Question

What pressure (in atm) is exerted by a column of isopropanol \(\left(\mathrm{C}_{3} \mathrm{H}_{7} \mathrm{OH}\right) 264 \mathrm{~m}\) high? The density of isopropanol is \(0.785 \mathrm{~g} / \mathrm{cm}^{3}\).

Short answer

The pressure exerted by the isopropanol is approximately 20.1 atm.

Step-by-step solution

Convert Density to kg/m³

The density of isopropanol is given in grams per cubic centimeter \(0.785 \text{ g/cm}^3\). To convert this to \(\text{kg/m}^3\): \[ 0.785 \text{ g/cm}^3 \times 1000 \frac{\text{kg}}{\text{g}} \times \left( \frac{100\text{ cm}}{1\text{ m}} \right)^3 = 785 \text{ kg/m}^3 \].

Calculate the Pressure Exerted

The formula for the pressure exerted by a liquid column is given by \(P = \rho gh\), where \(\rho\) is the density, \(g\) is the acceleration due to gravity (approx. \(9.81 \text{ m/s}^2\)), and \(h\) is the height of the column. Substitute the values into the equation: \[ P = 785 \text{ kg/m}^3 \times 9.81 \text{ m/s}^2 \times 264\text{ m} \].

Perform the Calculation

Calculate the pressure: \[ P = 785 \times 9.81 \times 264 = 2,037,384.4 \text{ Pa} \] (Pascals).

Convert Pressure to atms

1 atm is equivalent to 101,325 Pa. To convert from Pascals to atmospheres: \[ P = \frac{2,037,384.4 \text{ Pa}}{101,325 \text{ Pa/atm}} \approx 20.1 \text{ atm} \].

Key concepts

Density Conversion

Density conversion is an essential step when calculating pressure from a column of liquid, as it ensures compatibility of units within the formula. In this exercise, the density of isopropanol is initially given in grams per cubic centimeter ( 0.785 ext{ g/cm}^3 ). In scientific calculations, however, it's more practical to use kilograms per cubic meter (kg/m³). This transformation involves two conversion steps:

• Convert grams to kilograms: 1 ext{ g} = 0.001 ext{ kg}
• Convert cubic centimeters to cubic meters: 1 ext{ cm}^3 = 1 imes 10^{-6} ext{ m}^3

Applying these conversions, we multiply the given density by 1000, resulting in 785 ext{ kg/m}^3. Performing such conversions is key for consistency when the density is used in subsequent physics equations.

Pressure Formula

The pressure exerted by a column of liquid is calculated using the formula\[ P = \rho gh \]where:

• \( \rho \) represents the density of the liquid
• \( g \) is the acceleration due to gravity, which on Earth is approximately 9.81 ext{ m/s}^2
• \( h \) is the height of the liquid column

This formula stems from the principles of hydrostatic pressure. It tells us that pressure increases with the liquid's density and the height of the column. Substituting the known values, you calculate the pressure in Pascals, a standard SI unit for pressure measurement. This understanding of the pressure formula is crucial for solving real-world problems related to atmospheric pressure and fluid dynamics.

Unit Conversion

Unit conversion plays a vital role in ensuring that we use appropriate and consistent units throughout calculations. In scientific computations, especially when dealing with pressure calculations, converting units correctly is imperative.When converting pressure from Pascals (the SI unit) to atmospheres (atm), we use the relationship:\[ 1 \text{ atm} = 101,325 \text{ Pa} \]In the given exercise, the pressure was computed as 2,037,384.4 ext{ Pa}. To express this in atmospheres, you divide the pressure in Pascals by 101,325 Pa. This results in approximately 20.1 ext{ atm}. Such conversions help in interpreting scientific results in more familiar or convenient units, often bridging the gap in understanding in various applications.

Isopropanol Density

Understanding the density of isopropanol is essential when addressing problems involving pressure calculations of liquid columns. Isopropanol, chemically known as ( C_{3}H_{7}OH ), is a solvent, and its density is 0.785 ext{ g/cm}^3. This low density implies that isopropanol is relatively light, making it important to always convert it into the SI unit 785 ext{ kg/m}^3 when applying it in formulas. Density reflects how much mass occupies a given volume and is pivotal for determining attributes like the pressure in fluids and buoyancy. Knowing and converting the density of materials like isopropanol allows for accurate and meaningful calculations in various scientific and engineering contexts.