must be significantly taller due to its lower density.

• Mercury barometer: A shorter column due to higher fluid density
• Water barometer: A much taller column because of lower fluid density

Both types of barometers rely on the same physical principles: balancing the weight of the fluid column against the atmospheric pressure. Understanding how barometers work helps in grasping how atmospheric pressure is measured and the impact of fluid density on the design of these measuring devices.

Atmospheric Pressure

Atmospheric pressure is the force per area exerted onto a surface by the weight of the air above that surface. This pressure affects everything from weather patterns to how we build tools like barometers.

At sea level, standard atmospheric pressure is about 101,325 Pascals or 1 atmosphere (atm). This value is crucial for calculating how high a fluid column must be in a barometer to balance this pressure. When selecting a fluid for a barometer, its density determines how tall the column must be. Regardless of the fluid, the atmospheric pressure remains constant, but the column height varies inversely with the fluid's density.

Being able to predict levels of atmospheric pressure is important in many fields, including meteorology, aviation, and oceanography. Measures of atmospheric pressure help us prepare for changes in weather and make strategic decisions in various industries.