Question

Calculate the height of a column of methanol \(\left(\mathrm{CH}_{3} \mathrm{OH}\right)\) that would be supported by atmospheric pressure. The density of methanol is \(0.787 \mathrm{~g} / \mathrm{cm}^{3}\).

Short answer

The height of the methanol column supported by atmospheric pressure is approximately 13.11 meters.

Step-by-step solution

Understand the Problem

We need to find the height of a column of methanol that can be supported by atmospheric pressure. Atmospheric pressure at sea level is approximately 101,325 Pa (Pascals). The density of methanol is given as 0.787 g/cm³. Our goal is to relate these values to find the height.

Convert Density Units

Convert the density of methanol from g/cm³ to kg/m³ for SI unit consistency. Since 1 g/cm³ is equal to 1000 kg/m³, the conversion for methanol's density is:\[0.787 \, \text{g/cm}^3 = 0.787 \times 1000 \, \text{kg/m}^3 = 787 \, \text{kg/m}^3\]

Apply the Pressure-Height-Density Formula

Use the formula for pressure exerted by a fluid column, which is given by:\[P = \rho \cdot g \cdot h\]where \(P\) is the pressure (101,325 Pa), \(\rho\) is the density (787 kg/m³), \(g\) is acceleration due to gravity (9.81 m/s²), and \(h\) is the height of the column. Rearrange this formula to solve for height \(h\):\[h = \frac{P}{\rho \cdot g}\]

Compute the Methanol Column Height

Substitute the known values into the equation:\[h = \frac{101,325}{787 \cdot 9.81}\]Calculate the above expression to find \(h\).

Calculate the Final Result

Computing the value gives:\[h = \frac{101,325}{7726.47} \approx 13.11 \, \text{m}\]

Final Step: Verify and Conclude

Check that the calculated height is reasonable given the context of standard atmospheric pressure. The column height is approximately 13.11 meters, which is consistent with the expectation for a liquid of similar density under atmospheric conditions.

Key concepts

Density Conversion

Density conversion is an important step when dealing with exercises in fluid dynamics, especially when working with different unit systems like the metric system and the SI units. In this exercise, the density of methanol is provided as 0.787 g/cm³, but to perform calculations using standard pressure units, it's crucial to convert this figure into kilograms per cubic meter (kg/m³). This is because 1 Pascal (Pa), the unit of pressure, is equivalent to 1 kg/m⋅s².
First, remember that 1 g/cm³ is equivalent to 1000 kg/m³. This conversion is due to the differences in the cubic measurements between centimeters and meters and the mass measurements between grams and kilograms.
So, to convert 0.787 g/cm³ to kg/m³, multiply by 1000:

• 0.787 g/cm³ × 1000 = 787 kg/m³

This conversion ensures all density units are compatible with SI units used in subsequent calculations.

Pressure-Height-Density Relationship

The pressure-height-density relationship is fundamental in fluid dynamics and helps to determine how forces are distributed in fluids. This relationship is captured by the formula:

• \[ P = \rho \cdot g \cdot h \]

Where:

• \( P \) is the pressure exerted by the fluid (Pa)
• \( \rho \) is the fluid density (kg/m³)
• \( g \) is the acceleration due to gravity, approximately 9.81 m/s²
• \( h \) is the height of the fluid column (m)

In this exercise, you rearrange the formula to find the height \( h \):

• \[ h = \frac{P}{\rho \cdot g} \]

This rearrangement helps determine the height of a fluid column that would exert a particular pressure, considering the fluid's density and gravity.
It's vital to ensure all variables are in the correct units to maintain consistency and accuracy in your calculations.

Atmospheric Pressure

Atmospheric pressure is a key concept in this exercise. It refers to the pressure exerted by the weight of the Earth's atmosphere. At sea level, this value is approximately 101,325 Pa (Pascals). This standard measurement is crucial when calculating the height of liquid columns supported by atmospheric pressure.
The importance of atmospheric pressure lies in its role in balancing fluid columns according to the pressure-height-density relationship. Understanding its constant value at sea level provides a baseline for comparison in many fluid dynamics problems.
Thus, in this scenario, the challenge is to find the height of a methanol column that atmospheric pressure could support. Using the known values for density and gravity, this allows you to solve for height using the pressure formula.

SI Units

SI units, or the International System of Units, form the basis for standardized measurements in science, including fluid dynamics. Using SI units allows for consistency and ease of communication across various scientific disciplines.
The key SI units in this exercise include:

• Pressure measured in Pascals (Pa)
• Density measured in kilograms per cubic meter (kg/m³)
• Height in meters (m)
• Acceleration due to gravity in meters per second squared (m/s²)

Converting all measurements to SI units is crucial for simplifying formulas and ensuring mathematical functions are dimensionally consistent.
Every calculation in this exercise relies on this consistency. Thus, ensuring that each quantity is expressed in its corresponding SI unit is integral to accurately computing results.