Question

What pressure (in atm) is exerted by a column of toluene \(\left(\mathrm{C}_{7} \mathrm{H}_{8}\right) 87 \mathrm{~m}\) high? The density of toluene is \(0.867 \mathrm{~g} / \mathrm{cm}^{3}\)

Short answer

The pressure exerted by the toluene column is approximately 7.32 atm.

Step-by-step solution

Convert Density to kg/m³

The given density of toluene is 0.867 g/cm³. We need to convert this to kg/m³. Since 1 g/cm³ = 1000 kg/m³, we have: \(0.867 \, \text{g/cm}^3 = 867 \, \text{kg/m}^3\).

Calculate the Hydrostatic Pressure

The pressure exerted by a liquid column is given by the formula: \(P = \rho gh\), where \(\rho\) is the density in kg/m³, \(g\) is the acceleration due to gravity (9.81 m/s²), and \(h\) is the height of the column in meters. Substitute \(\rho = 867\), \(g = 9.81\), and \(h = 87\) into the formula: \[ P = 867 \, \text{kg/m}^3 \times 9.81 \, \text{m/s}^2 \times 87 \, \text{m} = 741099.57 \, \text{Pa} \].

Convert Pressure from Pa to atm

1 atmosphere (atm) is equivalent to 101325 Pascal (Pa). To convert the pressure from Pascal to atm, use the formula: \( P_{\text{atm}} = \frac{P}{101325} \). Hence, \( P_{\text{atm}} = \frac{741099.57}{101325} \approx 7.32 \text{ atm} \).

Key concepts

Toluene Density

Toluene is a common organic solvent with interesting properties, one of which is its density. Knowing the density of a substance allows you to determine how much mass is in a given volume, which is crucial for calculating pressures in fluids. The density of toluene is given as 0.867 g/cm³. However, for calculations involving pressure over large areas or heights, the density is often converted to kg/m³.
To convert from g/cm³ to kg/m³, you use the conversion factor: 1 g/cm³ is equal to 1000 kg/m³. Thus, for toluene, this calculation looks like:

• 0.867 g/cm³ × 1000 kg/m³ = 867 kg/m³

This conversion is essential for using the density in formulas that require SI units, like the hydrostatic pressure formula, which is expressed in terms of kg/m³, m/s², and meters. Remember, converting units accurately is a foundational skill in physics and helps ensure precise calculations.

Pressure Conversion

Pressure is a measure of force applied over an area. In certain calculations, especially those involving fluids like toluene, it’s important to work with pressure units that are most applicable to the situation. The term ‘hydrostatic pressure’ refers to the pressure exerted by a static fluid due to the force of gravity. The hydrostatic pressure (Default: P = ρgh),where

• \( P \) is the pressure
• \( \rho \) is the fluid density
• \( g \) is the acceleration due to gravity
• \( h \) is the height of the fluid column

is crucial in determining how much pressure a liquid exerts at a given depth.
For our scenario with toluene, the hydrostatic pressure was calculated:
\[ 867 \, \text{kg/m}^3 \times 9.81 \, \text{m/s}^2 \times 87 \, \text{m} = 741099.57 \, \text{Pa} \]
An understanding of how different units interact is necessary to accurately convert and compare these values.

Pascal to Atmosphere Conversion

After calculating the pressure in units of Pascals from the hydrostatic column of toluene, it may be necessary to convert this to atmospheres for practical applications. Pascal (Pa) and atmosphere (atm) are both units of pressure, but they belong to different systems. One atmosphere is defined as 101325 Pascals.
To convert a pressure from Pascals to atmospheres, use the formula:

• \( P_{\text{atm}} = \frac{P_{\text{Pa}}}{101325} \)

For instance, converting 741099.57 Pa:

• \( P_{\text{atm}} = \frac{741099.57}{101325} \approx 7.32 \, \text{atm} \)

Understanding this conversion is handy when you need to interpret pressures in a context more familiar to everyday activities, like atmospheric pressure at sea level, making the information more relatable and easier to understand. Employing such conversions is a practical skill for scientists and engineers who often need to switch between different units to communicate their results effectively.