Question

Mercury has a density of \(13.55 \mathrm{~g} / \mathrm{cm}^{3}\). A barometer is constructed using an oil with a density of \(0.75 \mathrm{~g} / \mathrm{cm}^{3} .\) If the atmospheric pressure is 1.0 atm, calculate the height in meters of the oil column in the barometer.

Short answer

The height of the oil column is approximately 13.72 meters.

Step-by-step solution

Understand the Problem

We need to find the height of an oil column in a barometer that provides the same pressure as a mercury column. We are given the density of mercury and oil and the atmospheric pressure in atm.

Convert Atmospheric Pressure to Pascals

Atmospheric pressure of 1 atm is equivalent to 101,325 Pascals. This is a standard conversion needed to work within the SI units system.

Use the Pressure Formula for Fluids

The pressure exerted by a fluid column is given by the formula \(P = \rho gh\), where \(\rho\) is the density of the fluid, \(g\) is the acceleration due to gravity (approximated as 9.81 m/s²), and \(h\) is the height of the fluid column.

Calculate Mercury Column Height for 1 atm

Assuming mercury to form the column to balance 1 atm pressure, we rearrange the pressure formula to solve for height \(h\): \(h = \frac{P}{\rho g}\). Why? Because we have \(P = 101,325 \text{~Pa}\), \(\rho = 13,550 \text{~kg/m}^3\), and \(g = 9.81 \text{~m/s}^2\).

Solve for Mercury Column Height

Substitute the known values: \(h = \frac{101,325}{13,550 \, 9.81}\approx 0.760\text{m}\). This is the height of a mercury column that balances the atmospheric pressure.

Set Up Equation for Oil Column

For the oil column, use the same pressure formula \(P = \rho gh\), where \(\rho = 750 \text{~kg/m}^3\). Since pressure is equal to 1 atm, we equate the pressures: \(13,550 \times 9.81 \times 0.760 = 750 \times 9.81 \times h_o\) to solve for \(h_o\).

Calculate the Oil Column Height

Rearrange and solve: \(h_o = \frac{13,550 \times 0.760}{750}\), then calculating gives \(h_o \approx 13.72\text{m}\).

Conclusion

The height of the oil column is approximately 13.72 meters when it balances 1 atm of atmospheric pressure.

Key concepts

Barometer Calculations

Barometers are essential tools for measuring atmospheric pressure, often using a column of liquid, such as mercury or oil. Understanding how they work helps in many scientific and technical activities. Barometers rely on the principle that a fluid's weight at a certain height can balance atmospheric pressure. By using different fluids, the height of the column needed can vary significantly, depending on the fluid's density.

For example, in a barometer using mercury (which has a high density of 13.55 g/cm³), a relatively short column (about 0.760 meters) can balance the standard atmospheric pressure of 1 atm. However, when the fluid in the barometer is switched to one with a lower density, such as oil with a density of 0.75 g/cm³, a much taller column (around 13.72 meters) is required to achieve the same pressure. This difference occurs because less dense fluids exert less pressure at the same height compared to denser fluids.

When performing barometer calculations, it is crucial to understand unit conversions and the relationship between pressure, density, and height. These variables are combined in the formula for fluid pressure: \(P = \rho gh\), where \(P\) is the pressure, \(\rho\) is the fluid's density, \(g\) is the gravitational acceleration, and \(h\) is the height of the fluid column.

Fluid Column Pressure

The pressure exerted by a column of fluid is a fundamental concept when studying barometers and other applications involving liquids. This pressure arises due to the weight of the fluid above a specific point, pushing down due to gravity. The basic formula used to calculate the pressure of a fluid column is: \(P = \rho gh\). This equation links pressure (\(P\)), density (\(\rho\)), gravitational acceleration (\(g\)), and height (\(h\)).

Here's a simple breakdown:

• \(\rho\) is the density of the fluid, which tells us how much mass the fluid has in a given volume. Higher density fluids will exert more pressure at the same height.
• \(g\) is the gravitational acceleration, generally taken as 9.81 m/s² on Earth's surface.
• \(h\) is the height of the fluid column. Greater heights result in more pressure because there's more fluid weighing down.

For a mercury column that balances 1 atm of pressure, we see a relatively short height (around 0.760 meters). In contrast, a fluid like oil, which is less dense, needs a column height significantly higher, about 13.72 meters. Thus, understanding the relationship in the fluid column pressure formula helps us predict and calculate the necessary height of different fluid barometers.

Unit Conversion in Chemistry

Unit conversion is an integral part of chemistry and physics, especially when dealing with calculations involving pressure, volume, and temperature. Understanding how to properly convert units ensures accuracy and consistency in scientific equations and is particularly necessary when working with SI and non-SI units.

For example, when converting atmospheric pressure from atm to Pascals (Pa), we use the standard conversion: 1 atm equals 101,325 Pa. This conversion is important as it allows us to work with the International System of Units (SI), widely adopted in scientific work.

In the context of barometer calculations, proper unit conversion is necessary for dealing with fluid density units, often provided in g/cm³, which must be converted to kg/m³ when inserting into the pressure equation \(P = \rho gh\). The key steps include:

• 1 g/cm³ = 1000 kg/m³. This transformation allows the direct use in the SI pressure equation.
• Always ensure unit consistency throughout the calculations to avoid errors, whether computing pressures or heights of fluid columns.
• Having standard values at hand (like the conversion of 1 atm to 101,325 Pa) speeds up calculations and reduces potential mistakes.

Mastering these conversions is essential in chemistry, physics, and engineering contexts, enabling effective and reliable problem-solving.